testing_generation/Corpus/CORPUS.txt

<<START>> <<START>> <<START>> 

   Neural Ordinary Differential Equations

Ricky T. Q. Chen*, Yulia Rubanova*, Jesse Bettencourt*, David Duvenaud
      University of Toronto, Vector Institute
   {rtqichen, rubanova, jessebett, duvenaud}@cs.toronto.edu

                                   Abstract
We introduce a new family of deep neural network models. Instead of specifying a
discrete sequence of hidden layers, we parameterize the derivative of the hidden
state using a neural network. The output of the network is computed using a black-
box differential equation solver. These continuous-depth models have constant
memory cost, adapt their evaluation strategy to each input, and can explicitly trade
numerical precision for speed. We demonstrate these properties in continuous-depth
residual networks and continuous-time latent variable models. We also construct
continuous normalizing flows, a generative model that can train by maximum
likelihood, without partitioning or ordering the data dimensions. For training, we
show how to scalably backpropagate through any ODE solver, without access to its
internal operations. This allows end-to-end training of ODEs within larger models.

                               1   Introduction
                                                                                                         
Models such as residual networks, recurrent neural network decoders, and normalizing flows build 
complicated transformations by composing a sequence of transformations to a hidden state:    

                          <<FORMULA>>           (1)          
                                                                        
where t ∈ {0 . . . T } and ht ∈ R . These iterative updates can be seen as an Euler discretization of a
continuous transformation (Lu et al., 2017; Haber and Ruthotto, 2017; Ruthotto and Haber, 2018).                    
What happens as we add more layers and take smaller steps? In the limit, we parameterize the continuous     
dynamics of hidden units using an ordinary differential equation (ODE) specified by a neural network:       
Starting from the input layer h(0), we can define the output layer h(T ) to be the solution to this

                          <<FORMULA>>           (2)                                  

ODE initial value problem at some time T . This value can be computed by a black-box differential
equation solver, which evaluates the hidden unit dynamics f wherever necessary to determine the
solution with the desired accuracy. Figure 1 contrasts these two approaches.
Defining and evaluating models using ODE solvers has several benefits:
Memory efficiency In Section 2, we show how to compute gradients of a scalar-valued loss with
respect to all inputs of any ODE solver, without backpropagating through the operations of the solver.
Not storing any intermediate quantities of the forward pass allows us to train our models with constant
memory cost as a function of depth, a major bottleneck of training deep models.
32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montréal, Canada.
Adaptive computation Euler’s method is perhaps the simplest method for solving ODEs. There
have since been more than 120 years of development of efficient and accurate ODE solvers (Runge,
1895; Kutta, 1901; Hairer et al., 1987). Modern ODE solvers provide guarantees about the growth
of approximation error, monitor the level of error, and adapt their evaluation strategy on the fly to
achieve the requested level of accuracy. This allows the cost of evaluating a model to scale with
problem complexity. After training, accuracy can be reduced for real-time or low-power applications.
Scalable and invertible normalizing flows An unexpected side-benefit of continuous transforma-
tions is that the change of variables formula becomes easier to compute. In Section 4, we derive
this result and use it to construct a new class of invertible density models that avoids the single-unit
bottleneck of normalizing flows, and can be trained directly by maximum likelihood.
Continuous time-series models Unlike recurrent neural networks, which require discretizing
observation and emission intervals, continuously-defined dynamics can naturally incorporate data
which arrives at arbitrary times. In Section 5, we construct and demonstrate such a model.
2    Reverse-mode automatic differentiation of ODE solutions
The main technical difficulty in training continuous-depth networks is performing reverse-mode
differentiation (also known as backpropagation) through the ODE solver. Differentiating through
the operations of the forward pass is straightforward, but incurs a high memory cost and introduces
additional numerical error.
We treat the ODE solver as a black box, and compute gradients using the adjoint sensitivity
method (Pontryagin et al., 1962). This approach computes gradients by solving a second, aug-
mented ODE backwards in time, and is applicable to all ODE solvers. This approach scales linearly
with problem size, has low memory cost, and explicitly controls numerical error.
Consider optimizing a scalar-valued loss function L(), whose input is the result of an ODE solver:
                                      
                                    <<FORMULA>>       (3)               

To optimize L, we require gradients with respect to θ. The first step is to determining how the gradient 
of the loss depends on the hidden state z(t) at each instant. This quantity is called the adjoint a(t) = ∂L/∂z(t). 
Its dynamics are given by another ODE, which can be thought of as the State instantaneous analog of the chain rule:
 Adjoint State

                                    <<FORMULA>>       (4)

We can compute ∂L/∂z(t0 ) by another call to an ODE solver. This solver must run backwards, starting from the initial 
value of ∂L/∂z(t1 ). One complication is that solving this ODE requires the knowing value of z(t) along its entire tra-
jectory. However, we can simply recompute z(t) backwards in time together with the adjoint, starting from its final 
value z(t1 ).

If the loss depends directly on the state at multi- Computing the gradients with respect to the pa-
ple observation times, the adjoint state must be parameters θ requires evaluating a third integral,
updated in the direction of the partial derivative of which depends on both z(t) and a(t):
the loss with respect to each observation.                     

                                  <<FORMULA>>         (5)

The vector-Jacobian products <<FORMULA>> and <<FORMULA>> in (4) and (5) can be efficiently evaluated by
automatic differentiation, at a time cost similar to that of evaluating f . All integrals for solving z, 
and <<FORMULA>> can be computed in a single call to an ODE solver, which concatenates the original state, the
adjoint, and the other partial derivatives into a single vector. Algorithm 1 shows how to construct the
necessary dynamics, and call an ODE solver to compute all gradients at once.

                                 <<ALGORITHM>>

Most ODE solvers have the option to output the state z(t) at multiple times. When the loss depends
on these intermediate states, the reverse-mode derivative must be broken into a sequence of separate
solves, one between each consecutive pair of output times (Figure 2). At each observation, the adjoint
must be adjusted in the direction of the corresponding partial derivative ∂L/∂z(ti ).
The results above extend those of Stapor et al. (2018, section 2.4.2). An extended version of
Algorithm 1 including derivatives w.r.t. t0 and t1 can be found in Appendix C. Detailed derivations
are provided in Appendix B. Appendix D provides Python code which computes all derivatives for
scipy.integrate.odeint by extending the autograd automatic differentiation package. This
code also supports all higher-order derivatives. We have since released a PyTorch (Paszke et al.,
2017) implementation, including GPU-based implementations of several standard ODE solvers at
github.com/rtqichen/torchdiffeq.

                   Replacing residual networks with ODEs for supervised learning

In this section, we experimentally investigate the training of neural ODEs for supervised learning.
Software To solve ODE initial value problems numerically, we use the implicit Adams method
implemented in LSODE and VODE and interfaced through the scipy.integrate package. Being
an implicit method, it has better guarantees than explicit methods such as Runge-Kutta but requires
solving a nonlinear optimization problem at every step. This setup makes direct backpropagation
through the integrator difficult. We implement the adjoint sensitivity method in Python’s autograd
framework (Maclaurin et al., 2015). For the experiments in this section, we evaluated the hidden
state dynamics and their derivatives on the GPU using Tensorflow, which were then called from the
Fortran ODE solvers, which were called from Python autograd code.

Model Architectures We experiment with a small residual network which downsamples the et al. (1998).
input twice then applies 6 standard residual blocks He et al. (2016b), which are replaced by an ODESolve 
module in the ODE-Net variant. We also test a network with the same architecture but where gradients are 
backpropagated directly through a Runge-Kutta integrator, re-ferred to as RK-Net. Table 1 shows test error,
number of parameters, and memory cost. L denotes the number of layers in the ResNet, and L̃ is the number 
of function evaluations that the ODE solver
requests in a single forward pass, which can be interpreted as an implicit number of layers. We find
that ODE-Nets and RK-Nets can achieve around the same performance as the ResNet.
Error Control in ODE-Nets ODE solvers can approximately ensure that the output is within a
given tolerance of the true solution. Changing this tolerance changes the behavior of the network.
We first verify that error can indeed be controlled in Figure 3a. The time spent by the forward call is
proportional to the number of function evaluations (Figure 3b), so tuning the tolerance gives us a
                                                      3
trade-off between accuracy and computational cost. One could train with high accuracy, but switch to
a lower accuracy at test time.
         Figure 3: Statistics of a trained ODE-Net. (NFE = number of function evaluations.)
Figure 3c) shows a surprising result: the number of evaluations in the backward pass is roughly
half of the forward pass. This suggests that the adjoint sensitivity method is not only more memory
efficient, but also more computationally efficient than directly backpropagating through the integrator,
because the latter approach will need to backprop through each function evaluation in the forward
pass.
Network Depth It’s not clear how to define the ‘depth‘ of an ODE solution. A related quantity is
the number of evaluations of the hidden state dynamics required, a detail delegated to the ODE solver
and dependent on the initial state or input. Figure 3d shows that he number of function evaluations
increases throughout training, presumably adapting to increasing complexity of the model.

                     4    Continuous Normalizing Flows

The discretized equation (1) also appears in normalizing flows (Rezende and Mohamed, 2015) and
the NICE framework (Dinh et al., 2014). These methods use the change of variables theorem to
compute exact changes in probability if samples are transformed through a bijective function f :

                      <<FORMULA>>                             (6)

An example is the planar normalizing flow (Rezende and Mohamed, 2015):

                     <<FORMULA>>                             (7)

Generally, the main bottleneck to using the change of variables formula is computing of the deter-
minant of the Jacobian ∂f/∂z, which has a cubic cost in either the dimension of z, or the number
of hidden units. Recent work explores the tradeoff between the expressiveness of normalizing flow
layers and computational cost (Kingma et al., 2016; Tomczak and Welling, 2016; Berg et al., 2018).
Surprisingly, moving from a discrete set of layers to a continuous transformation simplifies the
computation of the change in normalizing constant:
Theorem 1 (Instantaneous Change of Variables). Let z(t) be a finite continuous random variable
with probability p(z(t)) dependent on time. Let dz  dt = f (z(t), t) be a differential equation describing
a continuous-in-time transformation of z(t). Assuming that f is uniformly Lipschitz continuous in z
and continuous in t, then the change in log probability also follows a differential equation,

                                 <<FORMULA>>                 (8)

Proof in Appendix A. Instead of the log determinant in (6), we now only require a trace operation.
Also unlike standard finite flows, the differential equation f does not need to be bijective, since if
uniqueness is satisfied, then the entire transformation is automatically bijective.
As an example application of the instantaneous change of variables, we can examine the continuous
analog of the planar flow, and its change in normalization constant:

                               <<FORMULA>>                            (9)

Given an initial distribution p(z(0)), we can sample from p(z(t)) and evaluate its density by solving
this combined ODE.
Using multiple hiddenP units with  P linear cost While det is not a linear function, the trace function
is, which implies tr( n Jn ) = n tr(Jn ). Thus if our dynamics is given by a sum of functions then
the differential equation for the log density is also a sum:


                              <<FORMULA>>                              (10)
 
This means we can cheaply evaluate flow models having many hidden units, with a cost only linear in
the number of hidden units M . Evaluating such ‘wide’ flow layers using standard normalizing flows
costs O(M 3 ), meaning that standard NF architectures use many layers of only a single hidden unit.
Time-dependent dynamics We can specify the parameters of a flow as a function of t, making the
differential equation f (z(t), t) change with t. This is parameterization is a kind of hypernetwork       
(Ha et al., 2016). We also introduce a gating mechanism for each hidden unit, 

                              <<FORMULA>>

where σn (t) ∈ (0, 1) is a neural network that learns when the dynamic fn (z) should be applied. We
call these models continuous normalizing flows (CNF).

4.1    Experiments with Continuous Normalizing Flows

We first compare continuous and discrete planar flows at learning to sample from a known distribution.
We show that a planar CNF with M hidden units can be at least as expressive as a planar NF with
K = M layers, and sometimes much more expressive.
Density matching We configure the CNF as described above, and train for 10,000 iterations
using Adam (Kingma and Ba, 2014). In contrast, the NF is trained for 500,000 iterations using
RMSprop (Hinton et al., 2012), as suggested by Rezende and Mohamed (2015). For this task, we
minimize KL (q(x)kp(x)) as the loss function where q is the flow model and the target density p(·)
can be evaluated. Figure 4 shows that CNF generally achieves lower loss.
Maximum Likelihood Training A useful property of continuous-time normalizing flows is that
we can compute the reverse transformation for about the same cost as the forward pass, which cannot
be said for normalizing flows. This lets us train the flow on a density estimation task by performing
maximum likelihood estimation, which maximizes Ep(x) [log q(x)] where q(·) is computed using
the appropriate change of variables theorem, then afterwards reverse the CNF to generate random
samples from q(x).
For this task, we use 64 hidden units for CNF, and 64 stacked one-hidden-unit layers for NF. Figure 5
shows the learned dynamics. Instead of showing the initial Gaussian distribution, we display the
transformed distribution after a small amount of time which shows the locations of the initial planar
flows. Interestingly, to fit the Two Circles distribution, the CNF rotates the planar flows so that
the particles can be evenly spread into circles. While the CNF transformations are smooth and
interpretable, we find that NF transformations are very unintuitive and this model has difficulty fitting
the two moons dataset in Figure 5b.

               5  A generative latent function time-series model

Applying neural networks to irregularly-sampled data such as medical records, network traffic, or
neural spiking data is difficult. Typically, observations are put into bins of fixed duration, and the
latent dynamics are discretized in the same way. This leads to difficulties with missing data and ill-
defined latent variables. Missing data can be addressed using generative time-series models (Álvarez
and Lawrence, 2011; Futoma et al., 2017; Mei and Eisner, 2017; Soleimani et al., 2017a) or data
imputation (Che et al., 2018). Another approach concatenates time-stamp information to the input of
an RNN (Choi et al., 2016; Lipton et al., 2016; Du et al., 2016; Li, 2017).
We present a continuous-time, generative approach to modeling time series. Our model represents
each time series by a latent trajectory. Each trajectory is determined from a local initial state, zt0 , and
a global set of latent dynamics shared across all time series. Given observation times t0 , t1 , . . . , tN
and an initial state zt0 , an ODE solver produces zt1 , . . . , ztN , which describe the latent state at each
observation.We define this generative model formally through a sampling procedure:
                             <<FORMULA>>                                    (11)
                             <<FORMULA>>                                    (12)
                             <<FORMULA>>                                    (13)
Function f is a time-invariant function that takes the value z at the current time step and outputs the
gradient: ∂z(t)/∂t = f (z(t), θf ). We parametrize this function using a neural net. Because f is time-

invariant, given any latent state z(t), the entire latent trajectory is uniquely defined. Extrapolating
this latent trajectory lets us make predictions arbitrarily far forwards or backwards in time.
Training and Prediction We can train this latent-variable model as a variational autoen-
coder (Kingma and Welling, 2014; Rezende et al., 2014), with sequence-valued observations. Our
recognition net is an RNN, which consumes the data sequentially backwards in time, and out-
puts qφ (z0 |x1 , x2 , . . . , xN ). A detailed algorithm can be found in Appendix E. Using ODEs as a
generative model allows us to make predictions for arbitrary time points t1 ...tM on a continuous
timeline.
Poisson Process likelihoods The fact that an observation oc-
curred often tells us something about the latent state. For ex-
ample, a patient may be more likely to take a medical test if           
they are sick. The rate of events can be parameterized by a
function of the latent state: p(event at time t| z(t)) = λ(z(t)).
Given this rate function, the likelihood of a set of indepen-
dent observation times in the interval [tstart , tend ] is given by an                        t
inhomogeneous Poisson process (Palm, 1943):                                

We can parameterize λ(·) using another neural network. Con-
veniently, we can evaluate both the latent trajectory and the
Poisson process likelihood together in a single call to an ODE solver. Figure 7 shows the event rate
learned by such a model on a toy dataset.
A Poisson process likelihood on observation
times can be combined with a data likelihood to
jointly model all observations and the times at
which they were made.

5.1   Time-series Latent ODE Experiments 

We investigate the ability of the latent ODE
model to fit and extrapolate time series. The
recognition network is an RNN with 25 hidden
units. We use a 4-dimensional latent space. We
parameterize the dynamics function f with a
one-hidden-layer network with 20 hidden units.
The decoder computing p(xti |zti ) is another              
neural network with one hidden layer with 20                       
hidden units. Our baseline was a recurrent neu-                   
ral net with 25 hidden units trained to minimize                  
negative Gaussian log-likelihood. We trained a                     
second version of this RNN whose inputs were
concatenated with the time difference to the next
observation to aid RNN with irregular observations.
Bi-directional spiral dataset We generated neural network. (b): Reconstructions and extrapo-
a dataset of 1000 2-dimensional spirals, each lations by a latent neural ODE. Blue curve shows
starting at a different point, sampled at 100 model prediction. Red shows extrapolation. (c) A
equally-spaced timesteps. The dataset contains projection of inferred 4-dimensional latent ODE
two types of spirals: half are clockwise while trajectories onto their first two dimensions. Color
the other half counter-clockwise. To make the indicates the direction of the corresponding trajec-
task more realistic, we add gaussian noise to the tory. The model has learned latent dynamics which
observations.                                     
                                                       
progression through time, starting at purple and ending at red. Note that the trajectories on the left
are counter-clockwise, while the trajectories on the right are clockwise.
Time series with irregular time points To generate irregular timestamps, we randomly sample
points from each trajectory without replacement (n = {30, 50, 100}). We report predictive root-
mean-squared error (RMSE) on 100 time points extending beyond those that were used for training.
Table 2 shows that the latent ODE has substantially lower predictive RMSE.

We observed that reconstructions and extrapolations are consistent with the ground truth
regardless of number of observed points and despite the noise.
Latent space interpolation Figure 8c shows latent trajectories projected onto the first two dimen-
sions of the latent space. The trajectories form two separate clusters of trajectories, one decoding to
clockwise spirals, the other to counter-clockwise. Figure 9 shows that the latent trajectories change
smoothly as a function of the initial point z(t0 ), switching from a clockwise to a counter-clockwise
spiral.

                        6    Scope and Limitations

Minibatching The use of mini-batches is less straightforward than for standard neural networks.
One can still batch together evaluations through the ODE solver by concatenating the states of each
batch element together, creating a combined ODE with dimension D × K. In some cases, controlling
error on all batch elements together might require evaluating the combined system K times more
often than if each system was solved individually. However, in practice the number of evaluations did
not increase substantially when using minibatches.
Uniqueness When do continuous dynamics have a unique solution? Picard’s existence theo-
rem (Coddington and Levinson, 1955) states that the solution to an initial value problem exists and is
unique if the differential equation is uniformly Lipschitz continuous in z and continuous in t. This
theorem holds for our model if the neural network has finite weights and uses Lipshitz nonlinearities,
such as tanh or relu.
Setting tolerances Our framework allows the user to trade off speed for precision, but requires
the user to choose an error tolerance on both the forward and reverse passes during training. For
sequence modeling, the default value of 1.5e-8 was used. In the classification and density estimation
experiments, we were able to reduce the tolerance to 1e-3 and 1e-5, respectively, without degrading
performance.
Reconstructing forward trajectories Reconstructing the state trajectory by running the dynamics
backwards can introduce extra numerical error if the reconstructed trajectory diverges from the
original. This problem can be addressed by checkpointing: storing intermediate values of z on the
forward pass, and reconstructing the exact forward trajectory by re-integrating from those points. We
did not find this to be a practical problem, and we informally checked that reversing many layers of
continuous normalizing flows with default tolerances recovered the initial states.
                                                     8
                        7    Related Work

The use of the adjoint method for training continuous-time neural networks was previously pro-
posed (LeCun et al., 1988; Pearlmutter, 1995), though was not demonstrated practically. The
interpretation of residual networks He et al. (2016a) as approximate ODE solvers spurred research
into exploiting reversibility and approximate computation in ResNets (Chang et al., 2017; Lu et al.,
2017). We demonstrate these same properties in more generality by directly using an ODE solver.
Adaptive computation One can adapt computation time by training secondary neural networks
to choose the number of evaluations of recurrent or residual networks (Graves, 2016; Jernite et al.,
2016; Figurnov et al., 2017; Chang et al., 2018). However, this introduces overhead both at training
and test time, and extra parameters that need to be fit. In contrast, ODE solvers offer well-studied,
computationally cheap, and generalizable rules for adapting the amount of computation.
Constant memory backprop through reversibility Recent work developed reversible versions
of residual networks (Gomez et al., 2017; Haber and Ruthotto, 2017; Chang et al., 2017), which gives
the same constant memory advantage as our approach. However, these methods require restricted
architectures, which partition the hidden units. Our approach does not have these restrictions.
Learning differential equations Much recent work has proposed learning differential equations
from data. One can train feed-forward or recurrent neural networks to approximate a differential
equation (Raissi and Karniadakis, 2018; Raissi et al., 2018a; Long et al., 2017), with applica-
tions such as fluid simulation (Wiewel et al., 2018). There is also significant work on connecting
Gaussian Processes (GPs) and ODE solvers (Schober et al., 2014). GPs have been adapted to fit
differential equations (Raissi et al., 2018b) and can naturally model continuous-time effects and
interventions (Soleimani et al., 2017b; Schulam and Saria, 2017). Ryder et al. (2018) use stochastic
variational inference to recover the solution of a given stochastic differential equation.
Differentiating through ODE solvers The dolfin library (Farrell et al., 2013) implements adjoint
computation for general ODE and PDE solutions, but only by backpropagating through the individual
operations of the forward solver. The Stan library (Carpenter et al., 2015) implements gradient
estimation through ODE solutions using forward sensitivity analysis. However, forward sensitivity
analysis is quadratic-time in the number of variables, whereas the adjoint sensitivity analysis is
linear (Carpenter et al., 2015; Zhang and Sandu, 2014). Melicher et al. (2017) used the adjoint
method to train bespoke latent dynamic models.
In contrast, by providing a generic vector-Jacobian product, we allow an ODE solver to be trained
end-to-end with any other differentiable model components. While use of vector-Jacobian products
for solving the adjoint method has been explored in optimal control (Andersson, 2013; Andersson
et al., In Press, 2018), we highlight the potential of a general integration of black-box ODE solvers
into automatic differentiation (Baydin et al., 2018) for deep learning and generative modeling.
8    Conclusion
We investigated the use of black-box ODE solvers as a model component, developing new models
for time-series modeling, supervised learning, and density estimation. These models are evaluated
adaptively, and allow explicit control of the tradeoff between computation speed and accuracy.
Finally, we derived an instantaneous version of the change of variables formula, and developed
continuous-time normalizing flows, which can scale to large layer sizes.
9    Acknowledgements
We thank Wenyi Wang and Geoff Roeder for help with proofs, and Daniel Duckworth, Ethan Fetaya,
Hossein Soleimani, Eldad Haber, Ken Caluwaerts, Daniel Flam-Shepherd, and Harry Braviner for
feedback. We thank Chris Rackauckas, Dougal Maclaurin, and Matthew James Johnson for helpful
discussions. We also thank Yuval Frommer for pointing out an unsupported claim about parameter
efficiency.
                                                    9
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         Appendix A   Proof of the Instantaneous Change of Variables Theorem

Theorem (Instantaneous Change of Variables). Let z(t) be a finite continuous random variable with probability
p(z(t)) dependent on time. Let dz/dt = f (z(t), t) be a differential equation describing a continuous-in-time
transformation of z(t). Assuming that f is uniformly Lipschitz continuous in z and continuous in t, then the
change in log probability also follows a differential equation:

                                 <<FORMULA>>

Proof. To prove this theorem, we take the infinitesimal limit of finite changes of log p(z(t)) through time. First
we denote the transformation of z over an ε change in time as
                                 <<FORMULA>>                                           (14)
We assume that f is Lipschitz continuous in z(t) and continuous in t, so every initial value problem has a unique
solution by Picard’s existence theorem. We also assume z(t) is bounded. These conditions imply that f , Tε , and
 ∂T are all bounded. In the following, we use these conditions to exchange limits and products.

                                 <<FORMULA>>

We can write the differential equation <<FORMULA>> using the discrete change of variables formula, and the
definition of the derivative:

                                 <<FORMULA>>                                                                    (15)

                                 <<FORMULA>>                                                                    (16)

                                 <<FORMULA>>                                           (by L’Hôpital’s rule)    (17)

                                 <<FORMULA>>                                                                    (18)

                                 <<FORMULA>>                                                                    (19)
    
                                 <<FORMULA>>                                                                    (20)

The derivative of the determinant can be expressed using Jacobi’s formula, which gives

                                 <<FORMULA>>                                                                    (21)

                                 <<FORMULA>>                                                                    (22)
    
                                 <<FORMULA>>                                                                    (23)


Substituting Tε with its Taylor series expansion and taking the limit, we complete the proof.

                                 <<FORMULA>>                                                                    (24)

                                 <<FORMULA>>                                                                    (25)

                                 <<FORMULA>>                                                                    (26)
    
                                 <<FORMULA>>                                                                    (27)


                                          A.1     Special Cases

Planar CNF. Let f (z) = uh(wz + b), then  ∂z = u ∂h  ∂z. Since the trace of an outer product is the inner
product, we have

                                <<FORMULA>>                                                                     (28)

This is the parameterization we use in all of our experiments.
Hamiltonian CNF. The continuous analog of NICE (Dinh et al., 2014) is a Hamiltonian flow, which splits

the data into two equal partitions and is a volume-preserving transformation, implying that ∂t = 0. We
can verify this. Let

                               <<FORMULA>>                                 (29)

Then because the Jacobian is all zeros on its diagonal, the trace is zero. This is a volume-preserving flow.
A.2     Connection to Fokker-Planck and Liouville PDEs
The Fokker-Planck equation is a well-known partial differential equation (PDE) that describes the probability
density function of a stochastic differential equation as it changes with time. We relate the instantaneous change
of variables to the special case of Fokker-Planck with zero diffusion, the Liouville equation.
As with the instantaneous change of variables, let z(t) ∈ RD evolve through time following dz(t)/dt = f (z(t), t).
Then Liouville equation describes the change in density of z–a fixed point in space–as a PDE,

                              <<FORMULA>>                                    (30)

However, (30) cannot be easily used as it requires the partial derivatives of p(z,t)/∂z, which is typically approximated
using finite difference. This type of PDE has its own literature on efficient and accurate simulation (Stam, 1999).
Instead of evaluating p(·, t) at a fixed point, if we follow the trajectory of a particle z(t), we obtain

                              <<FORMULA>>

partial derivative from first argument, z(t) partial derivative from second argument, t

                              <<FORMULA>>                                      (31)

We arrive at the instantaneous change of variables by taking the log,

                              <<FORMULA>>                                      (32)

While still a PDE, (32) can be combined with z(t) to form an ODE of size D + 1,

                              <<FORMULA>>                                       (33)

Compared to the Fokker-Planck and Liouville equations, the instantaneous change of variables is of more
practical impact as it can be numerically solved much more easily, requiring an extra state of D for following
the trajectory of z(t). Whereas an approach based on finite difference approximation of the Liouville equation
would require a grid size that is exponential in D.
Appendix B             A Modern Proof of the Adjoint Method
We present an alternative proof to the adjoint method (Pontryagin et al., 1962) that is short and easy to follow.
                                                                         14
B.1        Continuous Backpropagation

Let z(t) follow the differential equation dt = f (z(t), t, θ), where θ are the parameters. We will prove that if
we define an adjoint state

                                                                <<FORMULA>>                                                            (34)

then it follows the differential equation

                                                               <<FORMULA>>                                                             (35)

For ease of notation, we denote vectors as row vectors, whereas the main text uses column vectors.
The adjoint state is the gradient with respect to the hidden state at a specified time t. In standard neural networks,
the gradient of a hidden layer ht depends on the gradient from the next layer ht+1 by chain rule

                                                                 <<FORMULA>>                                                            (36)

With a continuous hidden state, we can write the transformation after an ε change in time as

                                                                 <<FORMULA>>                                                            (37)
 
                                                                 <<FORMULA>>                                                            (38)

The proof of (35) follows from the definition of derivative:

              <<FORMULA>>                                                                                                               (39)

              <<FORMULA>>                                                                                     (by Eq 38)                (40)

              <<FORMULA>>                                                                        (Taylor series around z(T))            (41)

              <<FORMULA>>                                                                                                               (42)

              <<FORMULA>>                                                                                                               (43)

             <<FORMULA>>                                                                                                                (44)
  
             <<FORMULA>>                                                                                                                (45)

We pointed out the similarity between adjoint method and backpropagation (eq. 38). Similarly to backpropaga-
tion, ODE for the adjoint state needs to be solved backwards in time. We specify the constraint on the last time
point, which is simply the gradient of the loss wrt the last time point, and can obtain the gradients with respect to
the hidden state at any time, including the initial value.

                                 <<FORMULA>>                   (46)

Here we assumed that loss function L depends only on the last time point tN . If function L depends also on
intermediate time points t1 , t2 , . . . , tN −1 , etc., we can repeat the adjoint step for each of the intervals [tN −1 , tN ],
[tN −2 , tN −1 ] in the backward order and sum up the obtained gradients.
B.2        Gradients wrt. θ and t
We can generalize (35) to obtain gradients with respect to θ–a constant wrt. t–and and the initial and end times,
t0 and tN . We view θ and t as states with constant differential equations and write

                                  <<FORMULA>>                                                 (47)

We can then combine these with z to form an augmented state1 with corresponding differential equation and
adjoint state,

                                    <<FORMULA>>                 (48)

Note this formulates the augmented ODE as an autonomous (time-invariant) ODE, but the derivations in the
previous section still hold as this is a special case of a time-variant ODE. The Jacobian of f has the form

                                      <<FORMULA>>                                (49)

where each 0 is a matrix of zeros with the appropriate dimensions. We plug this into (35) to obtain

                                    <<FORMULA>>                                  (50)

The first element is the adjoint differential equation (35), as expected. The second element can be used to obtain
the total gradient with respect to the parameters, by integrating over the full interval and setting aθ (tN ) = 0.

                                       <<FORMULA>>                               (51)

Finally, we also get gradients with respect to t0 and tN , the start and end of the integration interval.

                                       <<FORMULA>>                               (52)

Between (35), (46), (51), and (52) we have gradients for all possible inputs to an initial value problem solver.

            Appendix C              Full Adjoint sensitivities algorithm

This more detailed version of Algorithm 1 includes gradients with respect to the start and end times of integration.
Algorithm 2 Complete reverse-mode derivative of an ODE initial value problem

Input: dynamics parameters θ, start time t0 , stop time t1 , final state z(t1 ), loss gradient ∂L/∂z(t1 )

                  <<ALGORITHM>>

Note that we’ve overloaded t to be both a part of the state and the (dummy) independent variable. The
distinction is clear given context, so we keep t as the independent variable for consistency with the rest of the
text.

                     Appendix D                Autograd Implementation

                        <<ALGORITHM>>

                     Appendix E               Algorithm for training the latent ODE model

To obtain the latent representation zt0 , we traverse the sequence using RNN and obtain parameters of distribution
q(zt0 |{xti , ti }i , θenc ). The algorithm follows a standard VAE algorithm with an RNN variational posterior and
an ODESolve model:
                                       <<ALGORITHM>>
                                        <<FORMULA>>                         (53)
                                       <<ALGORITHM>>

                     Appendix F               Extra Figures

                                       <<FIGURE>>

<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>> 

              Learning differential equations that are easy to solve

                      Jacob Kelly∗                                Jesse Bettencourt∗
         University of Toronto, Vector Institute         University of Toronto, Vector Institute
             jkelly@cs.toronto.edu                          jessebett@cs.toronto.edu
             Matthew James Johnson                               David Duvenaud
               Google Brain                                 University of Toronto, Vector Institute

            mattjj@google.com                           duvenaud@cs.toronto.edu

                              Abstract


Differential equations parameterized by neural networks become expensive to solve
numerically as training progresses. We propose a remedy that encourages learned
dynamics to be easier to solve. Specifically, we introduce a differentiable surrogate
for the time cost of standard numerical solvers, using higher-order derivatives
of solution trajectories. These derivatives are efficient to compute with Taylor-
mode automatic differentiation. Optimizing this additional objective trades model
performance against the time cost of solving the learned dynamics. We demonstrate
our approach by training substantially faster, while nearly as accurate, models in
supervised classification, density estimation, and time-series modelling tasks.

                    1       Introduction

Differential equations describe a system’s behavior by specifying its instantaneous dynamics. 
Historically, differential equations have been derived from theory, such as Newtonian mechanics, 
Maxwell’s equations, or epidemiological models of infectious disease, with parameters inferred 
from observations. Solutions to these equations usually cannot be expressed in closed-form, 
requiring numerical approximation. Recently, ordinary differential equations parameterized by 
millions of learned parameters, called neural ODEs, have been fit for latent time series models, 
density models, or as a replacement for very deep neural networks (Rubanova et al., 2019; Grath-
wohl et al., 2019; Chen et al., 2018). These models are not constrained to match a theoretical 
model,and sometimes substantially different dynamics can give nearly indistinguishable predictions. 
This raises the possibility that we can find nearly equivalent models that are substantially easier
and faster to solve. Yet standard training methods have no way to penalize the complexity of the 
dynamics being learned.                                                  

                          <<FIGURE>>

Equal Contribution. Code available at: github.com/jacobjinkelly/easy-neural-ode

How can we learn dynamics that are faster to solve numerically without substantially changing their
predictions? Much of the computational advantages of a continuous-time formulation come from
using adaptive solvers, and most of the time cost of these solvers comes from repeatedly evaluating
the dynamics function, which in our settings is a moderately-sized neural network. So, we’d like to
reduce the number of function evaluations (NFE) required for these solvers to reach a given error
tolerance. Ideally, we would add a term penalizing the NFE to the training objective, and let a
gradient-based optimizer trade off between solver cost and predictive performance. But because NFE
is integer-valued, we need to find a differentiable surrogate.
The NFE taken by an adaptive solver depends on how far it can extrapolate the trajectory forward
without introducing too much error. For example, for a standard adaptive-step Runge-Kutta solver
with order m, the step size is approximately inversely proportional to the norm of the local mth total
derivative of the solution trajectory with respect to time. That is, a larger mth derivative leads to a
smaller step size and thus more function evaluations. Thus, we propose to minimize the norm of this
total derivative during training, as a way to control the time required to solve the learned dynamics.
In this paper, we investigate the effect of this speed regularization in various models and solvers.
We examine the relationship between the solver order and the regularization order, and characterize
the tradeoff between speed and performance. In most instances, we find that solver speed can be
approximately doubled without a substantial increase in training loss. We also provide an extension
to the JAX program transformation framework that provides Taylor-mode automatic differentiation,
which is asymptotically more efficient for computing the required total derivatives than standard
nested gradients.
Our work compares against and generalizes that of Finlay et al. (2020), who proposed regularizing
dynamics in the FFJORD density estimation model, and showed that it stabilized dynamics enough
in that setting to allow the use of fixed-step solvers during training.
2    Background
An ordinary differential equation (ODE) specifies the instantaneous change of a vector-valued state
<<FORMULA>>, computing the state at a later time:

                                    <<FORMULA>>

is called an initial value problem (IVP). For example, f could describe the equations of motion for a
particle, or the transmission and recovery rates for a virus across a population. Usually, the required
integral has no analytic solution, and must be approximated numerically.
Adaptive-step Runge-Kutta ODE Solvers Runge-Kutta methods (Runge, 1895; Kutta, 1901)
approximate the solution trajectories of ODEs through a series of small steps, starting at time t0 .
At each step, they choose a step size h, and fit a local approximation to the solution, ẑ(t), using
several evaluations of f . When h is sufficiently small, the numerical error of a mth-order method
is bounded by kẑ(t + h) − z(t + h)k ≤ chm+1 for some constant c (Hairer et al., 1993). So, for a
mth-order method, the local error grows approximately in proportion to the size of the mth coefficient
in the Taylor expansion of the true solution. All else being equal, controlling this coefficient for all
dimensions of z(t) will allow larger steps to be taken without surpassing the error tolerance.
Neural Ordinary Differential Equations The dynamics function f can be a moderately-sized
neural network, and its parameters θ trained by gradient descent. Solving the resulting IVP is
analogous to evaluating a very deep residual network in which the number of layers corresponds
to the number of function evaluations of the solver (Chang et al., 2017; Ruthotto & Haber, 2018;
Chen et al., 2018). Solving such continuous-depth models using adaptive numerical solvers has
several computational advantages over standard discrete-depth network architectures. However, this
approach is often slower than using a fixed-depth network, due to an inability to control the number
of steps required by an adaptive-step solver.

                        3   Regularizing Higher-Order Derivatives for Speed

The ability of Runge-Kutta methods to take large and accurate steps is limited by the Kth-order
Taylor coefficients of the solution trajectory. We would like these coefficients to be small. Specifically,
we propose to regularize the squared norm of the Kth-order total derivatives of the state with respect
to time, integrated along the entire solution trajectory:

                                     <<FORMULA>>                                  (1)

where k·k2 is the squared `2 norm, and the dependence on the dynamics parameters θ is implicit
through the solution z(t) integrating dz(t)

                                       <<dt = f (z(t), t, θ)>>. 

During training, we weigh this regularization term by a hyperparameter λ and add it to our original loss 
to get our regularized objective:

                                     <<FORMULA>>                                   (2)

What kind of solutions are allowed when RK = 0? For K = 0,

                                     <<FORMULA>>

we have kz(t)k2 = 0, so the only possible solution is z(t) = 0.                  

For K = 1, we have kf (z(t), t)k2 = 0, so all solutions are
constant, flat trajectories. For K = 2 solutions are straight-line               
trajectories. Higher values of K shrink higher derivatives, but
don’t penalize lower-order dynamics. For instance, a quadratic                      
trajectory will have R3 = 0. Setting the Kth order dynamics to
exactly zero everywhere automatically makes all higher orders                        
zero as well. Figure 1 shows that regularizing R3 on a toy 1D
neural ODE reduces NFE.     

                                    <<FIGURE>>
                                                                                     
Which orders should we regularize? We propose matching the
order of the regularizer to that of the solver being used. We        
conjecture that regularizing dynamics of lower orders than that     
of the solver restricts the model unnecessarily, and that let-     
ting the lower orders remain unregularized should not increase    
NFE very much. Figure 2 shows empirically which orders           
of Runge-Kutta solvers can efficiently solve which orders of       
toy polynomial trajectories. We test these conjectures on real      
models and datasets in section 6.2.                             

The solution trajectory and our regularization term can be computed in a single call to an ODE solver
by augmenting the system with the integrand in eq. (1).

            4   Efficient Higher Order Differentiation with Taylor Mode

The number of terms in higher-order forward derivatives grows exponentially in K, becoming
prohibitively expensive for K = 5, and causing substantial slowdowns even for K = 2 and K = 3.
Luckily, there exists a generalization of forward-mode automatic differentiation (AD), known as
Taylor mode, which can compute the total derivative exactly for a cost of only O(K 2 ). We found
that this asymptotic improvement reduced wall-clock time by an order of magnitude, even for K as
low as 3.
First-order forward-mode AD Standard forward-mode AD computes, for a function f (x) and
an input perturbation vector v, the product ∂f  ∂x v. This Jacobian-vector product, or JVP, can be
computed efficiently without explicitly instantiating the Jacobian. This implicit computation of JVPs
is straightforward whenever f is a composition of operations for which which implicit JVP rules are
known.
Higher-order Jacobian-vector products Forward-mode AD can be generalized to higher orders
                                                                                            K
to compute Kth-order Jacobians contracted K times against the perturbation vector: ∂∂xKf v ⊗K .
Similarly, this can also be computed without representing any Jacobian matrices explicitly.

A naïve approach to higher-order forward mode is to recursively apply first-order forward mode.
                                                             K
Specifically, nesting JVPs K times gives the right answer: <<FORMULA>> but
causes an unnecessary exponential slowdown, costing O(exp(K)). This is because expressions that
appear in lower derivatives also appear in higher derivatives, but the work to compute is not shared
across orders.
Taylor Mode Taylor-mode AD generalizes             Function               Taylor propagation rule
first-order forward mode to compute the first    <<y = z + cw>>                  <<y[k] = z[k] + cw[k]>>
K derivatives exactly with a time cost of only                                     <<Pk>>
O(K 2 ) or O(K log K), depending on the op-       <<y =z∗w>>                 << y[k] =  h j=0   z[j] w[k−j] i>>
                                                                                       <<Pk−1>>
erations involved. Instead of providing rules      <<y = z/w>>         <<y[k] = w10 zk − j=0 z[j] w[k−j]>>
for propagating perturbation vectors, one pro-                                     <<Pk>>
                                                 <<y = exp(z)>>                <<ỹ[k] = j=1 y[k−j] z̃[j]>>
vides rules for propagating truncated Taylor                                       <<Pk>>
series. Some example rules are shown in ta-      <<s = sin(z)>>                <<s̃[k] = j=1 z̃[j] c[k−j]>>
                                                                                   <<Pk>>
ble 1. For more details see the Appendix and     <<c = cos(z)>>              <<c̃[k] = j=1 −z̃[j] s[k−j]>>
Griewank & Walther (2008, Chapter 12). We
provide an open source implementation of Table 1: Rules for propagating Taylor polynomial
Taylor mode AD in the JAX Python library coefficients through standard functions. These rules
(Bradbury et al., 2018).                       generalize standard first-order derivatives. Notation
                                               <<z[i] = i!1 zi>> and <<ỹ[i] = i!i zi>>.

                     5     Experiments
                                                                                    
We consider three different tasks in which continuous-
                                                                      
depth or continuous time models might have computa-                     
                                                                     
                                                                                 
tional advantages over standard discrete-depth models:
supervised learning, continuous generative modeling of                              
time-series (Rubanova et al., 2019), and density estima-                             
tion using continuous normalizing flows (Grathwohl et al.,
2019). Unless specified otherwise, we use the standard
                                                                                     
dopri5 Runge-Kutta 4(5) solver (Dormand & Prince,
1980; Shampine, 1986).                                                             <<FIGURE>>      
                                                                                                   
5.1   Supervised Learning                                             Figure 3: Number of function evalua-
                                                                      tions (NFE) and training error during
We construct a model for MNIST classification: it takes in            training. Speed regularization (solid)
as input a flattened MNIST image and integrates it through            decreases the NFE throughout training
dynamics given by a simple MLP, then applies a linear                 without substantially changing the train-
classification layer. In fig. 3 we compare the NFE and                ing error.
training error of a model with and without regularizing
R3 .
                                                                  
5.2   Continuous Generative Time Series Models

As in Rubanova et al. (2019), we use the Latent ODE        
architecture for modelling trajectories of ICU patients
using the PhysioNet Challenge 2012 dataset (Silva
et al., 2012). This variational autoencoder architec-            
ture uses an RNN recognition network, and models                     
the state dynamics using an ODE in a latent space.
In the supervised learning setting described in the
previous section only the final state affects model pre- Figure 4: Regularizing dynamics in a la-
dictions. In contrast, time-series models’ predictions tent ODE modeling PhysioNet clinical data.
also depend on the value of the trajectory at all inter- Shown are a representative 2-dimensional
mediate times when observations were made. So, we slice of 20 dimensional dynamics. We re-
might expect speed regularization to be ineffective duce average NFE from 281 to 90 while only
due to these extra constraints on the dynamics. How- incurring an 8% increase in loss.
ever, fig. 4 shows that, without changing their overall
shape the latent dynamics can be adjusted to reduce their NFE by a factor of 3.
                                                   
5.3                       Density Estimation with Continuous Normalizing Flows

Our third task is unsupervised density estimation, using a scalable variant of continuous normalizing
flows called FFJORD (Grathwohl et al., 2019). We fit the MINIBOONE tabular dataset from
Papamakarios et al. (2017) and the MNIST image dataset (LeCun et al., 2010). We use the respective
singe-flow architectures from Grathwohl et al. (2019).
Grathwohl et al. (2019) noted that the NFE required to numerically integrate their dynamics could
become prohibitively expensive throughout training. Table 2 shows that we can reduce NFE by 38%
for only a 0.6% increase in log-likelihood measured in bits/dim.
How to train your Neural ODE We compare against the approach of Finlay et al. (2020), who
design two regularization terms specifically for stabilizing the dynamics of FFJORD models:
       
                        <<FORMULA>>

The first term is designed to encourage straight-line paths, and the second, stochastic, term is designed
to reduce overfitting. Finlay et al. (2020) used fixed-step solvers during training for some datasets.
We compare these two regularization on training with each of adaptive and fixed-step solvers, and
evaluated using an adaptive solver, in section 6.3.
6                        Analysis and Discussion
6.1                       Trading off function evaluations for loss
What does the trade off between accuracy and speed look like? Ideally, we could reduce the solver
time a lot without substantially reducing model performance. Indeed, this is demonstrated in all three
settings we explored. Figure 5 shows that generally, model performance starts getting substantially
worse only after a 50% reduction in solver speed when controlling R2 .

               <<FIGURE>>

Figure 5: Tuning the regularization of R2 trades off between training loss and solver speed in three
different applications of neural ODEs. Horizontal axes show average number of function evaluations,
and vertical axes show unregularized training loss, both at the end of training.

6.2 Order of regularization vs. order of solver

Which order of total derivatives should we regularize for a particular solver? As mentioned earlier,
we conjecture that the best choice would be to match the order of the solver being used. Regularizing
too low an order might needlessly constrain the dynamics and make it harder to fit the data, while
regularizing too high an order might leave the dynamics difficult to solve for a lower-order solver.
However, we also expect that optimizing higher-order derivatives might be challenging, since these
higher derivatives can change quickly even for small changes to the dynamics parameters.
Figures 6 and 7 investigate this question on the task of MNIST classification. Figure 6 compares the
effectiveness of regularizing different orders when using a solver of a particular order. For a 2nd
order solver, regularizing K = 2 produces a strictly better trade-off between performance and speed,
as expected. For higher-order solvers, including ones with adaptive order, we found that regularizing
orders above K = 3 gave little benefit.

               <<FIGURE>>

Figure 7 investigates the relationship between RK and the quantity it is meant to be a surrogate
for: NFE. We observe a clear monotonic relationship between the two, for all orders of solver and
regularization.

               6.3          Do we reduce training time?

Our approach produces models that are fastest to evaluate at test time. However, when we train
with adaptive solvers we do not improve overall training time, due to the additional expense of
computing our regularizer. Training with a fixed-grid solver is faster, but can be unstable if dynamics
are unregularized. Finlay et al. (2020)’s regularization and ours allow us to use fixed grid solvers and
reduce training time. However, ours is 2.4× slower than Finlay et al. (2020) for FFJORD because
their regularization re-uses terms already computed in the FFJORD training objective. For objectives
where these cannot be re-used, like MNIST classification, our method is 1.7× slower, but achieves
better test-time NFE.

               6.4       Are we making the solver overconfident?

Because we optimize dynamics in a way specifically designed to make the solver take longer steps,
we might fear that we are “adversarially attacking” our solver, making it overconfident in its ability
to extrapolate. Figure 8c shows that this is not the case for MNIST classification.

               6.5       Does speed regularization overfit?

Finlay et al. (2020) motivated one of their regularization terms by the possibility of overfitting: having
faster dynamics only for the examples in the training set, but still low on the test set. However, they
did not check whether overfitting was occurring. In fig. 8b we confirm that our regularized dynamics
have nearly identical average solve time on a held-out test set, on MNIST classification.

                             7        Related Work

Although the field of numerical ODE solvers is extremely mature, as far as we know, there has
been almost no work specifically on tuning differential equations to be faster to solve. The closest

                                 <<FIGURE>>

Figure 8: Figure 8c We observe that the actual solver error is about equally well-calibrated for
regularized dynamics as random dynamics, indicating that regularization does not make the solver
overconfident. Figure 8b: There is negligible overfitting of solver speed. ??: Speed regularization
does not usefully improve generalization. For large λ, our method reduces overfitting, but increases
overall test error due to under-fitting.

related work is Grathwohl et al. (2019) who mention attempting to use weight decay and spectral
normalization to reduce NFE, and of course Finlay et al. (2020), who, among other contributions,
introduced the use of fixed-step solvers for stable training.
Stabilizing dynamics Simard et al. (1991) regularized the dynamics of discrete-time recurrent
neural networks to improve their stability, by constraining the norm of the Jacobian of the dynamics
function in the direction of its largest eigenvalue. However, this approach has an O(D3 ) time cost.
De Brouwer et al. (2019) introduced a parameterization of neural ODEs analogous to instantaneous
Gated Recurrent Unit (GRU) recurrent neural network architectures in order to stabilize training
dynamics. Dupont et al. (2019) provided theoretical arguments that adding extra dimensions to the
state of a neural ODE should make training easier, and showed that this helped reduce NFE during
training.
Gradually increasing depth Chang et al. (2017) noted the connection between residual networks
and ODEs, and took advantage of this connection to gradually make resnets deeper during training,
in order to save time. One can view the increase in NFE while neural ODEs as an automatic, but
uncontrolled, version of their method. Their results suggest we might benefit from introducing a
speed regularization schedule that gradually tapers off during training.
Gradient Regularization Novak et al. (2018); Drucker & LeCun (1992) regularized the gradients
of neural networks to improve generalization.
Table 2: Density Estimation on MNIST using FFJORD. For adaptive solvers, indicated by ∞ Steps,
our approach is slowest to train, but requires the fewest NFE once trained. For fixed-step solvers our
approach achieves lower bits/dim and NFE when comparing across fixed-grid solvers using the same
number of steps. Fixed step solvers that diverged due to instability are indicated by NaN bits/dim.
  
                        8    Scope

The initial speedups obtained in this paper are not yet enough to make neural ODEs competitive with
standard fixed-depth architectures in terms of speed for standard supervised learning. However, there
are many applications where continuous-depth architectures provide a unique advantage. Besides
density models such as FFJORD and time series models, continuous-depth architectures have been
applied in solving mean-field games (Ruthotto et al., 2019), image segmentation (Pinckaers & Litjens,
2019), image super-resolution (Scao, 2020), and molecular simulations (Wang et al., 2020). These
applications, which already use continuous-time models, could benefit from the speed regularization
proposed in this paper.
While we investigated only ODEs in this paper, this approach could presumably be extended straight-
forwardly to neural stochastic differential equations fit by adaptive solvers (Li et al., 2020) and other
flavors of parametric differential equations fit by gradient descent (Rackauckas et al., 2019).

                      9    Limitations

Hyperparameters The hyperparameter λ needs to be chosen to balance speed and training loss.
One the other hand, neural ODEs don’t require choosing the outer number of layers, which needs to
be chosen separately for each stack of layers in standard architectures.
One also needs to choose solver order and tolerances, and these can substantially affect solver speed.
We did not investigate loosening tolerances, or modifying other parameters of the solver. The default
tolerance of 1.4e-8 for both atol and rtol behaved well in all our experiments.
One also needs to choose K. Higher K seems to generally work better, but is slower per step at
training time. In principle, if one can express their utility explicitly in terms of training loss and NFE,
it may be possible to tune λ automatically during training using the predictable relationship between
RK and NFE shown in fig. 7.
Slower overall training Although speed regularization reduces the overall NFE during training, it
makes each step more expensive. In our density estimation experiments (table 2), the overall effect
was about about 70% slower training, compared to no regularization, when using adaptive solvers.
However, test-time evaluation is much faster, since there is no slowdown per step.
10     Conclusions
This paper is an initial attempt at controlling the integration time of differential equations by regular-
izing their dynamics. This is an almost unexplored problem, and there are almost certainly better
quantities to optimize than the ones examined in this paper.
Based on these initial experiments, we propose three practical takeaways:
       1. Across all tasks, tuning the regularization usually gave at least a 2x speedup without
          substantially hurting model performance.
       2. Overall training time with speed regularization is in general about 30% to 50% slower with
          adaptive solvers.
       3. For standard solvers, regularizing orders higher than R2 or R3 provided little additional
          benefit.
Future work It may be possible to adapt solver architectures to take advantage of flexibility in
choosing the dynamics. Standard solver design has focused on robustly and accurately solving a
given set of differential equations. However, in a learning setting, we could consider simply rejecting
some kinds of dynamics as being too difficult to solve, analogous to other kinds of constraints we put
on models to encourage statistical regularization.
                                                   
                        Acknowledgements

We thank Barak Perlmutter, Ken Jackson, Ricky T.Q. Chen, Will Grathwohl, Chris Finlay, and
Chris Rackauckas for feedback and helpful discussions. Resources used in preparing this research
were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and
companies sponsoring the Vector Institute.

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                  Appendix A            Taylor-mode Automatic Differentiation

                              A.1    Taylor Polynomials

To clarify the relationship between the presentation in Chapter 13 of Griewank & Walther (2008) and
our results we give the distinction between the Taylor coefficients and derivative coefficients, also
known, unhelpfully, as Tensor coefficients.

For a sufficiently smooth vector valued function f : Rn → Rm and the polynomial

                         << x(t) = x[0] + x[1] t + x[2] t2 + x[3] t3 + · · · + x[d] td ∈ Rn>>                                 (5)

we are interested in the d-truncated Taylor expansion

                          <<y(t) = f (x(t)) + O(td+1 )>>                                                                      (6)
                               
                          <<≡ y[0] + y[1] t + y[2] t + y[3] t + · · · + y[d] t ∈ R >>                                    (7)

with the notation that <<FORMULA>> is the Taylor coefficient, which is the normalized derivative coefficient.

The Taylor coefficients of the expansion, y[j] , are smooth functions of the i ≤ j coefficients x[i],

                                       <<FORMULA>>                                                                        (8)

                                       <<FORMULA>>                                                                        (9)

                                       <<FORMULA>>                                                                       (10)

                                       <<FORMULA>>                                                                       (11)

These, as given in Griewank & Walther (2008), are written in terms of the normalized, Taylor
coefficients. This obscures their direct relationship with the derivatives, which we make explicit.
Consider the polynomial eq. (5) with Taylor coefficients expanded so their normalization is clear.
Further, let’s use suggestive notation that these coefficients correspond to the higher derivatives of 
x with respect to t, making x(t) a Taylor polynomial. That is <<FORMULA>>.
                                       <<FORMULA>>                                                                       (12)

                                       <<FORMULA>>                                                                       (13)

                                       <<FORMULA>>                                                                       (14)

Again, we are interested in the polynomial eq. (7), but with the normalization terms explicit

                                       <<FORMULA>>                                                                       (15)

Now we can expand the expressions for the Taylor coefficients y[i] to expressions for derivative
coefficients yi = i!y[i].

The coefficients of the Taylor expansion, yj , are smooth functions of the i ≤ j coefficients xi,

                                       <<FORMULA>>                                                                       (16)

                                       <<FORMULA>>                                                                       (17)

                                       <<FORMULA>>                                                                       (18)

                                       <<FORMULA>>                                                                       (19)

                                       <<FORMULA>>                                                                       (20)

                                       <<FORMULA>>                                                                       (21)

Therefore, eqs. (16), (17), (19) and (21) show that the derivative coefficient yi are exactly the ith
order higher derivatives of the composition f (x(t)) with respect to t. The key insight to this exercise
is that by writing the derivative coefficients explicitly we reveal that the expressions for the terms,
eqs. (16) to (18) and (20), involve terms previously computed for lower order terms.
In general, it will be useful to consider that the yk derivative coefficients is a function of all lower
order input derivatives

                                             <<yk = yk (x0 , . . . , xk )>>.                                  (22)

We provide the API to compute this in JAX by indexing the k-output of jet

                                      <<yk = jet(f, x0 , (x1 , . . . , xk ))[k]>>.

                  A.2    Relationship with Differential Equations

                           A.2.1    Autonomous Form

We can transform the initial value problem

                                     <<FORMULA>>                                (23)

into an autonomous dynamical system by augmenting the system to include the independent variable
with trivial dynamics Hairer et al. (1993):

                               <<FORMULA>>                              (24)

We do this for notational convenience, as well it disambiguates that derivatives with respect to t are 
meant in the “total" sense. This is aleviates the potential ambiguity of ∂t f (x(t), t) which could mean
both the derivative with respect to the second argument and the derivative through x(t) by the chain
rule <<FORMULA>>.

            A.2.2    Taylor Coefficients for ODE Solution with jet

Recall that jet gives us the coefficients for yi as a function of f and the coefficients xj≤i . We
can use jet and the relationship xk+1 = yk to recursively compute the coefficients of the solution
polynomial.

                     Algorithm 1 Taylor Coefficients for ODE Solution by Recursive Jet

                                    <<ALGORITHM>>

                        A.3    Regularizing Taylor Terms

Computing the Taylor coefficients for the ODE solution as in algorithm 1 will give a local approx-
imation to the ODE solution. If infinitely many Taylor coefficients could be computed this would
give the exact solution. The order of the final Taylor coefficient, determining the truncation of the
polynomial, gives the order of the approximation.
If the higher order Taylor coefficients of the solution are large, then truncation will result in a local
approximation that quickly diverts from the solution. However, if the higher Taylor coefficients are
small then the local approximation will remain close to the solution. This motivates our regularization
method. The effect of our regularizer on the Taylor expansion of a solution to a neural ODE can be
seen in fig. 9.

                  Appendix B         Experimental Details

Experiments were conducted using GPU-based ODE solvers. Training gradients were computed
using the adjoint method, in which the trajectory is reconstructed backwards in time to save memory,
for backpropagation. As in Finlay et al. (2020), we normalize our regularization term in eq. (1) by
the dimension of the vector-valued trajectory z(t) so that we may choose λ free of scaling by the
dimension of the problem.

               B.1    Efficient computation of the gradient of regularization term

To optimize our regularized objective, we must compute its gradient. We use the adjoint method
as described in Chen et al. (2018) to differentiate through the solution to the ODE. In particular, to
optimize our model we only need to compute the gradient of the regularization term. The adjoint
method gives the gradient of the ODE solution as a solution to an augmented ODE.

                               <<FIGURE>>

Figure 9: Left: The dynamics and a trajectory of a neural ODE trained on a toy supervised learning
problem. The dynamics are poorly approximated by a 6th-order local Taylor series, and requires 92
NFE by a solve by a 5th-order Runge-Kutta solver. Right: Regularizing the 6th-order derivatives of
trajectories gives dynamics that are easier to solve numerically, requiring only 68 NFE.

                     B.2   Supervised Learning

The dynamics function f : Rd × R → Rd is given by an MLP as follows

                                            <<z1 = σ(x)>>
                                        <<h1 = W1 [z1 ; t] + b1>>
                                            <<z2 = σ(h1 )>>
                                        <<y = W2 [z2 ; t] + b2>>

Where <<[·; ·]>> denotes concatenation of a scalar onto a column vector. The parameters are <<W1 ∈
R^h×d>>, <<b1 ∈ R^h>> and <<W2 ∈ R^d×h>> , <<b2 ∈ R^d>> . Here we use 100 hidden units, i.e.<< h = 100>>. We have
<<d = 784>>, the dimension of an MNIST image.
We train with a batch size of 100 for 160 epochs. We use the standard training set of 60,000 images,
and the standard test set of 10,000 images as a validation/test set. We optimize our model using SGD
with momentum with β = 0.9. Our learning rate schedule is 1e-1 for the first 60 epochs, 1e-2 until
epoch 100, 1e-3 until epoch 140, and 1e-4 for the final 20 epochs.
B.3   Continuous Generative Modelling of Time-Series
The PhysioNet dataset consists of observations of 41 distinct traits over a time period of 48 hours.
We remove the parameters “Age”, “Gender”, “Height”, and “ICUType” as these attributes do not vary
in time. We also quantize the measurements for each attribute by the hour by averaging multiple
measurements within the same hour. This leaves 49 unique time stamps (the extra time stamp for
observations at exactly the endpoint of the 48 hour observation period). We report all our losses on
this quantized data. We performed this rather coarse quantization for computational reasons having
to do with our particular implementation of this model. The validation split was obtained by taking
a random split of 20% of the trajectories from the full dataset. In total there are 8000 trajectories.
Code is included for processing the dataset, and links to downloading the data may be found in the
code for Rubanova et al. (2019). All other experimental details may be found in the main body and
appendices of Rubanova et al. (2019).

                     B.4   Continuous Normalizing Flows

For the model trained on the MINIBOONE tabular dataset from Papamakarios et al. (2017), we used
the same architecture as in Table 4 in the appendix of Grathwohl et al. (2019). We chose the number
of epochs and a learning rate schedule based on manual tuning on the validation set, in contrast
to Grathwohl et al. (2019) who tuned these automatically using early stopping and an automatic
heuristic for the learning rate decay using evaluation on a validation set. In particular, we trained for
500 epochs with a learning rate of 1e-3 for the first 300 epochs, 1e-4 until epoch 425, and 1e-5
for the remaining 75 epochs. The number of epochs and learning rate schedule was determined by
evaluating the model on the validation set every 10 epochs, and decaying the learning rate by a factor
of 10 once the loss on the validation set stopped improving for several evaluations, with the goal of
matching or improving upon the log-likelihood reported in Grathwohl et al. (2019). The data was
obtained as made available from Papamakarios et al. (2017), which was already processed and split
into train/validation/test. In particular, the training set has 29556 examples, the validation set has
3284 examples, and the test set has 3648 examples, which consist of 43 features.
It is important to note that we implemented a single-flow model for the MNIST dataset, while the
original comparison in Finlay et al. (2020) was on a multi-flow model. This accounts for discrepancy
in bits/dim and NFE reported in Finlay et al. (2020).
All other experimental details are as in Grathwohl et al. (2019).

                              B.5   Hardware

MNIST Supervised learning, Physionet Time-series, and MNIST FFJORD experiments were trained
and evaluated on NVIDIA Tesla P100 GPU. Tabular data FFJORD experiments were evaluated on
NVIDIA Tesla P100 GPU but trained on NVIDIA Tesla T4 GPU. All experiments except for MNIST
FFJORD were trained with double precision for purposes of reproducibility.

                     Appendix C         Additional Results

                               C.1   Overfitting of NFE


                                    <<FIGURE>>

                Figure 10: The difference in NFE is tracked by the variance of NFE.

In fig. 10 we note that there is a striking correspondence in the variance of NFE across individual
examples (in both the train set (dark red) and test set (light red)) and the absolute difference in NFE
between examples in the training set and test set. This suggests that any difference in the average
NFE between training examples and test examples is explained by noise in the estimate of the true
average NFE. It is also interesting that speed regularization does not have a monotonic relationship
with the variance of NFE, and we speculate as to how this might interact between the correspondence
of NFE for a particular example and the difficulty in the model correctly classifying it.

                     C.2         Trading off function evaluations with a surrogate loss

In fig. 11 and fig. 12 we confirm that our method poses a suitable tradeoff not only on the loss being
optimized, but also on the potentially non-differentiable loss which we truly care about. On MNIST,
we get a similar pareto curve when plotting classification error as opposed to cross-entropy loss, and
similarly on the time-series modelling task we see that we get a similar pareto curve on MSE loss as
compared to IWAE loss. The pareto curves are plotted for R3 , R2 respectively.

                                                   <<FIGURE>>

                                         Figure 11: MNIST Classification                                                                                                 
                                         
                                                   <<FIGURE>>

                                         Figure 12: Physionet Time-Series

                                    C.3         Wall-clock Time

We include additional tables with wall-clock time and training with fixed grid solvers in table 3 and
table 4.


                           Appendix D          Comparison to How to Train Your Neural ODE

The terms from Finlay et al. (2020) are

                                    <<FORMULA>>

and an estimate of
                                    <<FORMULA>>

                       Table 3: Classification on MNIST

                                    <<TABLE>>

These are combined with a weighted average and integrated along the solution trajectory.
These terms are motivated by the expansion

                                    <<FORMULA>>

Namely, eq. (3) regularizes the first total derivative of the solution, f (z(t), t), along the trajectory, and
eq. (4) regularizes a stochastic estimate of the Frobenius norm of the spatial derivative, ∇z f (z(t), t),
along the solution trajectory.
In contrast, R2 regularizes the norm of the second total derivative directly. In particular, this takes
into account the ∂f ∂t term. In other words, this accounts for the explicit dependence of f on time,
while eq. (3) and eq. (4) capture only the implicit dependence on time through z(t).
Even in the case of an autonomous system, that is, where ∂f    ∂t is identically 0 and the dynamics f only
depend implicitly on time, these terms still differ. Namely, R2 integrates the following along the
solution trajectory:

                                       <<FORMULA>>

while Finlay et al. (2020) penalizes the respective norms of the matrix ∇z f (z(t), t) and vector
f (z(t), t) separately.

                     Table 4: Density Estimation on Tabular Data (MINIBOONE)

                                       <<TABLE>>

<<END>> <<END>> <<END>>


<<START> <<START>> <<START>>


          How to Train Your Neural ODE: the World of Jacobian and Kinetic Regularization

                Chris Finlay 1 Jörn-Henrik Jacobsen 2 Levon Nurbekyan 3 Adam M Oberman 1
                                                                                                                        
                    Abstract

         Training neural ODEs on large datasets has not
         been tractable due to the necessity of allowing
         the adaptive numerical ODE solver to refine its
         step size to very small values. In practice this
         leads to dynamics equivalent to many hundreds
         or even thousands of layers. In this paper, we
         overcome this apparent difficulty by introducing
         a theoretically-grounded combination of both op-
         timal transport and stability regularizations which
         encourage neural ODEs to prefer simpler dynam-
         ics out of all the dynamics that solve a problem
         well. Simpler dynamics lead to faster conver-
         gence and to fewer discretizations of the solver,
         considerably decreasing wall-clock time without
         loss in performance. Our approach allows us to
         train neural ODE-based generative models to the
         same performance as the unregularized dynamics,                 
         with significant reductions in training time. This
         brings neural ODEs closer to practical relevance             
         in large-scale applications.

                                       <<FIGURE>>   

                        Figure 1. Optimal transport map and a generic normalizing flow.

                                                                      Indeed, it was observed that there is a striking similarity
 1. Introduction                                                      between ResNets and the numerical solution of ordinary
                                                                        differential equations (E, 2017; Haber & Ruthotto, 2017;
Recent research has bridged dynamical systems, a                     Ruthotto & Haber, 2018; Chen et al., 2018; 2019). In these
workhorse of mathematical modeling, with neural networks,            works, deep networks are interepreted as discretizations of
the defacto function approximator for high dimensional data.         an underlying dynamical system, where time indexes the
The great promise of this pairing is that the vast mathemat-         “depth” of the network and the parameters of the discretized
ical machinery stemming from dynamical systems can be                dynamics are learned. An alternate viewpoint was taken by
leveraged for modelling high dimensional problems in a               neural ODEs (Chen et al., 2018), where the dynamics of
dimension-independent fashion.                                       the neural network are approximated by an adaptive ODE
Connections between neural networks and ordinary differ-             solver on the fly. This latter approach is quite compelling
ential equations (ODEs) were almost immediately noted                as it does not require specifying the number of layers of the
after residual networks (He et al., 2016) were first proposed.       network beforehand. Furthermore, it allows the learning of
                                                                     homeomorphisms without any structural constraints on the
                                                                     function computed by the residual block.
                                                                     Neural ODEs have shown great promise in the physical sciences 
                                                                     (Köhler et al., 2019), in modeling irregular time series
                                                                     (Rubanova et al., 2019), mean field games (Ruthotto et al.,
                                                                     2019), continuous-time modeling (Yildiz et al., 2019; Kanaa
                                                                     et al., 2019), and for generative modeling through normaliz-
                                                                     ing flows with free-form Jacobians (Grathwohl et al., 2019).


Recent work has even adapted neural ODEs to the stochas-        based on (ODE) which abstain from a priori fixing step-size.
tic setting (Li et al., 2020). Despite these successes, some    Chen et al.’s method is a continuous-time generalization of
hurdles still remain. In particular, although neural ODEs are   residual networks, where the dynamics are generated by an
memory efficient, they can take a prohibitively long time to    adaptive ODE solver that chooses step-size on-the-fly.
train, which is arguably one of the main stumbling blocks
                                                                Because of their adaptive nature, neural ODEs can be more
towards their widespread adoption.
                                                                flexible than ResNets in certain scenarios, such as when
In this work we reduce the training time of neural ODEs         trading between model speed and accuracy. Moreover given
by regularizing the learned dynamics, complementing other       a fixed network depth, the memory footprint of neural ODEs
recent approaches to this end such as augmented neural          is orders of magnitude smaller than a standard ResNet dur-
ODEs (Dupont et al., 2019). Without further constraints on      ing training. They therefore show great potential on a host
their dynamics, high dimensional neural ODEs may learn          of applications, including generative modeling and density
dynamics which minimize an objective function, but which        estimation. An apparent drawback of neural ODEs is their
generate irregular solution trajectories. See for example       long training time: although a learned function f (· ; θ) may
Figure 1b, where an unregularized flow exhibits undesirable     generate a map that solves a problem particularly well, the
properties due to unnecessarily fluctuating dynamics. As        computational cost of numerically integrating (ODE) may
a solution, we propose two theoretically motivated regular-     be so prohibitive that it is not tractable in practice. In this
ization terms arising from an optimal transport viewpoint       paper we demonstrate this need not be so: with proper reg-
of the learned map, which encourage well-behaved dynam-         ularization, it is possible to learn f (· ; θ) so that (ODE) is
ics (see 1a left). We empirically demonstrate that proper       easily and quickly solved.
regularization leads to significant speed-up in training time
without loss in performance, thus bringing neural ODEs          2.1. FFJORD
closer to deployment on large-scale datasets. Our methods
are validated on the problem of generative modelling and        In density estimation and generative modeling, we wish
density estimation, as an example of where neural ODEs          to estimate an unknown data distribution p(x) from which
have shown impressive results, but could easily be applied      we have drawn N samples. Maximum likelihood seeks to
elsewhere.                                                      approximate p(x) with a parameterized distribution pθ (x)
                                                                by minimizing the Kullback-Leibler divergence between the
In summary, our proposed regularized neural ODE (RN-            two, or equivalently minimizing
ODE) achieves the same performance as the baseline, while
reducing the wall-clock training time by many hours or even                                    
days.                                                                               <<FORMULA>>              (1)
                                                                                             
2. Neural ODEs & Continuous normalizing                         Continuous normalizing flows (Grathwohl et al., 2019; Chen
    flows                                                       et al., 2018) parameterize pθ (x) using a vector field f :
                                                                Rd × R 7→ Rd as follows. Let z(x, T ) be the solution map
Neural ODEs simplify the design of deep neural networks         given by running the dynamics (ODE) for fixed time T .
by formulating the forward pass of a deep network as the        Suppose we are given a known distribution q at final time T ,
solution of a ordinary differential equation. Initial work      such as the normal distribution. Change of variables tells us
along these lines was motivated by the similarity of the eval-  that the distribution pθ (x) may be evaluated through
uation of one layer of a ResNet and the Euler discretization
of an ODE. Suppose the block in the t-th layer of a ResNet          <<log pθ (x) = log q (z(x, T )) + log det | ∇ z(x, T )|>>          (2)
is given by the function f (x, t; θ), where θ are the block’s
parameters. Then the evaluation of this layer of the ResNet     Evaluating the log determinant of the Jacobian is difficult.
is simply xt+1 = xt + f (xt , t; θ). Now, instead consider      Grathwohl et al. (2019) exploit the following identity from
the following ODE                                               fluid mechanics (Villani, 2003, p 114)

                <<FORMULA>>                        (ODE)                 <<log det | ∇ z(x, t)| = div (f ) (z(x, t), t))>>       (3)

The Euler discretization of this ODE with step-size <<τ>> is        where <<div(·)>> is the divergence operator, <<div(f ) (x) =
<<zt+1 = zt + τ f (zt , t; θ)>>, which is nearly identical to the      i ∂xi fi (x)>>. By the fundamental theorem of calculus, we
forward evaluation of the ResNet’s layer (setting step-size          1
                                                                       In the normalizing flow literature divergence is typically writ-
<<τ = 1>> gives equality). Armed with this insight, Chen et al.     ten explicitly as the trace of the Jacobian, however we use div (·)
(2018) suggested a method for training neural networks          which is more common elsewhere.

                                                         <<FIGURE>>
Figure 2. Log-likelihood (measured in bits/dim) on the validation set as a function of wall-clock time. Rolling average of three hours, with
90% confidence intervals.

may then rewrite (2) in integral form                                    From this simple motivating example, the need for regular-
                                                                         ity of the vector field is apparent. Without placing demands
                                                                         on the vector field f , it is entirely possible that the learned
 <<log pθ (x) = log q (z(x, T )) + div (f ) (z(x, s), s) ds>>
                                                                         dynamics will be poorly conditioned. This is not just a theo-
                                                               (4)       retical exercise: because the dynamics must be solved with
Remark 2.1 (Divergence trace estimate). In (Grathwohl                    a numerical integrator, poorly conditioned dynamics will
et al., 2019), the divergence is estimated using an unbiased             lead to difficulties during numerical integration of (ODE).
Monte-Carlo trace estimate (Hutchinson, 1990; Avron &                    Indeed, later we present results demonstrating a clear corre-
Toledo, 2011),                                                           lation between the number of time steps an adaptive solver
                                                                         takes to solve (ODE), and the regularity of f .       
             <<FORMULA>>             (5)                                 How can the regularity of the vector field be measured? One
                                                                         motivating approach is to measure the force experienced by
                                                                         a particle z(t) under the dynamics generated by the vector
By using the substitution (4), the task of maximizing log-               field f , which is given by the total derivative of f with
likelihood shifts from choosing pθ to minimize (1), to learn-            respect to time
ing the flow generated by a vector field f . This results in a
normalizing flow with a free-form Jacobian and reversible                      
dynamics, and was named FFJORD by Grathwohl et al..                                          <<FORMULA>>                      (6)
                                                                                      
2.2. The need for regularity                                                                  <<FORMULA>>                  (7)

The vector field learned through FFJORD that maximizes                  Well conditioned flows will place constant, or nearly con-
the log-likelihood is not unique, and raises troubling prob-             stant, force on particles as they travel. Thus, in this work we
lems related to the regularity of the flow. For a simple                 propose regularizing the dynamics with two penalty terms,
example, refer to Figure 1, where we plot two normaliz-                  one term regularizing f and the other ∇ f . The first penalty,
ing flows, both mapping a toy one-dimensional distribution               presented in Section 3, is a measure of the distance travelled
to the unit Gaussian, and where both maximize the log-                   under the flow f , and can alternately be interpreted as the
likelihood of exactly the same sample of particles. Figure               kinetic energy of the flow. This penalty term is based off
1a presents a “regular” flow, where particles travel in straight         of numerical methods in optimal transport, and encourages
lines that travel with constant speed. In contrast, Figure 1b            particles to travel in straight lines with constant speed. The
shows a flow that still maximizes the log-likelihood, but                second penalty term, discussed in Section 4, performs regu-
that has undesirable properties, such as rapidly varying local           larization on the Jacobian of the vector field. Taken together
trajectories and non-constant speed.                                     the two terms ensure that the force experienced by a particle

under the flow is constant or nearly so.                         3.1. Linking normalizing flows to optimal transport

These two regularizers will promote dynamics that follow         Now suppose we wish to minimize (18a), with q(z) a unit
numerically easy-to-integrate paths, thus greatly speeding       normal distribution, and p(x) a data distribution, unknown
up training time.                                                to us, but from which we have drawn N samples, and which
                                                                 we model as a discrete distribution of Dirac masses. Enforc-
3. Optimal transport maps &                                      ing the initial condition is trivial because we have sampled
                                                                 from p directly. The continuity equation (18b) need not be
   Benamou-Brenier
                                                                 enforced because we are tracking a finite number of sam-
There is a remarkable similarity between density estimation      pled particles. However the final time condition ρT = q
using continuous time normalizing flows, and the calcula-        cannot be implemented directly, since we do not have di-
tion of the optimal transport map between two densities          rect control on the form ρT (z) takes. Instead, introduce
using the Benamou-Brenier formulation (Benamou & Bre-            a Kullback-Leibler term to (18a) penalizing discrepancy
nier, 2000; Santambrogio, 2015). While a review of optimal       between ρT and q. This penalty term has an elegant simpli-
transport theory is far outside the scope of this paper, here    fication when p(x) is modeled as a distribution of a finite
we provide an informal summary of key ideas relevant to          number of masses, as is done in generative modeling. Set-
continuous normalizing flows. The quadratic-cost optimal         ting ρ0 = pθ a brief derivation yields
transport map between two densities p(x) and q(x) is a map       
z : Rd 7→ Rd minimizing the transport cost                       
                                                                                <<FORMULA>>                 (10)

              <<FORMULA>>                  (8)
                                                                 With this simplification (18a) becomes

subject to the constraint that A q(z) dz = z−1 (A) p(x) dx, 
in other words that the measure of any set A is preserved                      
under the map z. In a seminal work, Benamou & Brenier                            <<FORMULA>>                (11)
(2000) showed that rather than solving for minimizers of (8)
directly, an indirect (but computationally efficient) method
is available by writing z(x, T ) as the solution map of a
flow under a vector field f (as in (ODE)) for time T , by        For further details on this derivation consult the supplemen-
minimizing                                                       tary materials.
                                                                 The connection between the Benamou-Brenier formulation
                      <<FORMULA>>                          (9a)   of the optimal transport problem on a discrete set of points
                                                                 and continuous normalizing flows is apparent: the optimal
                                                                 transport problem (11) is a regularized form of the continu-
                      <<FORMULA>>                          (9b)  ous normalizing flow optimization problem (1). We there-
                      <<ρ0 (x) = p>>,                      (9c)  fore expect that adding a kinetic energy regularization term
                      <<ρT (z) = q>>.                      (9d)  to FFJORD will encourage solution trajectories to prefer
                                                                 straight lines with constant speed.
 The objective function (18a) is a measure of the kinetic
energy of the flow. The constraint (18b) ensures probability
mass is conserved. The latter two constraints guarantee the      4. Unbiased Frobenius norm regularization of
learned distribution agrees with the source p and target q.          the Jacobian
Note that the kinetic energy (18a) is an upper bound on the
                                                                 Refering to equation (7), one can see that even if f is regu-
transport cost, with equality only at optimality.
                                                                 larized to be small, via a kinetic energy penalty term, if the
The optimal flow f minimizing (18) has several particularly      Jacobian is large then the force experienced by a particle
appealing properties. First, particles induced by the opti-      may also still be large. As a result, the error of the numerical
mal flow f travel in straight lines. Second, particles travel    integrator can be large, which may lead an adaptive solver
with constant speed. Moreover, under suitable conditions         to make many function evaluations. This relationship is
on the source and target distributions, the optimal solution     apparent in Figure 3, where we empirically demonstrate the
map is unique (Villani, 2008). Therefore the solution map        correlation between the number of function evaluations of
z(x, t) is entirely characterized by the initial and final posi- f taken by the adaptive solver, and the size of the Jacobian
tions: z(x, t) = (1 − Tt )z(x, 0) + Tt z(x, T ). Consequently,   norm of f . The correlation is remarkably strong: dynamics
given an optimal f it is extraordinarily easy to solve (ODE)     governed by a poorly conditioned Jacobian matrix require
numerically with minimal computational effort.                   the adaptive solver to take many small time steps.


Algorithm 1 RNODE: regularized neural ODE training of
FFJORD
         <<ALGORITHM>>

                                                                                 <<FIGURE>>

                                                                 Figure 3. Number of function evaluations vs Jacobian Frobenius
                                                                norm of flows on CIFAR10 during training with vanilla FFJORD,
                                                                 using an adaptive ODE solver.
\
                                                                 Avron & Toledo, 2011). For real matrix B, an unbiased
                     <<FORMULA>>                                 estimate of the trace is given by

                                                                                    <<FORMULA>>                 (14)

                                                                 where <<FORMULA>> is drawn from a unit normal distribution. 
                                                                 Thus the squared Frobenius norm can be easily estimated by 
                                                                 setting B = AAT.
Moreover, in particle-based methods, the kinetic energy          Turning to the Jacobian <<FORMULA>> of a vector valued func-
term forces dynamics to travel in straight lines only on         tion f : Rd 7→ Rd , recall that the vector-Jacobian product
data seen during training, and so the regularity of the map      <<FORMULA>> may be quickly computed through reverse-mode
is only guaranteed on trajectories taken by training data.       automatic differentiation. Therefore an unbiased Monte-
The issue here is one of generalization: the map may be          Carlo estimate of the Frobenius norm of the Jacobian is
irregular on off-distribution or perturbed images, and cannot    readily available
be remedied by the kinetic energy term during training alone.
In the context of generalization, Jacobian regularization is                    <<FORMULA>>                         (15)
analagous to gradient regularization, which has been shown       
to improve generalization (Drucker & LeCun, 1992; Novak                         <<FORMULA>>                         (16)
et al., 2018).

For these reasons, we also propose regularizing the Jacobian     Conveniently, in the FFJORD framework the quantity
through its Frobenius norm. The Frobenius norm k · kF of a       <<FORMULA>> must be computed during the estimate of the prob-
real matrix A can be thought of as the `2 norm of the matrix     ability distribution under the flow, in the Monte-Carlo esti-
A vectorized                                                     mate of the divergence term (5). Thus Jacobian Frobenius
                      <<FORMULA>>                         (12)   norm regularization is available with essentially no extra
                                                                 computational cost.
Equivalently it may be computed as
                                                                 5. Algorithm description
                               
                     <<kAkF = tr(AAT)>>                   (13)   All together, we propose modifying the objective function
                                                                 of the FFJORD continuous normalizing flow (Grathwohl
and is the Euclidean norm of the singular values of a matrix.    et al., 2019) with the two regularization penalties of Sec-
In trace form, the Frobenius norm lends itself to estimation     tions 3 & 4. The proposed method is called RNODE, short
using a Monte-Carlo trace estimator (Hutchinson, 1990;           for regularized neural ODE. Pseudo-code of the method is

                                                         <<TABLE>>

Table 1. Log-likelihood (in bits/dim) and training time (in hours) on validation images with uniform dequantization. Results on clean
images are found in the supplemental materials. For comparison we report both the results of the original FFJORD paper (Grathwohl
et al., 2019) and our own independent run of FFJORD (“vanilla”) on CIFAR10 and MNIST. Vanilla FFJORD did not train on ImageNet64
(denoted by “x”). Also reported are results for other flow-based generative modeling papers. Our method (FFJORD with RNODE) has
comparable log-likelihood as FFJORD but is significantly faster.
   

                                                              <<FIGURE>>

Figure 4. Quality of generated samples samples on 5bit CelebA-HQ64 with RNODE. Here temperature annealing (Kingma & Dhariwal,
2018) with T = 0.7 was used to generate visually appealing images. For full sized CelebA-HQ256 samples, consult the supplementary
materials.

presented in Algorithm 1. The optimization problem to be                   Here E, l, and n are respectively the kinetic energy, the
solved is                                                                  log determinant of the Jacobian, and the integral of the
                                                                           Frobenius norm of the Jacobian.
                                                                           Both the divergence term and the Jacobian Frobenius norm
                                                                           are approximated with Monte-Carlo trace estimates. In our
                     <<FORMULA>>                                           implementation, the Jacobian Frobenius estamate reuses
                                                                           the computatian T ∇ f from the divergence estimate for
                                                                           efficiency. We remark that the kinetic energy term only
                     <<FORMULA>>                                           requires the computation of a dot product. Thus just as
                                                                           in FFJORD, our implementation scales linearly with the
                     <<FORMULA>>             (17)                          number of time steps taken by the ODE solver.

                                                                           Gradients of the objective function with respect to the net-
where z(x, t) is determined by numerically solving (ODE).                  work parameters are computed using the adjoint sensitivity
Note that we take the mean over number of samples and                      method (Pontryagin et al., 1962; Chen et al., 2018).
input dimension. This is to ensure that the choice of regu-
larization strength λK and λJ is independent of dimension
size and sample size.                                                      6. Experimental design
To compute the three integrals and the log-probability under               Here we demonstrate the benefits of regularizing neural
q of z(x, T ) at final time T , we augment the dynamics of                 ODEs on generative models, an application where neu-
the ODE with three extra terms, so that the entire system                  ral ODEs have shown strong empirical performance. We
solved by the numerical integrator is                                      use four datasets: CIFAR10 (Krizhevsky & Hinton, 2009),
                                                                           MNIST (LeCun & Cortes, 1998), downsampled ImageNet
                                                                           (64x64) (van den Oord et al., 2016), and 5bit CelebA-HQ
                                                                           (256x256) (Karras et al., 2017). We use an identical neural
        <<FORMULA>>                                   (RNODE)              architecture to that of Grathwohl et al. (2019). The dynamics
                                                                           (Kingma & Dhariwal, 2018) trained with 40 GPUs for a week;
                                                                           in contrast we train with four GPUs in just under a week.
 
                                                      <<FIGURE>>

Figure 5. Ablation study of the effect of the two regularizers, comparing two measures of flow regularity during training with a fixed
step-size ODE solver. Figure 5a: mean Jacobian Frobenius norm as a function of training epoch. Figure 5b: mean kinetic energy of the
flow as a function of training epoch. Figure 5c: number of function evaluations.

are defined by a neural network <<f (z, t; θ(t)) : Rd × R+ 7→          step size by a factor of two until the discrete dynamics were
Rd>> where <<θ(t)>> is piecewise constant in time. On MNIST we         stable and achieved good performance. The Runge-Kutta
use 10 pieces; CIFAR10 uses 14; downsampled ImageNet                   4(5) adaptive solver was used on the two larger datasets. We
uses 18; and CelebA-HQ uses 26 pieces. Each piece is a                 have also observed that RNODE improves the training time
4-layer deep convolutional network comprised of 3x3 ker-               of the adaptive solvers as well, requiring many fewer func-
nels and softplus activation functions. Intermediary layers            tion evaluations; however in Python we have found that the
have 64 hidden dimensions, and time t is concatenated to               fixed grid solver is typically quicker at a specified number
the spatial input z. The integration time of each piece is             of function evaluations. At test time RNODE uses the same
[0, 1]. Weight matrices are chosen to imitate the multi-scale          adaptive solver as FFJORD.
architecture of Real NVP (Dinh et al., 2017), in that im-
                                                                       We always initialize RNODE so that <<f(z, t) = 0>>; thus train-
ages are ‘squeezed’ via a permutation to halve image height
                                                                       ing begins with an initial identity map. This is done by zero-
and width but quadruple the number of channels. Diver-
                                                                       ing the parameters of the last layer in each piece (block),
gence of f is estimated using the Gaussian Monte-Carlo
                                                                       following Goyal et al. (2017). The identity map is an ap-
trace estimator with one sample of fixed noise per solver
                                                                       propriate choice because it has zero transport cost and zero
time-step.
                                                                       Frobenius norm. Moreover the identity map is trivially
On MNIST and CIFAR10 we train with a batch size of                     solveable for any numerical solver, thus training begins
200 and train for 100 epochs on a single GPU3 , using the              without any effort required on the solver’s behalf.
Adam optimizer (Kingma & Ba, 2015) with a learning rate
                                                                       On all datasets we set both the kinetic energy regularization
of 1e−3. On the two larger datasets, we train with four
                                                                       coefficient λK and the Jacobian norm coefficient λJ to 0.01.
GPUs, using a per-GPU batch size of respectively 3 and 50
for CelebA-HQ and ImageNet. Data is preprocessed by per-
turbing with uniform noise followed by the logit transform.            7. Results
The reference implementation of FFJORD solves the dy-                  A comparison of RNODE against FFJORD and other flow-
namics using a Runge-Kutta 4(5) adaptive solver (Dormand               based generative models is presented in Table 1. We report
& Prince, 1980) with error tolerances 1e−5 and initial step            both our running of “vanilla” FFJORD and the results as
size 1e−2. We have found that using less accurate solvers              originally reported in (Grathwohl et al., 2019). We highlight
on the reference implementation of FFJORD results in nu-               that RNODE runs roughly 2.8x faster than FFJORD on both
merically unstable training dynamics. In contrast, a simple            datasets, while achieving or surpassing the performance of
fixed-grid four stage Runge-Kutta solver suffices for RN-              FFJORD. This can further be seen in Figure 2 where we plot
ODE during training on MNIST and CIFAR10, using a                      bits per dimension ( − d1 log2 p(x), a normalized measure
step size of 0.25. The step size was determined based on               of log-likelihood) on the validation set as a function of
a simple heuristic of starting with 0.5 and decreasing the             training epoch, for both datasets. Visual inspection of the
sample quality reveals no qualitative difference between

                                          <<FIGURE>>

          Figure 6. Quality of generated samples samples with and without regularization on MNIST, left, and CIFAR10, right.

regularized and unregularized approaches; refer to Figure 6.         encourages flows to travel a minimal distance. In addition,
Generated images for downsampled ImageNet and CelebA-                we see that the Jacobian norm alone also has a beneficial
HQ are deferred to the supplementary materials; we provide           effect on the distance particles travel. Overall, the results
smaller generated images for networks trained on CelebA-             support our theoretical reasoning empirically.
HQ 64x64 in Figure 4.
Surprisingly, our run of “vanilla” FFJORD achieved slightly          8. Previous generative flows inspired by
better performance than the results reported in (Grathwohl               optimal transport
et al., 2019). We suspect the discrepancy in performance
and run times between our implementation of FFJORD and               Zhang et al. (2018) define a neural ODE flow where the
that of the original paper is due to batch size: Grathwohl           dynamics are given as the gradient of a scalar potential func-
et al. use a batch size of 900 and train on six GPUs, whereas        tion. This interpretation has deep connections to optimal
on MNIST and CIFAR10 we use a batch size of 200 and                  transport: the optimal transport map is the gradient of a
train on a single GPU.                                               convex potential function. Yang & Karniadakis (2019) con-
                                                                     tinue along these lines, and define an optimal transport again
We were not able to train vanilla FFJORD on ImageNet64,              as a scalar potential gradient. Yang & Karniadakis (2019)
due to numerical underflow in the adaptive solver’s time step.       enforce that the learned map is in fact an optimal trans-
This issue cannot be remedied by increasing the solver’s             port map by penalizing their objective function with a term
error tolerance, for this would bias the log-likelihood esti-        measuring violations of the continuity equation. Ruthotto
mates on validation.                                                 et al. (2019) place generative flows within a broader context
                                                                     of mean field games, and as an example consider a neural
7.1. Ablation study on MNIST                                         ODE gradient potential flow solving the optimal transport
                                                                     problem in up to 100 dimensions. We also note the recent
In Figure 5, we compare the effect of each regularizer by
                                                                     work of Twomey et al. (2019), who proposed regularizing
itself on the training dynamics with the fixed grid ODE
                                                                     neural ODEs with an Euler-step discretization of the kinetic
solver on the MNIST dataset. Without any regularization at
                                                                     energy term to enforce ‘straightness’, although connections
all, training dynamics are numerically unstable and fail after
                                                                     to optimal transport were not discussed.
just under 50 epochs. This is precisely when the Jacobian
norm grows large; refer to Figure 5a. Figure 5a demonstrates         When a flow is the gradient of a scalar potential, the change
that each regularizer by itself is able to control the Jacobian      of variables formula (4) simplifies so that the divergence
norm. The Jacobian regularizer is better suited to this task,        term is replaced by the Laplacian of the scalar potential.
although it is interesting that the kinetic energy regularizer       Although mathematically parsimonious and theoretically
also improves the Jacobian norm. Unsurprisingly Figure 5b            well-motivated, we chose not to implement our flow as the
demonstrates the addition of the kinetic energy regularizer          gradient of a scalar potential function due to computational
                       How to Train Your Neural ODE: the World of Jacobian and Kinetic Regularization
constraints: such an implementation would require ‘triple       through CIFAR, and companies sponsoring the Vector Insti-
backprop’ (twice to compute or approximate the Laplacian,       tute (www.vectorinstitute.ai/#partners).
and once more for the parameter gradient). Ruthotto et al.
(2019) circumvented this problem by utilizing special struc-    References
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Acknowledgements                                                   2019.     URL http://papers.nips.cc/paper/
                                                                   9183-residual-flows-for-invertible-
C. F. and A. O. were supported by a grant from the Innova-         generative-modeling.
tive Ideas Program of the Healthy Brains and Healthy Lives
initiative (HBHL) through McGill University.                    Dinh, L., Sohl-Dickstein, J., and Bengio, S. Den-
L. N. was supported by AFOSR MURI FA9550-18-1-0502,                sity estimation using real NVP. In 5th International
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A. Details of Section 3.1: Benamou-Brenier                         Hence, multiplying the objective function in (20) by λ and
    formulation in Lagrangian coordinates                          ignoring the f -independent term Ex∼p log p(x) we obtain
                                                                   an equivalent objective function
The Benamou-Brenier formulation of the optimal transporta-                   
tion (OT) problem in Eulerian coordinates is                        
                                                                      <<FORMULA>>                      (21)

                  <<FORMULA>>                             (18a)

                                                                   Finally, if we assume that {xi }N  i=1 are iid sampled from p,
                  <<FORMULA>>                             (18b)    we obtain the empirical objective function

                  <<ρ0 (x) = p>>,                        (18c)         

                  <<ρT (z) = q>>.                      (18d)                 <<FORMULA>>                         (22)

 The connection between continuous normalizing flows
(CNF) and OT becomes transparent once we rewrite (18) in
Lagrangian coordinates. Indeed, for regular enough velocity
                                                                   B. Additional results
fields f one has that the solution of the continuity equation      Here we present additional generated samples on the two
(18b), (18c) is given by ρt = z(·, t)]p where z is the flow        larger datasets considered, CelebA-HQ and ImageNet64. In
                                                                   addition bits/dim on clean images are reported in Table 2.
         <<FORMULA>>

The relation ρt = z(·, t)]p means that for arbitrary test
function φ we have that

               <<φ(x)ρt (x, t)dx = φ(z(x, t))p(x)dx>>

Therefore (18) can be rewritten as

   <<min      kf (z(x, t), t)k2 p(x) dxdt>>               (19a)

   <<subject to         ż(x, t) = f (z(x, t), t)>>,       (19b)

                      <<z(x, 0) = x>>,                     (19c)

                      <<z(·, T )]p = q>>.                  (19d)

Note that ρt is eliminated in this formulation. The terminal
condition (18d) is trivial to implement in Eulerian coordi-
nates (grid-based methods) but not so simple in Lagrangian
ones (19d) (grid-free methods). To enforce (19d) we intro-
duce a penalty term in the objective function that measures
the deviation of z(·, T )]p from q. Thus, the penalized ob-
jective function is
               <<FORMULA>>          (20)
where λ > 0 is the penalization strength. Next, we observe
that this objective function can be written as an expectation
with respect to x ∼ p. Indeed, the Kullback-Leibler di-
vergence is invariant under coordinate transformations, and
therefore

         <<FORMULA>>
              
                  <<FIGURE>>

Figure 7. Quality of FFJORD RNODE generated images on ImageNet-64.

               <<FIGURE>>

Figure 8. Quality of FFJORD RNODE generated images on CelebA-HQ. We use temperature annealing, as described in (Kingma &
Dhariwal, 2018), to generate visually appealing images, with T = 0.5, . . . , 1.

Table 2. Additional results and model statistics of FFJORD RNODE. Here we report validation bits/dim on both validation images, and on
validation images with uniform variational dequantization (ie perturbed by uniform noise). We also report number of trainable model
parameters.
                          <<TABLE>>

<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>

                     A guide to convolution arithmetic for   deep
                                      learning

                               The authors of this guide would like to thank David Warde-Farley,
                             Guillaume Alain  and  Caglar Gulcehre for   their valuable feedback. We
                             are likewise grateful to all those who helped improve this tutorial with
                             helpful comments, constructive criticisms  and  code contributions. Keep
                             them coming!
                               Special thanks to Ethan Schoonover, creator of the Solarized color
                             scheme, 1 whose colors were used for   the figures.

                                                    Feedback
                               Your feedback is welcomed! We did our best to be as precise, infor-
                             mative  and  up to the point as possible, but should there be any thing you
                             feel might be an error or could be rephrased to be more precise or com-
                             prehensible, please don’t refrain from contacting us. Likewise, drop us a
                             line if you think there is something that might fit this technical report
                              and  you would like us to discuss – we will make our best effort to update
                             this document.

                                            Source code  and  animations
                               The code used to generate this guide along with its figures is available
                             on GitHub. 2 There the reader can also find an animated version of the
                             figures.


                      1 Introduction 5
                        1.1 Discrete convolutions . . . . . . . . . . . . . . . . . . . . . . . . .6
                        1.2 Pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

                      2 Convolution arithmetic 12
                        2.1 No zero padding, unit strides . . . . . . . . . . . . . . . . . . . .12
                        2.2 Zero padding, unit strides . . . . . . . . . . . . . . . . . . . . . .13
                            2.2.1 Half (same) padding . . . . . . . . . . . . . . . . . . . . .13
                            2.2.2 Full padding . . . . . . . . . . . . . . . . . . . . . . . . .13
                        2.3 No zero padding, non-unit strides . . . . . . . . . . . . . . . . . .15
                        2.4 Zero padding, non-unit strides . . . . . . . . . . . . . . . . . . . .15

                      3 Pooling arithmetic 18

                      4 Transposed convolution arithmetic 19
                        4.1 Convolution as a matrix operation . . . . . . . . . . . . . . . . .20
                        4.2 Transposed convolution . . . . . . . . . . . . . . . . . . . . . . .20
                        4.3 No zero padding, unit strides, transposed . . . . . . . . . . . . .21
                        4.4 Zero padding, unit strides, transposed . . . . . . . . . . . . . . .22
                            4.4.1 Half (same) padding, transposed . . . . . . . . . . . . . .22
                            4.4.2 Full padding, transposed . . . . . . . . . . . . . . . . . . .22
                        4.5 No zero padding, non-unit strides, transposed . . . . . . . . . . .24
                        4.6 Zero padding, non-unit strides, transposed . . . . . . . . . . . . .24

                      5 Miscellaneous convolutions 28
                        5.1 Dilated convolutions . . . . . . . . . . . . . . . . . . . . . . . . .28


                                                                        Chapter 1


                      Introduction


                      Deep convolutional neural networks (CNNs) have been at the heart of spectac-
                      ular advances in deep learning. Although CNNs have been used as early as the
                      nineties to solve character recognition tasks (Le Cunet al., 1997), their current
                      widespread application is due to much more recent work, when a deep CNN
                      was used to beat state-of-the-art in the ImageNet image classification challenge
                      (Krizhevskyet al., 2012).
                        Convolutional neural networks therefor  e constitute a very useful tool for   ma-
                      chine learning practitioners. However, learning to use CNNs for   the first time
                      is generally an intimidating experience. A convolutional layer’s output shape
                      is affected by the shape of its input as well as the choice of kernel shape, zero
                      padding  and  strides,  and  the relationship between these properties is not triv-
                      ial to infer. This contrasts with fully-connected layers, whose output size is
                      independent of the input size. Additionally, CNNs also usually feature apool-
                      ingstage, adding yet another level of complexity with respect to fully-connected
                      networks. Finally, so-called transposed convolutional layers (also known as frac-
                      tionally strided convolutional layers) have been employed in more  and  more work
                      as of late (Zeileret al., 2011; Zeiler  and  Fergus, 2014; Longet al., 2015; Rad-
                      for  det al., 2015; Visinet al., 2015; Imet al., 2016),  and  their relationship with
                      convolutional layers has been explained with various degrees of clarity.
                        This guide’s objective is twofold:

                        1.Explain the relationship between convolutional layers  and  transposed con-
                          volutional layers.
                        2.Provide an intuitive underst and ing of the relationship between input shape,
                          kernel shape, zero padding, strides  and  output shape in convolutional,
                          pooling  and  transposed convolutional layers.

                        In order to remain broadly applicable, the results shown in this guide are
                      independent of implementation details  and  apply to all commonly used machine
                      learning frameworks, such as Theano (Bergstraet al., 2010; Bastienet al., 2012),



                      Torch (Collobertet al., 2011), Tensorflow (Abadiet al., 2015)  and  Caffe (Jia et al., 2014).

                        This chapter briefly reviews the main building blocks of CNNs, namely dis-
                      crete convolutions  and  pooling. for   an in-depth treatment of the subject, see
                      Chapter 9 of the Deep Learning textbook (Goodfellowet al., 2016).


                      1.1 Discrete convolutions

                      The bread  and  butter of neural networks is affine transformations: a vector
                      is received as input  and  is multiplied with a matrix to produce an output (to
                      which a bias vector is usually added before passing the result through a non-
                      linearity). This is applicable to any  type of input, be it an image, a sound
                      clip or an unordered collection of features: whatever their dimensionality, their
                      representation can always be flattened into a vector before the transfomation.
                        Images, sound clips  and  many  other similar kinds of data have an intrinsic
                      structure. More formally, they share these important properties:

                        They are stored as multi-dimensional arrays.
                        They feature one or more axes for   which ordering matters (e.g., width  and 
                          height axes for   an image, time axis for   a sound clip).
                        One axis, called the channel axis, is used to access different views of the
                          data (e.g., the red, green  and  blue channels of a color image, or the left
                           and  right channels of a stereo audio track).

                        These properties are not exploited when an affine transformation is applied;
                      in fact, all the axes are treated in the same way  and  the topological information
                      is not taken into account. Still, taking advantage of the implicit structure of
                      the data may prove very h and y in solving some tasks, like computer vision  and 
                      speech recognition,  and  in these cases it would be best to preserve it. This is
                      where discrete convolutions come into play.
                        A discrete convolution is a linear transformation that preserves this notion
                      of ordering. It is sparse (only a few input units contribute to a given output
                      unit)  and  reuses parameters (the same weights are applied to multiple locations
                      in the input).
                        Figure 1.1 provides an example of a discrete convolution. The light blue
                      grid is called the input feature map. To keep the drawing simple, a single input
                      feature map is represented, but it is not uncommon to have multiple feature
                      maps stacked one onto another. 1 A kernel(shaded area) of value

                                            <<FIGURE>>

                           Figure 1.1: Computing the output values of a discrete convolution.



                                            <<FIGURE>>


                          Figure 1.2: Computing the output values of a discrete convolution for   N = 2, i1 =i2 = 5, k1 =k2 = 3, s1 =s2 = 2,  and  p1 =p2 = 1.







                      slides across the input feature map. At each location, the product between
                      each element of the kernel  and  the input element it overlaps is computed  and 
                      the results are summed up to obtain the output in the current location. The
                      procedure can be repeated using different kernels to for  m as many  output feature
                      maps as desired (Figure 1.3). The final outputs of this procedure are called
                      output feature maps.2 If there are multiple input feature maps, the kernel will
                      have to be 3-dimensional – or, equivalently each one of the feature maps will
                      be convolved with a distinct kernel –  and  the resulting feature maps will be
                      summed up elementwise to produce the output feature map.
                        The convolution depicted in Figure 1.1 is an instance of a 2-D convolution,
                      but it can be generalized to N-D convolutions. for   instance, in a 3-D convolu-
                      tion, the kernel would be a cuboid and  would slide across the height, width  and 
                      depth of the input feature map.
                        The collection of kernels defining a discrete convolution has a shape corre-
                      sponding to some permutation of(n;m;k 1 ;:::;k N ), where


                                      <<FORMULA>>

                        The following properties affect the output size oj of a convolutional layer
                      along axis j:

                                      <<FORMULA>>

                      for   instance, Figure 1.2 shows a 3x3 kernel applied to a 5x5 input padded
                      with a 1x1 border of zeros using 2x2 strides.
                        Note that strides constitute a for  m of subsampling. As an alternative to
                      being interpreted as a measure of how much the kernel is translated, strides can
                      also be viewed as how much of the output is retained. for   instance, moving
                      the kernel by hops of two is equivalent to moving the kernel by hops of one but
                      retaining only odd output elements (Figure 1.4).
                        1 An example of this is what was referred to earlier as channels for images  and  sound clips.
                        2 While there is a distinction between convolution  and  cross-correlation from a signal pro-
                      cessing perspective, the two become interchangeable when the kernel is learned. for   the sake
                      of simplicity  and  to stay consistent with most of the machine learning literature, the term
                      convolution will be used in this guide.

                                            <<FIGURE>>

                      Figure 1.3: A convolution mapping from two input feature maps to three output
                      feature maps using a32 3x3 collection of kernels w. In the left pathway,
                      input feature map 1 is convolved with kernel w1;1  and  input feature map 2 is
                      convolved with kernel w1;2 ,  and  the results are summed together elementwise
                      to for  m the first output feature map. The same is repeated for   the middle  and 
                      right pathways to for  m the second  and  third feature maps,  and  all three output
                      feature maps are grouped together to for  m the output.

                                            <<FIGURE>>

                      Figure 1.4: An alternative way of viewing strides. Instead of translating the
                       3x3 kernel by increments ofs= 2(left), the kernel is translated by increments
                      of1 and  only one ins= 2output elements is retained (right).


                                                1.2 Pooling

                      In addition to discrete convolutions themselves,pooling operations make up
                      another important building block in CNNs. Pooling operations reduce the size
                      of feature maps by using some function to summarize subregions, such as taking
                      the average or the maximum value.
                        Pooling works by sliding a window across the input  and  feeding the content
                      of the window to a pooling function. In some sense, pooling works very much
                      like a discrete convolution, but replaces the linear combination described by the
                      kernel with some other function. Figure 1.5 provides an example for   average
                      pooling,  and  Figure 1.6 does the same for   max pooling.
                        The following properties affect the output size j of a pooling layer along
                      axisj:

                                      <<FORMULA>>



                                                  <<FIGURE>>


                     Figure 1.5: Computing the output values of a  3x3  average pooling operation on a 5x5 input using 1x1 strides.

                                                  <<FIGURE>>


                     Figure 1.6: Computing the output values of a  3x3  max pooling operation on a 5X5 input using 1X1 strides.



                      Convolution arithmetic


                      The analysis of the relationship between convolutional layer properties is eased
                      by the fact that they don’t interact across axes, i.e., the choice of kernel size,
                      stride  and  zero padding along axis j only affects the output size of axis j.
                      Because of that, this chapter will focus on the following simplified setting:

                        2-D discrete convolutions (N= 2),
                        square inputs (i1 =i2 =i),
                        square kernel size (k1 =k2 =k),
                        same strides along both axes (s1 =s2 =s),
                        same zero padding along both axes (p1 =p2 =p).

                        This facilitates the analysis  and  the visualization, but keep in mind that the
                      results outlined here also generalize to the N-D  and  non-square cases.


                      2.1 No zero padding, unit strides

                      The simplest case to analyze is when the kernel just slides across every position
                      of the input (i.e.,s= 1 and p= 0). Figure 2.1 provides an example for  i= 4
                       and k= 3.
                        One way of defining the output size in this case is by the number of possible
                      placements of the kernel on the input. Let’s consider the width axis: the kernel
                      starts on the leftmost part of the input feature map  and  slides by steps of one
                      until it touches the right side of the input. The size of the output will be equal
                      to the number of steps made, plus one, accounting for   the initial position of the
                      kernel (Figure 2.8a). The same logic applies for   the height axis.
                        More formally, the following relationship can be inferred:
                        
                          Relationship 1.for   any i,k and p,  and  for  s= 1,

                                                     <<FORMULA>>



                      2.2 Zero padding, unit strides

                      To factor in zero padding (i.e., only restricting tos= 1), let’s consider its effect
                      on the effective input size: padding with p zeros changes the effective input size
                      from i to i+ 2p. In the general case, Relationship 1 can then be used to infer
                      the following relationship:

                          Relationship 2.for   any  i,k  and  p,  and  for   s= 1,

                                                <<FORMULA>>

                      Figure 2.2 provides an example for   i= 5,k= 4  and  p= 2.
                        In practice, two specific instances of zero padding are used quite extensively
                      because of their respective properties. Let’s discuss them in more detail.

                      2.2.1 Half (same) padding
                      Having the output size be the same as the input size (i.e.,o=i) can be a
                      desirable property:

                          Relationship 3.for   any  i  and  for   k o d (k= 2n+ 1; n2N),
                          s= 1  and  p=b k=2 c=n,

                                               <<FORMULA>> 

                      This is sometimes referred to as half(or same) padding. Figure 2.3 provides an
                      example for   i= 5,k= 3 and  (therefor  e) p= 1.

                      2.2.2 Full padding
                      While convolving a kernel generally decreases the output size with respect to
                      the input size, sometimes the opposite is required. This can be achieved with
                      proper zero padding:

                          Relationship 4.for   any  i  and  k,  and  for   p=kx1  and  s= 1,

                                                <<FORMULA>>


                                                <<FIGURE>>

                      Figure 2.1: (No padding, unit strides) Convolving a 3x3 kernel over a 4x4 
                      input using unit strides (i.e.,i= 4,k= 3,s= 1  and  p= 0).


                                                <<FIGURE>>

                      Figure 2.2: (Arbitrary padding, unit strides) Convolving a 4x4 kernel over a
                      5x5 input padded with a 2x2 border of zeros using unit strides (i.e.,i= 5,
                      k= 4,s= 1 and p= 2).


                                              <<FIGURE>>


                      Figure 2.3: (Half padding, unit strides) Convolving a 3x3 kernel over a 5x5 
                      input using half padding  and  unit strides (i.e.,i= 5,k= 3,s= 1  and  p= 1).


                                            <<FIGURE>>


                      Figure 2.4: (Full padding, unit strides) Convolving a 3x3 kernel over a 5x5 
                      input using full padding  and  unit strides (i.e.,i= 5,k= 3,s= 1  and  p= 2).


                      This is sometimes referred to as full padding, because in this setting every
                      possible partial or complete superimposition of the kernel on the input feature
                      map is taken into account. Figure 2.4 provides an example for i= 5,k= 3  and 
                      (therefore) p= 2.


                      2.3 No zero padding, non-unit strides

                      All relationships derived so far only apply for   unit-strided convolutions. Incorporating 
                      non unitary strides requires another inference leap. To facilitate
                      the analysis, let’s momentarily ignore zero padding (i.e.,s >1  and  p= 0).
                      Figure 2.5 provides an example for   i= 5,k= 3 and s= 2.
                        Once again, the output size can be defined in terms of the number of possible
                      placements of the kernel on the input. Let’s consider the width axis: the kernel
                      starts as usual on the leftmost part of the input, but this time it slides by steps
                      of sizes until it touches the right side of the input. The size of the output is
                      again equal to the number of steps made, plus one, accounting for   the initial
                      position of the kernel (Figure 2.8b). The same logic applies for   the height axis.
                        From this, the following relationship can be inferred:

                          Relationship 5.for   any  i,k  and  s,  and  for   p= 0,
                                               
                                               <<FORMULA>>

                      The floor function accounts for   the fact that sometimes the last possible step
                      does not coincide with the kernel reaching the end of the input, i.e., some input
                      units are left out (see Figure 2.7 for   an example of such a case).


                      2.4 Zero padding, non-unit strides

                      The most general case (convolving over a zero padded input using non-unit
                      strides) can be derived by applying Relationship 5 on an effective input of size
                      i+ 2p, in analogy to what was done for   Relationship 2:

                          Relationship 6.for   any i,k,p and s,
                                             
                                             <<FORMULA>>

                      As before, the floor function means that in some cases a convolution will produce
                      the same output size for   multiple input sizes. More specifically, ifi+ 2pkis
                      a multiple ofs, then any  input size j=i+a; a2 f0;:::; sx1 g will produce
                      the same output size. Note that this ambiguity applies only for   s >1.

                                            <<FIGURE>>

                        Figure 2.6 shows an example with i= 5,k= 3,s= 2  and  p= 1, while

                                            <<FIGURE>>

                      Figure 2.7 provides an example for   i= 6,k= 3,s= 2  and  p= 1. Interestingly,

                      despite having different input sizes these convolutions share the same output
                      size. While this doesn’t affect the analysis for   convolutions, this will complicate
                      the analysis in the case of transposed convolutions.


                                            <<FIGURE>>

                      Figure 2.5: (No zero padding, arbitrary strides) Convolving a 3x3 kernel over
                      a 5x5 input using 2x2 strides (i.e.,i= 5,k= 3,s= 2  and  p= 0).

                                            <<FIGURE>>

                      Figure 2.6: (Arbitrary padding  and  strides) Convolving a 3x3 kernel over a
                       5x5 input padded with a 1x1 border of zeros using 2x2 strides (i.e.,i= 5,
                      k= 3,s= 2  and  p= 1).

                                            <<FIGURE>>

                      Figure 2.7: (Arbitrary padding  and  strides) Convolving a 3x3 kernel over a
                      6x6 input padded with a 1x1 border of zeros using 2x2 strides (i.e.,i= 6,
                      k= 3,s= 2  and  p= 1). In this case, the bottom row  and  right column of the
                      zero padded input are not covered by the kernel.

                      (a) The kernel has to slide two steps (b) The kernel has to slide one step of
                      to the right to touch the right side of size two to the right to touch the right
                      the input ( and  equivalently downwards). side of the input ( and  equivalently down-
                      Adding one to account for   the initial ker- wards). Adding one to account for   the
                      nel position, the output size is 3x3. initial kernel position, the output size is 2x2.

                                            <<FIGURE>>

                                     Figure 2.8: Counting kernel positions.


                                                 Chapter 3

                      Pooling arithmetic

                      In a neural network, pooling layers provide invariance to small translations of
                      the input. The most common kind of pooling is max pooling, which consists
                      in splitting the input in (usually non-overlapping) patches  and  outputting the
                      maximum value of each patch. Other kinds of pooling exist, e.g., mean or
                      average pooling, which all share the same idea of aggregating the input locally
                      by applying a non-linearity to the content of some patches (Boureauet al.,
                      2010a,b, 2011; Saxeet al., 2011).
                        Some readers may have noticed that the treatment of convolution arithmetic
                      only relies on the assumption that some function is repeatedly applied onto
                      subsets of the input. This means that the relationships derived in the previous
                      chapter can be reused in the case of pooling arithmetic. Since pooling does not
                      involve zero padding, the relationship describing the general case is as follows:

                          Relationship 7.for   any  i,k  and  s,

                                              <<FORMULA>>

                      This relationship holds for any  type of pooling.


                                                Chapter 4

                      Transposed convolution arithmetic

                        
                      The need for   transposed convolutions generally arises from the desire to use a
                      transfor  mation going in the opposite direction of a normal convolution, i.e., from
                      something that has the shape of the output of some convolution to something
                      that has the shape of its input while maintaining a connectivity pattern that
                      is compatible with said convolution. for   instance, one might use such a trans-
                      for  mation as the decoding layer of a convolutional autoencoder or to project
                      feature maps to a higher-dimensional space.
                        Once again, the convolutional case is considerably more complex than the
                      fully-connected case, which only requires to use a weight matrix whose shape has
                      been transposed. However, since every convolution boils down to an efficient im-
                      plementation of a matrix operation, the insights gained from the fully-connected
                      case are useful in solving the convolutional case.
                        Like for   convolution arithmetic, the dissertation about transposed convolu-
                      tion arithmetic is simplified by the fact that transposed convolution properties
                      don’t interact across axes.
                        The chapter will focus on the following setting:

                        2-D transposed convolutions (N= 2),
                        square inputs (i1 =i2 =i),
                        square kernel size (k1 =k2 =k),
                        same strides along both axes (s1 =s2 =s),
                        same zero padding along both axes (p1 =p2 =p).

                      Once again, the results outlined generalize to the N-D  and  non-square cases.




                                                 4.1 Convolution as a matrix operation

                      Take for   example the convolution represented in Figure 2.1. If the input  and 
                      output were to be unrolled into vectors from left to right, top to bottom, the
                      convolution could be represented as a sparse matrix C where the non-zero elements 
                      are the elements w i;j of the kernel (with i  and  j being the row  and  column
                      of the kernel respectively):
                    
                                                <<FORMULA>>

                        This linear operation takes the input matrix flattened as a 16-dimensional
                      vector  and  produces a 4-dimensional vector that is later reshaped as the 2x2 
                      output matrix.
                        Using this representation, the backward pass is easily obtained by trans-
                      posingC; in other words, the error is backpropagated by multiplying the loss
                      withCT . This operation takes a 4-dimensional vector as input  and  produces
                      a 16-dimensional vector as output,  and  its connectivity pattern is compatible
                      withCby construction.
                        Notably, the kernel w defines both the matrices C  and  CT used for   the
                      for  ward  and  backward passes.


                      4.2 Transposed convolution

                      Let’s now consider what would be required to go the other way around, i.e.,
                      map from a 4-dimensional space to a 16-dimensional space, while keeping the
                      connectivity pattern of the convolution depicted in Figure 2.1. This operation
                      is known as a transposed convolution.
                        Transposed convolutions – also called fractionally strided convolutions or
                      deconvolutions 1 – work by swapping the for  ward  and  backward passes of a con-
                      volution. One way to put it is to note that the kernel defines a convolution, but
                      whether it’s a direct convolution or a transposed convolution is determined by
                      how the for  ward  and  backward passes are computed.
                        for   instance, although the kernel w defines a convolution whose for  ward  and 
                      backward passes are computed by multiplying with C  and  CT respectively, it
                      also defines a transposed convolution whose for  ward  and  backward passes are
                      computed by multiplying withCT  and  (CT )T =C respectively. 2
                        Finally note that it is always possible to emulate a transposed convolution
                      with a direct convolution. The disadvantage is that it usually involves adding
                        1 The term “deconvolution” is sometimes used in the literature, but we advocate against it
                      on the grounds that a deconvolution is mathematically defined as the inverse of a convolution,
                      which is different from a transposed convolution.
                        2 The transposed convolution operation can be thought of as the gradient of some convolution 
                        with respect to its input, which is usually how transposed convolutions are implemented
                      in practice.


                      many  columns  and  rows of zeros to the input, resulting in a much less efficient
                      implementation.
                        Building on what has been introduced so far, this chapter will proceed some-
                      what backwards with respect to the convolution arithmetic chapter, deriving the
                      properties of each transposed convolution by referring to the direct convolution
                      with which it shares the kernel,  and  defining the equivalent direct convolution.


                      4.3 No zero padding, unit strides, transposed

                      The simplest way to think about a transposed convolution on a given input is
                      to imagine such an input as being the result of a direct convolution applied on
                      some initial feature map. The transposed convolution can be then considered as
                      the operation that allows to recover the shape 3 of this initial feature map.
                        Let’s consider the convolution of a 3x3 kernel on a 4x4 input with unitary
                      stride  and  no padding (i.e.,i= 4,k= 3,s= 1  and  p= 0). As depicted in
                      Figure 2.1, this produces a 2x2 output. The transpose of this convolution will
                      then have an output of shape 4x4 when applied on a 2x2 input.
                        Another way to obtain the result of a transposed convolution is to apply an
                      equivalent – but much less efficient – direct convolution. The example described
                      so far could be tackled by convolving a 3x3 kernel over a 2x2 input padded
                      with a 2x2 border of zeros using unit strides (i.e.,i0 = 2,k0 =k,s0 = 1 and 
                      p0 = 2), as shown in Figure 4.1. Notably, the kernel’s  and  stride’s sizes remain
                      the same, but the input of the transposed convolution is now zero padded. 4
                        One way to understand  the logic behind zero padding is to consider the
                      connectivity pattern of the transposed convolution  and  use it to guide the design
                      of the equivalent convolution. for   example, the top left pixel of the input of the
                      direct convolution only contribute to the top left pixel of the output, the top
                      right pixel is only connected to the top right output pixel,  and  so on.
                        To maintain the same connectivity pattern in the equivalent convolution it is
                      necessary to zero pad the input in such a way that the first (top-left) application
                      of the kernel only touches the top-left pixel, i.e., the padding has to be equal to
                      the size of the kernel minus one.
                        Proceeding in the same fashion it is possible to determine similar observa-
                      tions for   the other elements of the image, giving rise to the following relationship:
                        3 Note that the transposed convolution does not guarantee to recover the input itself, as it
                      is not defined as the inverse of the convolution, but rather just returns a feature map that has
                      the same width  and  height.
                        4 Note that although equivalent to applying the transposed matrix, this visualization adds
                      a lot of zero multiplications in the for  m of zero padding. This is done here for   illustration
                      purposes, but it is inefficient,  and  software implementations will normally not perfor  m the
                      useless zero multiplications.

                      Relationship 8.A convolution described bys= 1,p= 0 and k
                          has an associated transposed convolution described byk0 =k,s0 =s
                           and p0 = kx1  and  its output size is

                                            <<FORMULA>>

                        Interestingly, this corresponds to a fully padded convolution with unit strides.


                      4.4 Zero padding, unit strides, transposed

                      Knowing that the transpose of a non-padded convolution is equivalent to con-
                      volving a zero padded input, it would be reasonable to suppose that the trans-
                      pose of a zero padded convolution is equivalent to convolving an input padded
                      withlesszeros.
                        It is indeed the case, as shown in Figure 4.2 for  i= 5,k= 4 and p= 2.
                        for  mally, the following relationship applies for   zero padded convolutions:

                          Relationship 9.A convolution described by s= 1,k and phas an
                          associated transposed convolution described by k0 =k,s0 =s and 
                          p0 =kp1 and  its output size is

                                           <<FORMULA>>

                      4.4.1 Half (same) padding, transposed
                      By applying the same inductive reasoning as befor  e, it is reasonable to expect
                      that the equivalent convolution of the transpose of a half padded convolution
                      is itself a half padded convolution, given that the output size of a half padded
                      convolution is the same as its input size. Thus the following relation applies:

                          Relationship 10.A convolution described byk= 2n+1; n2N,
                          s= 1 and p=bk=2c=nh as an associated transposed convolution
                          described byk0 =k,s0 =s and p0 =p and  its output size is

                                           <<FORMULA>>


                                           <<FIGURE>>

                        Figure 4.3 provides an example for   i= 5,k= 3 and  (therefor  e)p= 1.

                      4.4.2 Full padding, transposed
                      Knowing that the equivalent convolution of the transpose of a non-padded con-
                      volution involves full padding, it is unsurprising that the equivalent of the trans-
                      pose of a fully padded convolution is a non-padded convolution:

                                          <<FIGURE>>

                      Figure 4.1: The transpose of convolving a 3x3 kernel over a 4x4 input using
                      unit strides (i.e.,i= 4,k= 3,s= 1 and p= 0). It is equivalent to convolving
                      a 3x3 kernel over a 2x2 input padded with a 2x2 border of zeros using unit
                      strides (i.e.,i0 = 2,k0 =k,s0 = 1 and p0 = 2).

                                          <<FIGURE>>

                      Figure 4.2: The transpose of convolving a 4x4 kernel over a 5x5 input padded
                      with a 2x2 border of zeros using unit strides (i.e.,i= 5,k= 4,s= 1 and 
                      p= 2). It is equivalent to convolving a 4x4 kernel over a 6x6 input padded
                      with a 1x1 border of zeros using unit strides (i.e.,i0 = 6,k0 =k,s0 = 1 and 
                      p0 = 1).

                                           <<FIGURE>>

                      Figure 4.3: The transpose of convolving a 3x3 kernel over a 5x5 input using
                      half padding  and  unit strides (i.e.,i= 5,k= 3,s= 1 and p= 1). It is
                      equivalent to convolving a 3x3 kernel over a 5x5 input using half padding
                       and  unit strides (i.e.,i0 = 5,k0 =k,s0 = 1 and p0 = 1).



                       Relationship 11.A convolution described bys= 1,k and p= kx1 
                          has an associated transposed convolution described byk0 =k,s0 =s
                           and p0 = 0 and  its output size is

                                          <<FIGURE>>

                        Figure 4.4 provides an example for  i= 5,k= 3 and  (therefor  e)p= 2.


                      4.5 No zero padding, non-unit strides, transposed

                      Using the same kind of inductive logic as for   zero padded convolutions, one
                      might expect that the transpose of a convolution with s >1 involves an equiv-
                      alent convolution with s <1. As will be explained, this is a valid intuition,
                      which is why transposed convolutions are sometimes called fractionally strided
                      convolutions.
                        Figure 4.5 provides an example for  i= 5,k= 3 and s= 2which helps
                      understand  what fractional strides involve: zeros are inserted between input
                      units, which makes the kernel move around at a slower pace than with unit
                      strides. 5
                        for   the moment, it will be assumed that the convolution is non-padded
                      (p= 0)  and  that its input size i is such that  ixk  is a multiple ofs. In that
                      case, the following relationship holds:

                          Relationship 12.A convolution described byp= 0,k and s and 
                          whose input size is such that ixk is a multiple ofs, has an associated
                          transposed convolution described by~i0 ,k0 =k,s0 = 1 and p0 = kx1 ,
                          where~i0 is the size of the stretched input obtained by adding  sx1 
                          zeros between each input unit,  and  its output size is

                                            <<FORMULA>>

                      4.6 Zero padding, non-unit strides, transposed

                      When the convolution’s input sizeiis such thati+ 2pkis a multiple ofs,
                      the analysis can extended to the zero padded case by combining Relationship 9
                       and  Relationship 12:
                        5 Doing so is inefficient  and  real-world implementations avoid useless multiplications by
                      zero, but conceptually it is how the transpose of a strided convolution can be thought of.
 
                                          <<FIGURE>> 

                      Figure 4.4: The transpose of convolving a 3x3 kernel over a 5x5 input using
                      full padding  and  unit strides (i.e.,i= 5,k= 3,s= 1 and p= 2). It is equivalent
                      to convolving a 3x3 kernel over a77input using unit strides (i.e.,i0 = 7,
                      k0 =k,s0 = 1 and p0 = 0).

                                          <<FIGURE>>

                      Figure 4.5: The transpose of convolving a 3x3 kernel over a 5x5 input using
                       2x2 strides (i.e.,i= 5,k= 3,s= 2 and p= 0). It is equivalent to convolving
                      a 3x3 kernel over a 2x2 input (with1zero inserted between inputs) padded
                      with a 2x2 border of zeros using unit strides (i.e.,i0 = 2,~i0 = 3,k0 =k,s0 = 1
                       and p0 = 2).

                                        <<FIGURE>>

                      Figure 4.6: The transpose of convolving a 3x3 kernel over a 5x5 input padded
                      with a 1x1 border of zeros using 2x2 strides (i.e.,i= 5,k= 3,s= 2 and 
                      p= 1). It is equivalent to convolving a 3x3 kernel over a 3x3 input (with
                      1zero inserted between inputs) padded with a 1x1 border of zeros using unit
                      strides (i.e.,i0 = 3,~i0 = 5,k0 =k,s0 = 1 and p0 = 1).



                         Relationship 13.A convolution described byk,s and p and  whose
                          input sizeiis such tha ti+2pk is a multiple of s has an associated
                          transposed convolution described by~i0 ,k0 =k,s0 = 1 and p0 =
                          kp1, where ~i0 is the size of the stretched input obtained by
                          adding sx1 zeros between each input unit,  and  its output size is

                                          <<FORMULA>>


                                          <<FIGURE>>

                        Figure 4.6 provides an example for  i= 5,k= 3,s= 2 and p= 1.
                        The constraint on the size of the inputican be relaxed by introducing
                      another parametera2 f0;:::; sx1 gthat allows to distinguish between thes
                      different cases that all lead to the samei0 :

                          Relationship 14.A convolution described byk,s and phas an
                          associated transposed convolution described bya,~i0 ,k0 =k,s0 = 1
                           and p0 =kp1, where~i0 is the size of the stretched input obtained
                          by adding sx1 zeros between each input unit,  and a= (i+ 2pk)
                          modsrepresents the number of zeros added to the bottom  and  right
                          edges of the input,  and  its output size is

                                         <<FORMULA>>


                                         <<FIGURE>>

                        Figure 4.7 provides an example for  i= 6,k= 3,s= 2 and p= 1.

                                        <<FIGURE>>

                      Figure 4.7: The transpose of convolving a 3x3 kernel over a 6x6 input padded
                      with a 1x1 border of zeros using 2x2 strides (i.e.,i= 6,k= 3,s= 2 and 
                      p= 1). It is equivalent to convolving a 3x3 kernel over a 2x2 input (with
                      1zero inserted between inputs) padded with a 1x1 border of zeros (with an
                      additional border of size1added to the bottom  and  right edges) using unit
                      strides (i.e.,i0 = 3,~i0 = 5,a= 1,k0 =k,s0 = 1 and p0 = 1).


                                                 Chapter 5


                      Miscellaneous convolutions

                      5.1 Dilated convolutions

                      Readers familiar with the deep learning literature may have noticed the term
                      “dilated convolutions” (or “atrous convolutions”, from the French expressioncon-
                      volutions à trous) appear in recent papers. Here we attempt to provide an in-
                      tuitive underst and ing of dilated convolutions. for   a more in-depth description
                       and  to underst and  in what contexts they are applied, see Chenet al.(2014); Yu
                       and  Koltun (2015).
                        Dilated convolutions “inflate” the kernel by inserting spaces between the ker-
                      nel elements. The dilation “rate” is controlled by an additional hyperparameter
                      d. Implementations may vary, but there are usually dx1 spaces inserted between
                      kernel elements such thatd= 1corresponds to a regular convolution.
                        Dilated convolutions are used to cheaply increase the receptive field of output
                      units without increasing the kernel size, which is especially effective when multi-
                      ple dilated convolutions are stacked one after another. for   a concrete example,
                      see Oordet al.(2016), in which the proposed WaveNet model implements an
                      autoregressive generative model for   raw audio which uses dilated convolutions
                      to condition new audio frames on a large context of past audio frames.
                        To underst and  the relationship tying the dilation rated and  the output size
                      o, it is useful to think of the impact ofdon theeffective kernel size. A kernel
                      of sizekdilated by a factordhas an effective size

                                          <<FORMULA>>

                      This can be combined with Relationship 6 to for  m the following relationship for  
                      dilated convolutions:

                          Relationship 15.for any  i,k,p and s,  and  for   a dilation rated,

                                       <<FORMULA>>


                                        <<FIGURE>>
                      Figure 5.1: (Dilated convolution) Convolving a 3x3 kernel over a77input
                      with a dilation factor of 2 (i.e.,i= 7,k= 3,d= 2,s= 1 and p= 0).


                      Figure 5.1 provides an example for  i= 7,k= 3 and d= 2.



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<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>

                  A Survey of Model Compression and Acceleration for Deep Neural Networks

                 Yu Cheng, Duo Wang, Pan Zhou Member IEEE, and Tao Zhang Senior Member  IEEE

         Abstract—Deep convolutional neural networks (CNNs) have [2], [3]. It is also very time-consuming to train such a model
        recently achieved great success in many visual recognition tasks. to get reasonable performance. In architectures that rely only However, existing deep neural network models are computation- on fully-connected layers, the number of parameters can grow ally expensive and memory intensive, hindering their deployment
        in devices with low memory resources or in applications with to billions [4].

       strict latency requirements. Therefore, a natural thought is to   As larger neural networks with more layers and nodes
 
        without significantly decreasing the model performance. During becomes critical, especially for some real-time applications the past few years, tremendous progress has been made in such as online learning and incremental learning. In addi- this area. In this paper, we survey the recent advanced tech-
        niques for compacting and accelerating CNNs model developed. tion, recent years witnessed significant progress in virtual
        These techniques are roughly categorized into four schemes: reality, augmented reality, and smart wearable devices, cre-
        parameter pruning and sharing, low-rank factorization, trans- ating unprecedented opportunities for researchers to tackle
        ferred/compact convolutional filters, and knowledge distillation. fundamental challenges in deploying deep learning systems to Methods of parameter pruning and sharing will be described at portable devices with limited resources (e.g. memory, CPU, the beginning, after that the other techniques will be introduced.
        For each scheme, we provide insightful analysis regarding the energy, bandwidth). Efficient deep learning methods can have
        performance, related applications, advantages, and drawbacks significant impacts on distributed systems, embedded devices,
        etc. Then we will go through a few very recent additional and FPGA for Artificial Intelligence. For example, the ResNet-
        successful methods, for example, dynamic capacity networks and 50 [5] with 50 convolutional layers needs over 95MB memory stochastic depths networks. After that, we survey the evaluation for storage and over 3.8 billion floating number multiplications matrix, the main datasets used for evaluating the model per-
        formance and recent benchmarking efforts. Finally, we conclude when processing an image. After discarding some redundant
        this paper, discuss remaining challenges and possible directions weights, the network still works as usual but saves more than
        on this topic.                                   75% of parameters and 50% computational time. For devices
         Index Terms—Deep Learning, Convolutional Neural Networks, like cell phones and FPGAs with only several megabyte
        Model Compression and Acceleration,                  resources, how to compact the models used on them is also
                                                   important.
                                                     Achieving these goal calls for joint solutions from many
                                                     
                                                     I. INTRODUCTION                
                                                     
         disciplines, including but not limited to machine learning, op-
         In recent years, deep neural networks have recently received timization, computer architecture, data compression, indexing,
        lots of attention, been applied to different applications and and hardware design. In this paper, we review recent works
        achieved dramatic accuracy improvements in many tasks. on compressing and accelerating deep neural networks, which
        These works rely on deep networks with millions or even attracted a lot of attention from the deep learning community
        billions of parameters, and the availability of GPUs with and already achieved lots of progress in the past years.
        very high computation capability plays a key role in their   We classify these approaches into four categories: pa-
        success. For example, the work by Krizhevskyet al.[1] rameter pruning and sharing, low-rank factorization, trans-
        achieved breakthrough results in the 2012 ImageNet Challenge ferred/compact convolutional filters, and knowledge distil-
        using a network containing 60 million parameters with five lation. The parameter pruning and sharing based methods
        convolutional layers and three fully-connected layers. Usually, explore the redundancy in the model parameters and try to
        it takes two to three days to train the whole model on remove the redundant and uncritical ones. Low-rank factor-
        ImagetNet dataset with a NVIDIA K40 machine. Another ization based techniques use matrix/tensor decomposition to
        example is the top face verification results on the Labeled estimate the informative parameters of the deep CNNs. The
        Faces in the Wild (LFW) dataset were obtained with networks approaches based on transferred/compact convolutional filters
        containing hundreds of millions of parameters, using a mix design special structural convolutional filters to reduce the
        of convolutional, locally-connected, and fully-connected layers parameter space and save storage/computation. The knowledge
                                                   distillation methods learn a distilled model and train a more Yu Cheng is a Researcher from Microsoft AI & Research, One Microsoft
        Way, Redmond, WA 98052, USA.                         compact neural network to reproduce the output of a larger
         Duo Wang and Tao Zhang are with the Department of Automation, network.
        Tsinghua University, Beijing 100084, China.                     In Table I, we briefly summarize these four types of Pan Zhou is with the School of Electronic Information and Communi- methods. Generally, the parameter pruning & sharing, low- cations, Huazhong University of Science and Technology, Wuhan 430074,
        China.                                        rank factorization and knowledge distillation approaches can        IEEE SIGNAL PROCESSING MAGAZINE, SPECIAL ISSUE ON DEEP LEARNING FOR IMAGE UNDERSTANDING (ARXIV EXTENDED VERSION) 2


                                                TABLE I

                                            <<TABLE>>

        be used in DNN models with fully connected layers and
        convolutional layers, achieving comparable performances. On
        the other hand, methods using transferred/compact filters are
        designed for models with convolutional layers only. Low-rank
        factorization and transfered/compact filters based approaches
        provide an end-to-end pipeline and can be easily implemented
        in CPU/GPU environment, which is straightforward. while
        parameter pruning & sharing use different methods such as
        vector quantization, binary coding and sparse constraints to
        perform the task. Generally it will take several steps to achieve
        the goal.                                    
        
                                <<FIGURE>>
         Fig. 1. The three-stage compression method proposed in [10]: pruning, Regarding the training protocols, models based on param- quantization and encoding. The input is the original model and the output
       
        eter pruning/sharing low-rank factorization can be extracted is the compression model.
        from pre-trained ones or trained from scratch. While the
        transferred/compact filter and knowledge distillation models
        can only support train from scratch. These methods are inde- memory usage and float point operations with little loss in
        pendently designed and complement each other. For example, classification accuracy.
        transferred layers and parameter pruning & sharing can be   The method proposed in [10] quantized the link weights
        used together, and model quantization & binarization can be using weight sharing and then applied Huffman coding to the
        used together with low-rank approximations to achieve further quantized weights as well as the codebook to further reduce
        speedup. We will describe the details of each theme, their the rate. As shown in Figure 1, it started by learning the con-
        properties, strengths and drawbacks in the following sections. nectivity via normal network training, followed by pruning the
                                                   small-weight connections. Finally, the network was retrained
                                               to learn the final weights for the remaining sparse connections. 

              II. PARAMETER PRUNING AND SHARING 
                     
         This work achieved the state-of-art performance among allEarly works showed that network pruning is effective in parameter quantization based methods. It was shown in [11] reducing the network complexity and addressing the over- that Hessian weight could be used to measure the importancefitting problem [6]. After that researcher found pruning orig- of network parameters, and proposed to minimize Hessian-inally introduced to reduce the structure in neural networks weighted quantization errors in average for clustering networkand hence improve generalization, it has been widely studied parameters.to compress DNN models, trying to remove parameters which   In the extreme case of the 1-bit representation of eachare not crucial to the model performance. These techniques can weight, that is binary weight neural networks. There arebe further classified into three sub-categories: quantization and many works that directly train CNNs with binary weights, forbinarization, parameter sharing, and structural matrix.       instance, BinaryConnect [12], BinaryNet [13] and XNORNet-
                                                   works [14]. The main idea is to directly learn binary weights orA. Quantization and Binarization                    activation during the model training. The systematic study in
         Network quantization compresses the original network by [15] showed that networks trained with back propagation could
        reducing the number of bits required to represent each weight. be resilient to specific weight distortions, including binary
        Gonget al.[6] and Wu et al. [7] appliedk-means scalar weights.
        quantization to the parameter values. Vanhouckeet al.[8]   Drawbacks: the accuracy of the binary nets is significantly
        showed that 8-bit quantization of the parameters can result lowered when dealing with large CNNs such as GoogleNet.
        in significant speed-up with minimal loss of accuracy. The Another drawback of such binary nets is that existing bina-
        work in [9] used 16-bit fixed-point representation in stochastic rization schemes are based on simple matrix approximations
        rounding based CNN training, which significantly reduced and ignore the effect of binarization on the accuracy loss.        IEEE SIGNAL PROCESSING MAGAZINE, SPECIAL ISSUE ON DEEP LEARNING FOR IMAGE UNDERSTANDING (ARXIV EXTENDED VERSION) 3


         To address this issue, the work in [16] proposed a proximal connected layers, which is often the bottleneck in terms of
        Newton algorithm with diagonal Hessian approximation that memory consumption. These network layers use the nonlinear
        directly minimizes the loss with respect to the binary weights. transformsf(x;M) =(Mx), where()is an element-wise
        The work in [17] reduced the time on float point multiplication nonlinear operator,xis the input vector, andMis themn
        in the training stage by stochastically binarizing weights and matrix of parameters [29]. WhenMis a large general dense
        converting multiplications in the hidden state computation to matrix, the cost of storingmnparameters and computing
        significant changes.                              matrix-vector products inO(mn)time. Thus, an intuitive
                                                   way to prune parameters is to imposexas a parameterizedB. Pruning and Sharing                           structural matrix. Anmn matrix that can be described
         Network pruning and sharing has been used both to reduce using much fewer parameters thanmnis called a structured
        network complexity and to address the over-fitting issue. An matrix. Typically, the structure should not only reduce the
        early approach to pruning was the Biased Weight Decay memory cost, but also dramatically accelerate the inference
        [18]. The Optimal Brain Damage [19] and the Optimal Brain and training stage via fast matrix-vector multiplication and
        Surgeon [20] methods reduced the number of connections gradient computations.
        based on the Hessian of the loss function, and their work sug-   Following this direction, the work in [30], [31] proposed a
        gested that such pruning gave higher accuracy than magnitude- simple and efficient approach based on circulant projections,
                                                   while maintaining competitive error rates. Given a vectorr=based pruning such as the weight decay method. The training procedure of those methods followed the way training from   <<FORMULA>>, a circulant matrix R^2 R^dxd is defined
                                                   as: <<FORMULA>>
        scratch manner. A recent trend in this direction is to prune redundant, <<FORMULA>> non-informative weights in a pre-trained CNN model. For <<FORMULA>>
        example, Srinivas and Babu [21] explored the redundancy      <<FORMULA>> among neurons, and proposed a data-free pruning method to                       
        remove redundant neurons. Hanet al.[22] proposed to reduce                 <<FORMULA>>
        the total number of parameters and operations in the entire  thus the memory cost becomesO(d)instead of O(d^2) network. Chenet al.[23] proposed a HashedNets model that This circulant structure also enables the use of Fast Fourier used a low-cost hash function to group weights into hash Transform (FFT) to speed up the computation. Given ad-buckets for parameter sharing. The deep compression method dimensional vectorr, the above 1-layer circulant neural net-in [10] removed the redundant connections and quantized the work in Eq. 1 has time complexity ofO(dlogd).weights, and then used Huffman coding to encode the quan-   In [32], a novel Adaptive Fastfood transform was introducedtized weights. In [24], a simple regularization method based to reparameterize the matrix-vector multiplication of fully on soft weight-sharing was proposed, which included both connected layers. The Adaptive Fast food transform matrix quantization and pruning in one simple (re-)training procedure. R2Rnd was defined as:The above pruning schemes typically produce connections
        pruning in CNNs.                                              <<FORMULA>>            (2)
         There is also growing interest in training compact CNNs whereS,GandBare random diagonal matrices. 2
        with sparsity constraints. Those sparsity constraints are typ- <<FORMULA>> is a random permutation matrix, and H denotes
        ically introduced in the optimization problem asl0 orl1 - the Walsh-Hadamard matrix. Reparameterizing a fully con-
        norm regularizers. The work in [25] imposed group sparsity nected layer with d inputs and n outputs using the Adaptive
        constraint on the convolutional filters to achieve structured Fast food transform reduces the storage and the computational
        brain Damage, i.e., pruning entries of the convolution kernels costs from O(n^d) to O(n) and from O(n^d) to O(n*log(d)),
        in a group-wise fashion. In [26], a group-sparse regularizer respectively.
        on neurons was introduced during the training stage to learn   The work in [29] showed the effectiveness of the new
        compact CNNs with reduced filters. Wenet al.[27] added a notion of parsimony in the theory of structured matrices. Their
        structured sparsity regularizer on each layer to reduce trivial proposed method can be extended to various other structured
        filters, channels or even layers. In the filter-level pruning, all matrix classes, including block and multi-level Toeplitz-like
        the above works used l2-norm regularizers. The work in [28] [33] matrices related to multi-dimensional convolution [34].
        usedl1 -norm to select and prune unimportant filters.       Following this idea, [35] proposed a general structured effi-
         Drawbacks: there are some potential issues of the pruning cient linear layer for CNNs.
        and sharing. First, pruning with l1 or l2 regularization requires   Drawbacks: one problem of this kind of approaches is that
        more iterations to converge than general. In addition, all the structural constraint will hurt the performance since the
        pruning criteria require manual setup of sensitivity for layers, constraint might bring bias to the model. On the other hand,
        which demands fine-tuning of the parameters and could be how to find a proper structural matrix is difficult. There is no
        cumbersome for some applications.                   theoretical way to derive it out.

        C. Designing Structural Matrix                          
        
        III. LOW-RANK FACTORIZATION AND SPARSITY


         In architectures that contain fully-connected layers, it is   Convolution operations contribute the bulk of most com-
        critical to explore this redundancy of parameters in fully- putations in deep CNNs, thus reducing the convolution layer        IEEE SIGNAL PROCESSING MAGAZINE, SPECIAL ISSUE ON DEEP LEARNING FOR IMAGE UNDERSTANDING (ARXIV EXTENDED VERSION) 4


                                                                      TABLE II
                                                    COMPARISONS BETWEEN THE LOW -RANK MODELS AND THEIR BASELINES
                                                                   ON ILSVRC-2012.

                                                                    <<TABLE>>


                                       <<FIGURE>>

        Fig. 2. A typical framework of the low-rank regularization method. The left    
        is the original convolutional layer and the right is the low-rank constraint    
        convolutional layer with rank-K.                             
                                                      
        would improve the compression rate as well as the overall
        speedup. For the convolution kernels, it can be viewed as a
        4D tensor. Ideas based on tensor decomposition is derived by For instance, Mishaet al.[41] reduced the number of dynamic
        the intuition that there is a significant amount of redundancy parameters in deep models using the low-rank method. [42]
        in the 4D tensor, which is a particularly promising way to explored a low-rank matrix factorization of the final weight
        remove the redundancy. Regarding the fully-connected layer, layer in a DNN for acoustic modeling. In [3], Luet al.adopted
        it can be view as a 2D matrix and the low-rankness can also truncated SVD (singular value decomposition) to decompsite
        help.                                        the fully connected layer for designing compact multi-task
         It has been a long time for using low-rank filters to acceler- deep learning architectures.
        ate convolution, for example, high dimensional DCT (discrete   Drawbacks: low-rank approaches are straightforward for
        cosine transform) and wavelet systems using tensor products model compression and acceleration. The idea complements
        to be constructed from 1D DCT transform and 1D wavelets recent advances in deep learning, such as dropout, recti-
        respectively. Learning separable 1D filters was introduced fied units and maxout. However, the implementation is not
        by Rigamontiet al.[36], following the dictionary learning that easy since it involves decomposition operation, which
        idea. Regarding some simple DNN models, a few low-rank is computationally expensive. Another issue is that current
        approximation and clustering schemes for the convolutional methods perform low-rank approximation layer by layer, and
        kernels were proposed in [37]. They achieved 2speedup thus cannot perform global parameter compression, which
        for a single convolutional layer with 1% drop in classification is important as different layers hold different information.
        accuracy. The work in [38] proposed using different tensor Finally, factorization requires extensive model retraining to
        decomposition schemes, reporting a 4.5speedup with 1% achieve convergence when compared to the original model.
        drop in accuracy in text recognition.
         The low-rank approximation was done layer by layer. The   IV. T RANSFERRED /COMPACT CONVOLUTIONAL FILTERS
        parameters of one layer were fixed after it was done, and the   CNNs are parameter efficient due to exploring the trans-layers above were fine-tuned based on a reconstruction error lation invariant property of the representations to the input criterion. These are typical low-rank methods for compressing image, which is the key to the success of training very deep2D convolutional layers, which is described in Figure 2. Fol- models without severe over-fitting. Although a strong theory lowing this direction, Canonical Polyadic (CP) decomposition is currently missing, a large number of empirical evidenceof was proposed for the kernel tensors in [39]. Their work support the notion that both the translation invariant property used nonlinear least squares to compute the CP decomposition. and the convolutional weight sharing are important for good In [40], a new algorithm for computing the low-rank tensor predictive performance. The idea of using transferred convolu- decomposition for training low-rank constrained CNNs from tional filters to compress CNN models is motivated by recent scratch were proposed. It used Batch Normalization (BN) to works in [43], which introduced the equivariant group theory.transform the activation of the internal hidden units. In general, Letxbe an input,()be a network or layer and T() be the both the CP and the BN decomposition schemes in [40] (BN transform matrix. The concept of equivalence is defined as:Low-rank) can be used to train CNNs from scratch. However,
        there are few differences between them. For example, finding                <<FORMULA>>            (3) 
        the best low-rank approximation in CP decomposition is an ill-
        posed problem, and the best rank-K (K is the rank number) indicating that transforming the input x by the transform T()
        approximation may not exist sometimes. While for the BN and then passing it through the network or layer () should
        scheme, the decomposition always exists. We perform a simple give the same result as first mapping x through the network
        comparison of both methods shown in Table II. The actual and then transforming the representation. Note that in Eq.
        speedup and the compression rates are used to measure their (10), the transforms <<T()>> and <<T_0()>> are not necessarily the
        performances.                                  same as they operate on different objects. According to this
         As we mentioned before, the fully connected layers can theory, it is reasonable applying transform to layers or filters
        be viewed as a 2D matrix and thus the above mentioned () to compress the whole network models. From empirical
        methods can also be applied there. There are several classical observation, deep CNNs also benefit from using a large set of
        works on exploiting low-rankness in fully connected layers. convolutional filters by applying certain transformT()to a        IEEE SIGNAL PROCESSING MAGAZINE, SPECIAL ISSUE ON DEEP LEARNING FOR IMAGE UNDERSTANDING (ARXIV EXTENDED VERSION) 5


        small set of base filters since it acts as a regularizer for the                   TABLE III
        model.                                                  A SIMPLE COMPARISON OF DIFFERENT APPROACHES ON CIFAR-10 AND
         Following this direction, there are many recent reworks               
        proposed to build a convolutional layer from a set of base                        <<TABLE>>
        filters [43]–[46]. What they have in common is that the     
        transform T() lies in the family of functions that only operate      
        in the spatial domain of the convolutional filters. For example,      
        the work in [45] found that the lower convolution layers of     
        CNNs learned redundant filters to extract both positive and
        negative phase information of an input signal, and definedT()   Drawbacks: there are few issues to be addressed for ap-to be the simple negation function:                   proaches that apply transform constraints to convolutional fil-
                       
                       <<FORMULA>>             (4) 

        ters. First, these methods can achieve competitive performance x                 for wide/flat architectures (like VGGNet) but not thin/deepwhereWx is the basis convolutional filter andW is the filter x         ones (like GoogleNet, Residual Net). Secondly, the transferconsisting of the shifts whose activation is opposite to that assumptions sometimes are too strong to guide the learning,ofWx and selected after max-pooling operation. By doing making the results unstable in some cases.this, the work in [45] can easily achieve 2compression   Using a compact filter for convolution can directly reducerate on all the convolutional layers. It is also shown that the the computation cost. The key idea is to replace the loosenegation transform acts as a strong regularizer to improve and over-parametric filters with compact blocks to improve the classification accuracy. The intuition is that the learning the speed, which significantly accelerate CNNs on severalalgorithm with pair-wise positive-negative constraint can lead benchmarks. Decomposing33convolution into two11to useful convolutional filters instead of redundant ones.     convolutions was used in [48], which achieved significantIn [46], it was observed that magnitudes of the responses acceleration on object recognition. SqueezeNet [49] was pro-from convolutional kernels had a wide diversity of pattern posed to replace33convolution with11convolu-representations in the network, and it was not proper to discard tion, which created a compact neural network with about 50weaker signals with a single threshold. Thus a multi-bias non- fewer parameters and comparable accuracy when compared tolinearity activation function was proposed to generates more AlexNet.patterns in the feature space at low computational cost. The
        transformT()was define as:                           

                    <<FORMULA>>           (5) 

         V. KNOWLEDGE DISTILLATION   
         
         To the best of our knowledge, exploiting knowledge transfer
        where  were the multi-bias factors. The work in [47] con- (KT) to compress model was first proposed by Caruanaet
        side red a combination of rotation by a multiple of 90  and al.[50]. They trained a compressed/ensemble model of strong
        horizontal/vertical flipping with:                     classifiers with pseudo-data labeled, and reproduced the output
                                                   of the original larger network. But the work is limited to 
                                                   
                                                   <<FORMULA>>            (6) 

                                                   shallow models. The idea has been recently adopted in [51]
        whereWT was the transformation matrix which rotated the as knowledge distillation (KD) to compress deep and wide
        original filters with angle2 f90;180;270g. In [43], the networks into shallower ones, where the compressed model
        transform was generalized to any angle learned from data, and mimicked the function learned by the complex model. The
        was directly obtained from data. Both works [47] and [43] main idea of KD based approaches is to shift knowledge from
        can achieve good classification performance.             a large teacher model into a small one by learning the class
         The work in [44] definedT()as the set of translation distributions output via softmax.
        functions applied to 2D filters:                        The work in [52] introduced a KD compression framework,
                                                   which eased the training of deep networks by following a

                                                   <<FORMULA>>    (7) 

                                                   student-teacher paradigm, in which the student was penalized
        whereT(;x;y)denoted the translation of the first operand by according to a softened version of the teacher’s output. The
        (x;y)along its spatial dimensions, with proper zero padding framework compressed an ensemble of teacher networks into
        at borders to maintain the shape. The proposed framework a student network of similar depth. The student was trained
        can be used to 1) improve the classification accuracy as a to predict the output and the classification labels. Despite
        regularized version of maxout networks, and 2) to achieve its simplicity, KD demonstrates promising results in various
        parameter efficiency by flexibly varying their architectures to image classification tasks. The work in [53] aimed to address
        compress networks.                              the network compression problem by taking advantage of
         Table III briefly compares the performance of different depth neural networks. It proposed an approach to train thin
        methods with transferred convolutional filters, using VGGNet but deep networks, called FitNets, to compress wide and
        (16 layers) as the baseline model. The results are reported shallower (but still deep) networks. The method was extended
        on CIFAR-10 and CIFAR-100 datasets with Top-5 error. It is the idea to allow for thinner and deeper student models. In
        observed that they can achieve reduction in parameters with order to learn from the intermediate representations of teacher
        little or no drop in classification accuracy.               network, FitNet made the student mimic the full feature maps 


        of the teacher. However, such assumptions are too strict since layer with global average pooling [44], [62]. Network architec-
        the capacities of teacher and student may differ greatly.     ture such as GoogleNet or Network in Network, can achieve
         All the above approaches are validated on MNIST, CIFAR- state-of-the-art results on several benchmarks by adopting
        10, CIFAR-100, SVHN and AFLW benchmark datasets, and this idea. However, these architectures have not been fully
        experimental results show that these methods match or outper- optimized the utilization of the computing resources inside
        form the teacher’s performance, while requiring notably fewer the network. This problem was noted by Szegedyet al.[62]
        parameters and multiplications.                      and motivated them to increase the depth and width of the
         There are several extension along this direction of dis- network while keeping the computational budget constant.
        tillation knowledge. The work in [54] trained a parametric   The work in [63] targeted the Residual Network based
        student model to approximate a Monte Carlo teacher. The model with a spatially varying computation time, called
        proposed framework used online training, and used deep stochastic depth, which enabled the seemingly contradictory
        neural networks for the student model. Different from previous setup to train short networks and used deep networks at test
        works which represented the knowledge using the soften label time. It started with very deep networks, while during training,
        probabilities, [55] represented the knowledge by using the for each mini-batch, randomly dropped a subset of layers
        neurons in the higher hidden layer, which preserved as much and bypassed them with the identity function. Following this
        information as the label probabilities, but are more compact. direction, thew work in [64] proposed a pyramidal residual
        The work in [56] accelerated the experimentation process by networks with stochastic depth. In [65], Wuet al.proposed
        instantaneously transferring the knowledge from a previous an approach that learns to dynamically choose which layers
        network to each new deeper or wider network. The techniques of a deep network to execute during inference so as to best
        are based on the concept of function-preserving transfor- reduce total computation. Veitet al.exploited convolutional
        mations between neural network specifications. Zagoruyko networks with adaptive inference graphs to adaptively define
        et al.[57] proposed Attention Transfer (AT) to relax the their network topology conditioned on the input image [66].
        assumption of FitNet. They transferred the attention maps that   Other approaches to reduce the convolutional overheads in-are summaries of the full activations.                  clude using FFT based convolutions [67] and fast convolutionDrawbacks: KD-based approaches can make deeper models using the Winograd algorithm [68]. Zhaiet al.[69] proposed athinner and help significantly reduce the computational cost. strategy call stochastic spatial sampling pooling, which speed-However, there are a few disadvantages. One of those is that up the pooling operations by a more general stochastic version.KD can only be applied to classification tasks with softmax Saeedanet al.presented a novel pooling layer for convolu-loss function, which hinders its usage. Another drawback is tional neural networks termed detail-preserving pooling (DPP),the model assumptions sometimes are too strict to make the based on the idea of inverse bilateral filters [70]. Those worksperformance competitive with other type of approaches.     only aim to speed up the computation but not reduce the
                                                   memory storage.
                                                   
                                                   VI. OTHER TYPES OF APPROACHES

         We first summarize the works utilizing attention-based
        methods. Note that attention-based mechanism [58] can reduce    
        
                                                                                        VII. BENCHMARKS , EVALUATION AND DATABASES
        computations significantly by learning to selectively focus or   In the past five years the deep learning community had“attend” to a few, task-relevant input regions. The work in made great efforts in benchmark models. One of the most[59] introduced the dynamic capacity network (DCN) that well-known model used in compression and acceleration forcombined two types of modules: the small sub-networks with CNNs is Alexnet [1], which has been occasionally usedlow capacity, and the large ones with high capacity. The low- for assessing the performance of compression. Other popularcapacity sub-networks were active on the whole input to first standard models include LeNets [71], All-CNN-nets [72] andfind the task-relevant areas, and then the attention mechanism many others. LeNet-300-100 is a fully connected networkwas used to direct the high-capacity sub-networks to focus on with two hidden layers, with 300 and 100 neurons each.the task-relevant regions. By dong this, the size of the CNNs LeNet-5 is a convolutional network that has two convolutionalmodel has been significantly reduced.                  layers and two fully connected layers. Recently, more andFollowing this direction, the work in [60] introduced the more state-of-the-art architectures are used as baseline modelsconditional computation idea, which only computes the gra- in many works, including network in networks (NIN) [73],dient for some important neurons. It proposed a sparsely- VGG nets [74] and residual networks (ResNet) [75]. Table IVgated mixture-of-experts Layer (MoE). The MoE module summarizes the baseline models commonly used in severalconsisted of a number of experts, each a simple feed-forward typical compression methods.neural network, and a trainable gating network that selected
        a sparse combination of the experts to process each input. In   The standard criteria to measure the quality of model
        [61], dynamic deep neural networks (D2NN) were introduced, compression and acceleration are the compression and the
        which were a type of feed-forward deep neural network that speedup rates. Assume thatais the number of the parameters
        selected and executed a subset of D2NN neurons based on the in the original model Manda  is that of the compressed
        input.                                        model M , then the compression rate (M;M  ) of M over
         There have been other attempts to reduce the number of Mis                     aparameters of neural networks by replacing the fully connected                (M;M  ) =  :            (8)a        IEEE SIGNAL PROCESSING MAGAZINE, SPECIAL ISSUE ON DEEP LEARNING FOR IMAGE UNDERSTANDING (ARXIV EXTENDED VERSION) 7


                          TABLE IV                       or low rank factorization based methods. If you need
           SUMMARIZATION OF BASELINE MODELS USED IN DIFFERENT         end-to-end solutions for your problem, the low rank REPRESENTATIVE WORKS OF NETWORK COMPRESSION .          and transferred convolutional filters approaches could be
                                                                        considered.
                                                                         For applications in some specific domains, methods with low-rank factorization [40]           human prior (like the transferred convolutional filters, Network in network [73]      low-rank factorization [40]
                      <<TABLE>>                                          structural matrix) sometimes have benefits. For example,
                                                                           when doing medical images classification, transferred Residual networks [75]  compact filters [49], stochastic depth [63]       convolutional filters could work well as medical images parameter sharing [24]
                                                                           (like organ) do have the rotation transformation property.
                                                                            Usually the approaches of pruning & sharing could give parameter pruning [20], [22]          reasonable compression rate while not hurt the accuracy.
                                                       Thus for applications which requires stable model accu-
        Another widely used measurement is the index space saving     racy, it is better to utilize pruning & sharing.
        defined in several papers [30], [35] as                    If your problem involves small/medium size datasets, you
                                                       can try the knowledge distillation approaches. The com-aa
                     <<FORMULA>>           (9)     pressed student model can take the benefit of transferring a                    knowledge from teacher model, making it robust datasets
        where a and a are the number of the dimension of the index     which are not large.
        space in the original model and that of the compressed model,    As we mentioned before, techniques of the four groups
        respectively.                                      are orthogonal. It is reasonable to combine two or three
         Similarly, given the running timesofMands ofM ,     of them to maximize the performance. For some spe-
        the speedup rate <<FORMULA>> is defined as:                  cific applications, like object detection, which requires
                                 s                     both convolutional and fully connected layers, you can
                    <<FORMULA>>            (10)                    
                                                            compress the convolutional layers with low rank based
        Most work used the average training time per epoch to measure     method and the fully connected layers with a pruning
        the running time, while in [30], [35], the average testing time     technique.
        was used. Generally, the compression rate and speedup rate B. Technique Challengesare highly correlated, as smaller models often results in faster
        computation for both the training and the testing stages.       Techniques for deep model compression and acceleration
         Good compression methods are expected to achieve almost are still in the early stage and the following challenges still
        the same performance as the original model with much smaller need to be addressed.
        parameters and less computational time. However, for different    Most of the current state-of-the-art approaches are built
        applications with different CNN designs, the relation between     on well-designed CNN models, which have limited free-
        parameter size and computational time may be different.     dom to change the configuration (e.g., network structural,
        For example, it is observed that for deep CNNs with fully     hyper-parameters). To handle more complicated tasks,
        connected layers, most of the parameters are in the fully     it should provide more plausible ways to configure the
        connected layers; while for image classification tasks, float     compressed models.
        point operations are mainly in the first few convolutional layers    Pruning is an effective way to compress and acceler-
        since each filter is convolved with the whole image, which is     ate CNNs. The current pruning techniques are mostly
        usually very large at the beginning. Thus compression and     designed to eliminate connections between neurons. On
        acceleration of the network should focus on different type of     the other hand, pruning channel can directly reduce the
        layers for different applications.                         feature map width and shrink the model into a thinner
                                                       one. It is efficient but also challenging because removing
               VIII. D ISCUSSION AND CHALLENGES            channels might dramatically change the input of the
                                                       following layer.In this paper, we summarized recent efforts on compressing
        and accelerating deep neural networks (DNNs). Here we dis-    As we mentioned before, methods of structural matrix
                                                       and transferred convolutional filters impose prior humancuss more details about how to choose different compression     knowledge to the model, which could significantly affectapproaches, and possible challenges/solutions on this area.       the performance and stability. It is critical to investigate
                                                       how to control the impact of those prior knowledge.A. General Suggestions                              The methods of knowledge distillation provide many ben-
         There is no golden rule to measure which approach is the     efits such as directly accelerating model without special
        best. How to choose the proper method is really depending     hardware or implementations. It is still worthy developing
        on the applications and requirements. Here are some general     KD-based approaches and exploring how to improve their
        guidance we can provide:                             performances.
          If the applications need compacted models from pre-    Hardware constraints in various of small platforms (e.g.,
           trained models, you can choose either pruning & sharing     mobile, robotic, self-driving car) are still a major problem        IEEE SIGNAL PROCESSING MAGAZINE, SPECIAL ISSUE ON DEEP LEARNING FOR IMAGE UNDERSTANDING (ARXIV EXTENDED VERSION) 8


           to hinder the extension of deep CNNs. How to make full see more work for applications with larger deep nets (e.g.,
           use of the limited computational source and how to design video and image frames [88], [89]).
           special compression methods for such platforms are still
           challenges that need to be addressed.                         IX. ACKNOWLEDGMENTS
          Despite the great achievements of these compression ap-
           proaches, the black box mechanism is still the key barrier   The authors would like to thank the reviewers and broader
           to the adoption. Exploring the knowledge interpret-ability community for their feedback on this survey. In particular,
           is still an important problem.                    we would like to thank Hong Zhao from the Department of
                                                   Automation of Tsinghua University for her help on modifying
        C. Possible Solutions                             the paper. This research is supported by National Science
                                                   Foundation of China with Grant number 61401169.To solve the hyper-parameters configuration problem, we
        can rely on the recent learning-to-learn strategies [76], [77].
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          [86]M. Sandler, A. Howard, M. Zhu, A. Zhmoginov, and L.-C. Chen,                   Tao Zhang (taozhang@mail.tsinghua.edu.cn) ob-
              “Mobilenetv2: Inverted residuals and linear bottlenecks,” inThe IEEE                   tained his B.S., M.S., and Ph.D. degrees from Ts-
              Conference on Computer Vision and Pattern Recognition (CVPR), June                   inghua University, Beijing, China, in 1993, 1995,
              2018.                                                                and 1999, respectively, and another Ph.D. degree
          [87]J. Huang, V. Rathod, C. Sun, M. Zhu, A. Korattikara, A. Fathi, I. Fischer,                   from Saga University, Saga, Japan, in 2002, all in
              Z. Wojna, Y. Song, S. Guadarrama, and K. Murphy, “Speed/accuracy                   control engineering. He is currently a Professor with
              trade-offs for modern convolutional object detectors,” in2017 IEEE                   the Department of Automation, Tsinghua University.
              Conference on Computer Vision and Pattern Recognition, CVPR 2017,                   He serves the Associate Dean, School of Information
              Honolulu, HI, USA, July 21-26, 2017, 2017, pp. 3296–3297.                          Science and Technology and Head of the Department
          [88]Y. Cheng, Q. Fan, S. Pankanti, and A. Choudhary, “Temporal sequence                   of Automation. His current research interests include
              modeling for video event detection,” in The IEEE Conference on                   artificial intelligence, robotics, image processing,
              Computer Vision and Pattern Recognition (CVPR), June 2014.        control theory, and control of spacecraft.

<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>

            Analysis and Design of Echo State Networks

            Mustafa C. Ozturk
            can@cnel.ufl.edu

            Dongming Xu
            dmxu@cnel.ufl.edu

            Jose C. Principe
            principe@cnel.ufl.edu

            Computational NeuroEngineering Laboratory, Department of Electrical and
            Computer Engineering, University of Florida, Gainesville, FL 32611, U.S.A.


            The design of echo state network (ESN) parameters relies on the selec-
            tion of the maximum eigenvalue of the linearized system around zero
            (spectral radius). However, this procedure does not quantify in a sys-
            tematic manner the performance of the ESN in terms of approximation
            error. This article presents a functional space approximation framework
            to better understand the operation of ESNs and proposes an information-
            theoretic metric, the average entropy of echo states, to assess the richness
            of the ESN dynamics. Furthermore, it provides an interpretation of the
            ESN dynamics rooted in system theory as families of coupled linearized
            systems whose poles move according to the input signal dynamics. With
            this interpretation, a design methodology for functional approximation
            is put forward where ESNs are designed with uniform pole distributions
            covering the frequency spectrum to abide by the richness metric, irre-
            spective of the spectral radius. A single bias parameter at the ESN input,
            adapted with the modeling error, configures the ESN spectral radius to
            the input-output joint space. Function approximation examples compare
            the proposed design methodology versus the conventional design.


            1 Introduction

            Dynamic computational models require the ability to store and access the
            time history of their inputs and outputs. The most common dynamic neural
            architecture is the time-delay neural network (TDNN) that couples delay
            lines with a nonlinear static architecture where all the parameters (weights)
            are adapted with the backpropagation algorithm. The conventional delay
            line utilizes ideal delay operators, but delay lines with local first-order re-
            cursive filters have been proposed by Werbos (1992) and extensively stud-
            ied in the gamma model (de Vries, 1991; Principe, de Vries, & de Oliviera,
            1993). Chains of first-order integrators are interesting because they effec-
            tively decrease the number of delays necessary to create time embeddings


           (Principe, 2001). Recurrent neural networks (RNNs) implement a differ-
           ent type of embedding that is largely unexplored. RNNs are perhaps the
           most biologically plausible of the artificial neural network (ANN) models
           (Anderson, Silverstein, Ritz, & Jones, 1977; Hopfield, 1984; Elman, 1990),
           but are not well understood theoretically (Siegelmann & Sontag, 1991;
           Siegelmann, 1993; Kremer, 1995). One of the main practical problems with
           RNNs is the difficulty to adapt the system weights. Various algorithms,
           such as backpropagation through time (Werbos, 1990) and real-time recur-
           rent learning (Williams & Zipser, 1989), have been proposed to train RNNs;
           however, these algorithms suffer from computational complexity, resulting
           in slow training, complex performance surfaces, the possibility of instabil-
           ity, and the decay of gradients through the topology and time (Haykin,
           1998). The problem of decaying gradients has been addressed with spe-
           cial processing elements (PEs) (Hochreiter & Schmidhuber, 1997). Alter-
           native second-order training methods based on extended Kalman filtering
           (Singhal & Wu, 1989; Puskorius & Feldkamp, 1994; Feldkamp, Prokhorov,
           Eagen, & Yuan, 1998) and the multistreaming training approach (Feldkamp
           et al., 1998) provide more reliable performance and have enabled practical
           applications in identification and control of dynamical systems (Kechri-
           otis, Zervas, & Monolakos, 1994; Puskorius & Feldkamp, 1994; Delgado,
           Kambhampati, & Warwick, 1995).
             Recently,twonewrecurrentnetworktopologieshavebeenproposed:the
           echo state network (ESN) by Jaeger (2001, 2002a; Jaeger & Hass, 2004) and
           the liquid state machine (LSM) by Maass (Maass, Natschlager, & Markram,¨
           2002). ESNs possess a highly interconnected and recurrent topology of
           nonlinear PEs that constitutes a “reservoir of rich dynamics” (Jaeger, 2001)
           and contain information about the history of input and output patterns.
           The outputs of these internal PEs (echo states) are fed to a memoryless but
           adaptive readout network (generally linear) that produces the network out-
           put. The interesting property of ESN is that only the memoryless readout is
           trained, whereas the recurrent topology has fixed connection weights. This
           reduces the complexity of RNN training to simple linear regression while
           preserving a recurrent topology, but obviously places important constraints
           in the overall architecture that have not yet been fully studied. Similar ideas
           have been explored independently by Maass and formalized in the LSM
           architecture. LSMs, although formulated quite generally, are mostly im-
           plemented as neural microcircuits of spiking neurons (Maass et al., 2002),
           whereas ESNs are dynamical ANN models. Both attempt to model biolog-
           ical information processing using similar principles. We focus on the ESN
           formulation in this letter.

             The echo state condition is defined in terms of the spectral radius (the
           largest among the absolute values of the eigenvalues of a matrix, denoted
           by·) of the reservoir’s weight matrix (W<1). This condition states
           that the dynamics of the ESN is uniquely controlled by the input, and the
           effect of the initial states vanishes. The current design of ESN parameters           
           relies on the selection of spectral radius. However, there are many possible
           weight matrices with the same spectral radius, and unfortunately they do
           not all perform at the same level of mean square error (MSE) for functional
           approximation. A similar problem exists in the design of the LSM. LSMs
           have been shown to possess universal approximation given the separation
           property (SP) for the liquid (reservoir in ESNs) and the approximation
           property (AP) for the readout (Maass et al., 2002). SP is quantified by a
           kernel-quality measure proposed in Maass, Legenstein, and Bertschinger
           (2005) that is based on the rank of a matrix formed by the system states
           corresponding to different input signals. The kernel quality is a measure
           for the complexity and diversity of nonlinear operations carried out by the
           liquid on its input stream in order to boost the classification power of a
           subsequent linear decision hyperplane (Maass et al., 2005). A variation of
           SP has been proposed in Bertschinger and Natschlager (2004), and it has¨
           been argued that complex calculations can be best carried out by networks
           on the boundary between ordered and chaotic dynamics.

           In this letter,we are interested in studying the ESN for functional approx-
           imation (filters that map input function su(·) of time on output function sy(·)
           of time). We see two major shortcomings with the current ESN approach
           that uses echo state condition as a design principle. First, the impact of fixed
           reservoir parameters for function approximation means that the informa-
           tion about the desired response is conveyed only to the output projection.
           This is not optimal, and strategies to select different reservoirs for different
           applications have not been devised. Second, imposing a constraint only on
           the spectral radius is a weak condition to properly set the parameters of
           the reservoir, as experiments show (different randomizations with the same
           spectral radius perform differently for the same problem; see Figure 2).
             This letter aims to address these two problems by proposing a frame-
           work, a metric, and a design principle for ESNs. The framework is a signal
           processing interpretation of basis and projections in functional spaces to
           describe and understand the ESN architecture. According to this interpre-
           tation, the ESN states implement a set of basis functionals (representation
           space) constructed dynamically by the input, while the readout simply
           projects the desired response onto this representation space. The metric
           to describe the richness of the ESN dynamics is an information-theoretic
           quantity, the average state entropy (ASE). Entropy measures the amount of
           information contained in a given random variable (Shannon, 1948). Here,
           the random variable is the instantaneous echo state from which the en-
           tropy for the overall state (vector) is estimated. The probability density
           function (pdf) in a differential geometric framework should be thought of
           as a volume form; that is, in our case, the pdf of the state vector describes
           the metric of the state space manifold (Amari, 1990). Moreover, Cox (1946)
           established information as a coordinate free metric in the state manifold.
           Therefore, entropy becomes a global descriptor of information that quanti-
           fies the volume of the manifold defined by the random variable. Due to the
           time dependency of the states, the state entropy averaged over time (ASE)
           is an appropriate estimate of the volume of the state manifold.
             The design principle specifies that one should consider independently
           thecorrelationamongthebasisandthespectralradius.In the absence of any
           information about the desired response, the ESN states should be designed
           with the highest ASE, independent of the spectral radius. We interpret the
           ESN dynamics as a combination of time-varying linear systems obtained
           from the linearization of the ESN nonlinear PE in a small, local neighbor-
           hood of the current state. The design principle means that the poles of the
           linearized ESN reservoir should have uniform pole distributions to gener-
           ate echo states with the most diverse pole locations (which correspond to
           the uniformity of time constants). Effectively, this will create the least cor-
           related bases for a given spectral radius, which corresponds to the largest
           volume spanned by the basis set. When the designer has no other informa-
           tion about the desired response to set the basis, this principle distributes
           the system’s degrees of freedom uniformly in space. It approximates for
           ESNs the well-known property of orthogonal basis. The unresolved issue
           that ASE does not quantify is how to set the spectral radius, which depends
           again on the desired mapping. The concept of memory depth as explained
           in Principe et al. (1993) and Jaeger (2002a) is helpful in understanding the
           issues associated with the spectral radius. The correlation time of the de-
           sired response (as estimated by the first zero of the autocorrelation function)
           gives an indication of the type of spectral radius required (long correlation
           time requires high spectral radius). Alternatively, a simple adaptive bias is
           added at the ESN input to control the spectral radius integrating the infor-
           mation from the input-output joint space in the ESN bases. For sigmoidal
           PEs, the bias adjusts the operating points of the reservoir PEs, which has
           the net effect of adjusting the volume of the state manifold as required to
           approximate the desired response with a small error. This letter shows that
           ESNs designed with this strategy obtain systematically better results in a
           set of experiments when compared with the conventional ESN design.


           2 Analysis of Echo State Networks

              2.1 Echo States as Bases and Projections.Let us consider the ar-
           chitecture and recursive update equation of a typical ESN more closely.
           Consider the recurrent discrete-time neural network given in Figure 1
           with M input units, N internal PEs, and L output units. The value of
           the input unit at time n is <<u(n)=[u1 (n),u2 (n),...,uM (n)]^T>> , of internal
           units are <<x(n)=[x1 (n),x2 (n),...,xN (n)]^T>> , and of output units are <<y(n)=
           [y1 (n),y2 (n),...,yL (n)]^T>> . The connection weights are given in anN×M
           weight matrixWin =(win ) for connections between the input and the inter- ij 
           nalPEs,in an N×N matrix W=(wij ) for connections between the internal
           PEs, in an L×N matrix <<W_out =(w_out)>> for connections from PEs to the ij 
          Input Layer Dynamical Reservoir Read-out

                            <<FIGURE>>

           Figure 1: An echo state network (ESN). ESN is composed of two parts: a fixed-
           weight (W<1) recurrent network and a linear readout. The recurrent net-
           work is a reservoir of highly interconnected dynamical components, states of
           which are called echo states. The memoryless linear readout is trained to pro-
           duce the output.


           output units, and in an N× L matrix <<FORMULA>> for the connections ij that project back from the output to the internal PEs (Jaeger, 2001). The
           activation of the internal PEs (echo state) is updated according to

                             <<FORMULA>>,             (2.1)

           where f=(f1 ,f2 ,...,fN ) are the internal PEs’ activation functions.Here, all
          i ’s are hyperbolic tangent functions ( ex −  ). The output from the readout ex +e−x
           network is computed according to

               <<y(n+1)=f_out (W_out x(n+1))>>,                           (2.2)

           where <<f_out =(f_out ,f_out ,...,f_out )>> are the output unit’s nonlinear functions <<FORMULA>> (Jaeger, 2001, 2002a). 
           Generally, the readout is linear so f_out is identity.
             ESNs resemble the RNN architecture proposed in Puskorius and
           Feldkamp (1996) and also used by Sanchez (2004) in brain-machine
           interfaces. The critical difference is the dimensionality of the hidden re-
           current PE layer and the adaptation of the recurrent weights. We submit
           that the ideas of approximation theory in functional spaces (bases and pro-
           jections), so useful in adaptive signal processing (Principe, 2001), should
           be utilized to understand the ESN architecture. Let h(u(t)) be a real-valued
           function of a real-valued vector

              <<u(t)=[u1 (t),u2 (t),...,uM (t)] T>>.

           In functional approximation, the goal is to estimate the behavior ofh(u(t))
           as a combination of simpler functions ϕi (t), called the basis functionals,
           such that its approximant,hˆ(u(t)), is given by

                   <<FORMULA>>.

           Here,ai ’s are the projections ofh(u(t)) onto each basis function. One of
           the central questions in practical functional approximation is how to choose
           the set of bases to approximate a given desired signal. In signal processing,
           thechoicenormallygoesforacompletesetoforthogonalbasis,independent
           of the input. When the basis set is complete and can be made as large
           as required, fixed bases work wonders (e.g., Fourier decompositions). In
           neural computing, the basic idea is to derive the set of bases from the
           input signal through a multilayered architecture. For instance, consider a
           single hidden layer TDNN with NPEs and a linear output. The hidden-
           layer PE outputs can be considered a set of nonorthogonal basis functionals
           dependent on the input,

                    <<FORMULA>>

           bij ’s are the input layer weights, andgis the PE nonlinearity. The approxi-
           mation produced by the TDNN is then

                    <<FORMULA>>,                                (2.3)

           whereai ’s are the weights of the output layer. Notice that thebij ’s adapt
           the bases and theai ’s adapt the projection in the projection space. Here the
           goal is to restrict the number of bases (number of hidden layer PEs) because
           their number is coupled with the number of parameters to adapt, which
           has an impact on generalization and training set size, for example. Usually,
           since all of the parameters of the network are adapted, the best basis in the
           joint (input and desired signals) space as well as the best projection can be
           achieved and represents the optimal solution. The output of the TDNN is
           a linear combination of its internal representations, but to achieve a basis
           set (even if nonorthogonal), linear independence among theϕi (u(t))’s must
           be enforced. Ito, Shah and Pon, and others have shown that this is indeed
           the case (Ito, 1996; Shah & Poon, 1999), but a thorough discussion is outside
           the scope of this article.

             The ESN (and the RNN) architecture can also be studied in this frame-
           work. The states of equation 2.1 correspond to the basis set, which are
           recursively computed from the input, output, and previous states through
           Win ,W,andWback . Notice, however, that none of these weight matrices is
           adapted, that is, the functional bases in the ESN are uniquely defined by the
           input and the initial selection of weights. In a sense, ESNs are trading the
           adaptive connections in the RNN hidden layer by a brute force approach
           of creating fixed diversified dynamics in the hidden layer.
             For an ESN with a linear readout network, the output equation (y(n+
           1)=Wout x(n+1)) has the same form of equation 2.3, where theϕi ’s and
           ai ’s are replaced by the echo states and the readout weights, respectively.
           The readout weights are adapted in the training data, which means that the
           ESN is able to find the optimal projection in the projection space, just like
           the RNN or the TDNN.

             A similar perspective of basis and projections for information processing
           in biological networks has been proposed by Pouget and Sejnowski (1997).
           They explored the possibility that the response of neurons in parietal cortex
           serves as basis functions for the transformations from the sensory input
           to the motor responses. They proposed that “the role of spatial represen-
           tations is to code the sensory inputs and posture signals in a format that
           simplifies subsequent computation, particularly in the generation of motor
           commands”.

             The central issue in ESN design is exactly the nonadaptive nature of
           the basis set. Parameter sets in the reservoir that provide linearly inde-
           pendent states and possess a given spectral radius may define drastically
           different projection spaces because the correlation among the bases is not
           constrained. A simple experiment was designed to demonstrate that the se-
           lection of the ESN parameters by constraining the spectral radius is not the
           most suitable for function approximation. Consider a 100-unit ESN where
           the input signal is sin(2πn/10π). Mimicking Jaeger (2001), the goal is to let
           the ESN generate the seventh power of the input signal. Different realiza-
           tions of a randomly connected 100-unit ESN were constructed where the
           entries ofWare set to 0.4,−0.4, and 0 with probabilities of 0.025, 0.025,
           and 0.95, respectively. This corresponds to a spectral radius of 0.88. Input
           weights are set to+1or,−1 with equal probabilities, andWback is set to
           zero. Input is applied for 300 time steps, and the echo states are calculated
           using equation 2.1. The next step is to train the linear readout. One method

                                      <<FIGURE>>

           Figure 2: Performances of ESNs for different realizations ofWwith the same
           weight distribution. The weight values are set to 0.4,−0.4, and 0 with proba-
           bilities of 0.025, 0.025, and 0.95. All realizations have the same spectral radius
           of 0.88. In the 50 realizations, MSEs vary from 5.9×10 −9 to 8.9×10 −5 . Results
           show that for each set of random weights that provide the same spectral ra-
           dius, the correlation or degree of redundancy among the bases will change, and
           different performances are encountered in practice.


           to determine the optimal output weight matrix,Wout , in the mean square
           error (MSE) sense (where MSE is defined by <<FORMULA>>) is to use 2 the Wiener solution given by Haykin (2001):

                                        <<FORMULA>>

           Here,E[.] denotes the expected value operator, andddenotes the desired
           signal. Figure 2 depicts the MSE values for 50 different realizations of
           the ESNs. As observed, even though each ESN has the same sparseness
           and spectral radius, the MSE values obtained vary greatly among differ-
           ent realizations. The minimum MSE value obtained among the 50 realiza-
           tions is 5.9x10 −9 , whereas the maximum MSE is 8.9x10 −5 . This experiment    
           demonstrates that a design strategy that is based solely on the spectral
           radius is not sufficient to specify the system architecture for function ap-
           proximation. This shows that for each set of random weights that provide
           thesamespectralradius,thecorrelationordegreeofredundancyamongthe
           bases will change, and different performances are encountered in practice.

             2.2 ESN Dynamics as a Combination of Linear Systems.
             
           It is well known that the dynamics of a nonlinear system can be approximated by
           that of a linear system in a small neighborhood of an equilibrium point
           (Kuznetsov, Kuznetsov, & Marsden, 1998). Here, we perform the analysis
           with hyperbolic tangent nonlinearities and approximate the ESN dynam-
           ics by the dynamics of the linearized system in the neighborhood of the
           current system state. Hence, when the system operating point varies over
           time, the linear system approximating the ESN dynamics changes. We are
           particularly interested in the movement of the poles of the linearized ESN.
           Consider the update equation for the ESN without output feedback given
           by

               <<x(n+1)=f(Win u(n+1)+Wx(n))>>.

           Linearizing the system around the current statex(n), one obtains the
           Jacobian matrix, <<J(n+1)>>, defined by
                
                              <<FORMULA>>

           Here,net i(n) is the ith entry of the vector <<(W_in u(n+1)+Wx(n))>>, and w_ij
           denotes the (i,j)th entry of W. The poles of the linearized system at time
           n+1 are given by the eigenvalues of the Jacobian matrixJ(n+1). 1 As the
           amplitude of each PE changes, the local slope changes, and so the poles of
           A. The transfer function of a linear system <<x(n+1)=Ax(n)+Bu(n)>> is <<X(z) =(zI−U(z)A)−1>> 
           Adjoint <<(zI−A)>>. The poles of the transfer function can be obtained by solving <<det(zI−A)=0>>.
           The solution corresponds to the eigenvalues of A.     


           the linearized system are time varying, although the parameters of ESN are
           fixed. In order to visualize the movement of the poles, consider an ESN with
           100 states. The entries of the internal weight matrix are chosen to be 0,
           0.4 and −0.4 with probabilities 0.9, 0.05, and 0.05.W is scaled such that a
           spectral radius of 0.95 is obtained. Input weights are set to +1 or −1 with
           equal probabilities. A sinusoidal signal with a period of 100 is fed to the
           system, and the echo states are computed according to equation 2.1. Then
           the Jacobian matrix and the eigenvalues are calculated using equation 2.5.
           Figure 3 shows the pole tracks of the linearized ESN for different input
           values. A single ESN with fixed parameters implements a combination of
           many linear systems with varying pole locations, hence many different
           time constants that modulate the richness of the reservoir of dynamics as a
           function of input amplitude. Higher-amplitude portions of the signal tend
           to saturate the nonlinear function and cause the poles to shrink toward
           the origin of thez-plane (decreases the spectral radius), which results in a
           system with a large stability margin. When the input is close to zero, the
           poles of the linearized ESN are close to the maximal spectral radius chosen,
           decreasing the stability margin. When compared to their linear counterpart,
           an ESN with the same number of states results in a detailed coverage of
           thez-plane dynamics, which illustrates the power of nonlinear systems.
           Similar results can be obtained using signals of different shapes at the ESN
           input.
             A key corollary of the above analysis is that the spectral radius of an
           ESN can be adjusted using a constant bias signal at the ESN input without
           changing the recurrent connection matrix,W. The application of a nonzero
           constant bias will move the operating point to regions of the sigmoid func-
           tion closer to saturation and always decrease the spectral radius due to the
           shape of the nonlinearity. 2 The relevance of bias in terms of overall system
           performance has also been discussed in Jaeger (2002b) and Bertschinger
           and Natschlager (2004), but here we approach it from a system theory per-¨
           spective and explain its effect on reservoir dynamics.

           3 Average State Entropy as a Measure of the Richness of ESN Reservoir

           Previous research was aware of the influence of diversity of the recurrent
           layer outputs on the overall performance of ESNs and LSMs. Several met-
           rics to quantify the diversity have been proposed (Jaeger, 2001; Maass, et al.,


             2 Assume W has nondegenerate eigenvalues and corresponding linearly independent
           eigenvectors. Then consider the eigendecomposition of W, where <<FORMULA>>,Pis the
           eigenvectormatrixandDisthediagonalmatrixofeigenvalues <<FORMULA>> of W.SinceF(n)andD
           are diagonal, <<FORMULA>> is the eigendecomposition
           of <<J(n+1)>>. Here, each entry of <<FORMULA>>, is an eigenvalue of J. Therefore,
           <<FORMULA>> since <<FORMULA>>.

                              <<FIGURE>>

           Figure 3: The pole tracks of the linearized ESN with 100 PEs when the input
           goes through a cycle. An ESN with fixed parameters implements a combination
           of linear systems with varying pole locations. (A) One cycle of sinusoidal signal
           with a period of 100. (B–E) The positions of poles of the linearized systems
           when the input values are at B, C, D, and E in Figure 5A. (F) The cumulative
           pole locations show the movement of the poles as the input changes. Due to
           the varying pole locations, different time constants modulate the richness of
           the reservoir of dynamics as a function of input amplitude. Higher-amplitude
           signals tend to saturate the nonlinear function and cause the poles to shrink
           toward the origin of thez-plane (decreases the spectral radius), which results in
           a system with a large stability margin. When the input is close to zero, the poles
           ofthelinearizedESNareclosetothemaximalspectralradiuschosen,decreasing
           the stability margin. An ESN with more states results in a detailed coverage of
           thez-plane dynamics, which illustrates the power of nonlinear systems, when
           compared to their linear counterpart.

           Here, our approach of bases and projections leads to a new metric.
           We propose the instantaneous state entropy to quantify the distribution of
           instantaneous amplitudes across the ESN states. Entropy of the instanta-
           neous ESN states is appropriate to quantify performance in function ap-
           proximation because the ESN output is a mere weighted combination of
           the instantaneous value of the ESN states. If the echo state’s instantaneous
           amplitudes are concentrated on only a few values across the ESN state dy-
           namic range, the ability to approximate an arbitrary desired response by
           weighting the states is limited (and wasteful due to redundancy between
           the different states), and performance will suffer. On the other hand, if the
           ESN states provide a diversity of instantaneous amplitudes, it is much eas-
           ier to achieve the desired mapping. Hence, the instantaneous entropy of the
           states appears as a good measure to quantify the richness of dynamics with
           instantaneous mappers. Due to the time structure of signals, the average
           state entropy (ASE), defined as the state entropy averaged over time, will be
           the parameter used to quantify the diversity in the dynamical reservoir of
           the ESN. Moreover, entropy has been proposed as an appropriate measure
           of the volume of the signal manifold (Cox, 1946; Amari, 1990). Here, ASE
           measures the volume of the echo state manifold spanned by trajectories.
             Renyi’squadraticentropyisemployedherebecauseitisaglobalmeasure
           of information. In addition, an efficient nonparametric estimator of Renyi’s
           entropy,whichavoidsexplicitpdfestimation,hasbeendeveloped(Principe,
           Xu, & Fisher, 2000). Renyi’s entropy with parameterγfor a random variable
           X with a <<FORMULA>> is given by Renyi (1970):


                        <<FORMULA>>


           Renyi’s quadratic entropy is obtained forγ=2 (forγ→1, Shannon’s en-
           tropy is obtained). GivenNsamples{x1 ,x2 ,...,xN }drawn from the un-
           known pdf to be estimated, Parzen windowing approximates the underly-
           ing pdf by

                        <<FORMULA>>

           whereKσ is the kernel function with the kernel sizeσ. Then the Renyi’s
           quadratic entropy can be estimated by (Principe et al., 2000)

                          <<FORMULA>>


             The instantaneous state entropy is estimated using equation 3.1 where
           thesamplesaretheentriesofthestatevectorx(n)=[x1 (n),x2 (n),...,xN (n)] T
           of an ESN withNinternal PEs. Results will be shown with a gaussian kernel
           with kernel size chosen to be 0.3 of the standard deviation of the entries
           of the state vector. We will show that ASE is a more sensitive parameter to
           quantify the approximation properties of ESNs by experimentally demon-
           strating that ESNs with different spectral radius and even with the same
           spectral radius display different ASEs.

             Let us consider the same 100-unit ESN that we used in the previous
           section built with three different spectral radii 0.2, 0.5, 0.8 with an input
           signal of sin(2πn/20). Figure 4A depicts the echo states over 200 time ticks.
           The instantaneous state entropy is also calculated at each time step using
           equation 3.1 and plotted in Figure 4B. First, note that the instantaneous
           state entropy changes over time with the distribution of the echo states as
           we would expect, since state entropy is dependent on the input signal that
           also changes in this case. Second, as the spectral radius increases in the
           simulation, the diversity in the echo states increases. For the spectral radius
           of 0.2, echo state’s instantaneous amplitudes are concentrated on only a
           few values, which is wasteful due to redundancy between different states.
           In practice, to quantify the overall representation ability over time, we will
           use ASE, which takes values−0.735,−0.007, and 0.335 for the spectral
           radii of 0.2, 0.5, and 0.8, respectively. Moreover, even for the same spectral
           radius, several ASEs are possible. Figure 4C shows ASEs from 50 different
           realizations of ESNs with the same spectral radius of 0.5, which means that
           ASE is a finer descriptor of the dynamics of the reservoir. Although we
           have presented an experiment with sinusoidal signal, similar results are
           obtained for other inputs as long as the input dynamic range is properly
           selected.

             Maximizing ASE means that the diversity of the states over time is the
           largest and should provide a basis set that is as uncorrelated as possible.
           This condition is unfortunately not a guarantee that the ESN so designed
           will perform the best, because the basis set in ESNs is created independent
           of the desired response and the application may require a small spectral
           radius. However, we maintain that when the desired response is not ac-
           cessible for the design of the ESN bases or when the same reservoir is
           to be used for a number of problems, the default strategy should be to
           maximize the ASE of the state vector. The following section addresses
           the design of ESNs with high ASE values and a simple mechanism to
           adjust the reservoir dynamics without changing the recurrent connection
           weights.

           4 Designing Echo State Networks

             4.1 Design of the Echo State Recurrent Connections.According to the
           interpretation of ESNs as coupled linear systems, the design of the internal
           connection matrix, W, will be based on the distribution of the poles of the
           linearized system around zero state. Our proposal is to design the ESN
           such that the linearized system has uniform pole distribution inside the
           unit circle of thez-plane. With this design scenario, the system dynamics
           will include uniform coverage of time constants arising from the uniform
           distribution of the poles, which also decorrelates as much as possible the
           basis functionals. This principle was chosen by analogy to the identification
           oflinearsystemsusingKautzfilters(Kautz,1954),whichshowsthatthebest
           approximation of a given transfer function by a linear system with finite
           order is achieved when poles are placed in the neighborhood of the spectral
           resonances. When no information is available about the desired response,
           we should uniformly spread the poles to anticipate good approximation to
           arbitrary mappings.

             We again use a maximum entropy principle to distribute the poles inside
           the unit circle uniformly. The constraints of a circle as boundary conditions
           for discrete linear systems and complex conjugate locations are easy to
           include for the pole distribution (Thogula, 2003). The poles are first initial-
           ized at random locations; the quadratic Renyi’s entropy is calculated by
           equation 3.1, and poles are moved such that the entropy of the new dis-
           tribution is increased over iterations (Erdogmus & Principe, 2002). This
           method is efficient to find uniform coverage of the unit circle with an arbi-
           trary number of poles. The system with the uniform pole locations can be
           interpreted using linear system theory. The poles that are close to the unit
           circle correspond to many sharp bandpass filters specializing in different
           frequency regions, whereas the inner poles realize filters of larger frequency
           support. Moreover, different orientations (angles) of the poles create filters
           of different center frequencies.

             Now the problem is to construct an internal weight matrix from the pole
           locations (eigenvalues ofW). In principle, we would like to create a sparse

                                    <<FIGURE>>

           Figure 4: Examples of echo states and instantaneous state entropy. (A) Outputs
           ofechostates(100PEs)producedbyESNswithspectralradiusof0.2,0.5,and0.8,
           from top to bottom, respectively. The diversity of echo states increases when the
           spectral radius increases. Within the dynamic range of the echo states, systems
           with smaller spectral radius can generate only uneven representations, while
           forW=0.8, outputs of echo states almost uniformly distribute within their
           dynamic range. (B) Instantaneous state entropy is calculated using equation 3.1.
           Information contained in the echo states is changing over time according to the
           input amplitude. Therefore, the richness of representation is controlled by the
           input amplitude. Moreover, the value of ASE increases with spectral radius.
           (C) ASEs from 50 different realizations of ESNs with the same spectral radius
           of 0.5. The plot shows that ASE is a finer descriptor of the dynamics of the
           reservoir than the spectral radius. 

           matrix, so we started with the sparsest matrix (with an inverse), which is
           the direct canonical structure given by (Kailath, 1980)

                  <<FORMULA>>

           The characteristic polynomial of W_i's

                  <<FORMULA>>,                        (4.2)

           wherepi ’s are the eigenvalues andai ’s are the coefficients of the character-
           istic polynomial ofW. Here, we know the pole locations of the linear system
           obtained from the linearization of the ESN, so using equation 4.2, we can
           obtain the characteristic polynomial and constructWmatrix in the canon-
           ical form using equation 4.1. We will call the ESN constructed based on
           the uniform pole principle ASE-ESN. All other possible solutions with the
           same eigenvalues can be obtained byQ−1 WQ,whereQis any nonsingular
           matrix.

             To corroborate our hypothesis, we would like to show that the linearized
           ESN designed with the recurrent weight matrix having the eigenvalues
           uniformly distributed inside the unit circle creates higher ASE values for a
           given spectral radius compared to other ESNs with random internal con-
           nection weight matrices. We will consider an ESN with 30 states and use our
           procedure to create theWmatrix for ASE-ESN for different spectral radii
           between <<[0.1, 0.95]>>. Similarly, we constructed ESNs with sparse randomW
           matrices with different sparseness constraints. This corresponds to a weight
           distribution having the values 0, c and −c with probabilities <<p_1>> ,<<(1−p_1)/2>>,
           and <<(1−p_1)/2>>, wherep1 defines the sparseness ofWandcis a constant
           that takes a specific value depending on the spectral radius. We also created
           Wmatrices with values uniformly distributed between−1 and 1 (U-ESN)
           and scaled to obtain a given spectral radius (Jaeger & Hass, 2004). Then,
           for differentWin matrices, we run the ASE-ESNs with the sinusoidal input
           given in section 3 and calculate ASE. Figure 5 compares the ASE values
           averaged over 1000 realizations. As observed from the figure, the ASE-ESN
           with uniform pole distribution generates higher ASE on average for all
           spectral radii compared to ESNs with sparse and uniform random connec-
           tions. This approach is indeed conceptually similar to Jeffreys’ maximum
           entropy prior (Jeffreys, 1946): it will provide a consistently good response
           for the largest class of problems. Concentrating the poles of the linearized


                                    <<FIGURE>>

           Figure 5: Comparison of ASE values obtained for ASE-ESN havingWwith
           uniform eigenvalue distribution, ESNs with randomWmatrix, and U-ESN
           with uniformly distributed weights between−1 and 1. Randomly generated
           weights have sparseness of 0.07, 0.1, and 0.2. ASE values are calculated for the
           networks with spectral radius from 0.1 to 0.95. The ASE-ESN with uniform pole
           distribution generates a higher ASE on average for all spectral radii compared
           to ESNs with random connections.


           system in certain regions of the space provides good performance only if
           the desired response has energy in this part of the space, as is well known
           from the theory of Kautz filters (Kautz, 1954).

             4.2 Design of the Adaptive Bias.
             
           In conventional ESNs, only the output weights are trained, optimizing the 
           projections of the desired response onto the basis functions (echo states). 
           Since the dynamical reservoir is fixed,
           the basis functions are only input dependent. However, since function ap-
           proximation is a problem in the joint space of the input and desired signals,
           a penalty in performance will be incurred. From the linearization analysis
           that shows the crucial importance of the operating point of the PE non-
           linearity in defining the echo state dynamics, we propose to use a single
           external adaptive bias to adjust the effective spectral radius of an ESN. No-
           tice that according to linearization analysis, bias can reduce only spectral
           radius. The information for adaptation of bias is the MSE in training, which
           modulates the spectral radius of the system with the information derived
           from the approximation error. With this simple mechanism, some informa-
           tionfromtheinput-outputjointspaceisincorporatedinthedefinitionofthe
           projection space of the ESN. The beauty of this method is that the spectral
           radius can be adjusted by a single parameter that is external to the system
           without changing reservoir weights.

             The training of bias can be easily accomplished. Indeed, since the pa-
           rameter space is only one-dimensional, a simple line search method can be
           efficiently employed to optimize the bias. Among different line search al-
           gorithms, we will use a search that uses Fibonacci numbers in the selection
           of points to be evaluated (Wilde, 1964). The Fibonacci search method min-
           imizes the maximum number of evaluations needed to reduce the interval
           of uncertainty to within the prescribed length. In our problem, a bias value
           is picked according to Fibonacci search. For each value of bias, training
           data are applied to the ESN, and the echo states are calculated. Then the
           corresponding optimal output weights and the objective function (MSE)
           are evaluated to pick the next bias value.
             Alternatively, gradient-based methods can be utilized to optimize the
           bias, due to simplicity and low computational cost. System update equation
           with an external bias signal,b,isgivenby

               <<x(n+1)=f(W_in u(n+1)+Win b+Wx(n))>>.

           The update equation forbis given by

                <<FORMULA>>

             Here,Ois the MSE defined previously. This algorithm may suffer from
           similar problems observed in gradient-based methods in recurrent net-
           works training. However, we observed that the performance surface is
           rather simple. Moreover, since the search parameter is one-dimensional,
           the gradient vector can assume only one of the two directions. Hence, im-
           precision in the gradient estimation should affect the speed of convergence
           but normally not change the correct gradient direction.

           5 Experiments

           This section presents a variety of experiments in order to test the validity
           of the ESN design scheme proposed in the previous section.

             5.1 Short-Term Memory Capacity.

             This experiment compares the shortterm memory (STM) capacity of ESNs 
             with the same spectral radius using
           the framework presented in Jaeger (2002a). Consider an ESN with a sin-
           gle input signal, <<u(n)>>, optimally trained with the desired signal <<u(n−k)>>,
           for a given delayk. Denoting the optimal output signalyk (n), thek-delay     
           STM capacity of a network,MC k , is defined as a squared correlation coef-
           ficient betweenu <<(n−k)>> and <<FORMULA>> (Jaeger, 2002a). The STM capacity, MC,
           of the network is defined as  <<FORMULA>>. STM capacity measures how accu-
           rately the delayed versions of the input signal are recovered with optimally
           trained output units. Jaeger (2002a) has shown that the memory capacity
           for recalling an independent and identically distributed (i.i.d.) input by an
           Nunit RNN with linear output units is bounded by N.
             We use ESNs with 20 PEs and a single input unit. ESNs are driven
           by an i.i.d. random input signal,<<u(n)>>, that is uniformly distributed over
           [−0.5, 0.5]. The goal is to train the ESN to generate the delayed versions
           of the input, <<u(n−1),...,u(n−40)>>. We used four different ESNs: R-ESN,
           U-ESN, ASE-ESN, and BASE-ESN. R-ESN is a randomly connected ESN
           used in Jaeger (2002a) where the entries ofWmatrix are set to 0, 0.47,
           −0.47 with probabilities 0.8, 0.1, 0.1, respectively. This corresponds to a
           sparse connectivity of 20% and a spectral radius of 0.9. The entries ofWof
           U-ESN are uniformly distributed over [−1, 1] and scaled to obtain the spec-
           tral radius of 0.9. ASE-ESN also has a spectral radius of 0.9 and is designed
           with uniform poles. BASE-ESN has the same recurrent weight matrix as
           ASE-ESN and an adaptive bias at its input. In each ESN, the input weights
           are set to 0.1 or−0.1 with equal probability, and direct connections from the
           input to the output are allowed, whereasWback is set to 0 (Jaeger, 2002a).
           The echo states are calculated using equation 2.1 for 200 samples of the
           input signal, and the first 100 samples corresponding to initial transient
           are eliminated. Then the output weight matrix is calculated using equation
           2.4. For the BASE-ESN, the bias is trained for each task. All networks are
           run with a test input signal, and the corresponding output andMC k are
           calculated. Figure 6 shows thek-delay STM capacity (averaged over 100
           trials) of each ESN for delays 1,...,40 for the test signal. The STM capac-
           ities of R-ESN, U-ESN, ASE-ESN, and BASE-ESN are 13.09, 13.55, 16.70,
           and 16.90, respectively. First, ESNs with uniform pole distribution (ASE-
           ESN and BASE-ESN) haveMCs that are much longer than the randomly
           generated ESN given in Jaeger (2002a) in spite of all having the same spec-
           tral radius. In fact, the STM capacity of ASE-ESN is close to the theoretical
           maximumvalueofN=20.AcloserlookatthefigureshowsthatR-ESNper-
           forms slightly better than ASE-ESN for delays less than 9. In fact, for small
           k, large ASE degrades the performance because the tasks do not need long
           memory depth. However, the drawback of high ASE for smallkis recov-
           ered in BASE-ESN, which reduces the ASE to the appropriate level required
           for the task. Overall, the addition of the bias to the ASE-ESN increases the
           STM capacity from 16.70 to 16.90. On the other hand, U-ESN has slightly
           better STM compared to R-ESN with only three different weight values,
           although it has more distinct weight values compared to R-ESN. It is also
           significant to note that theMCwill be very poor for an ESN with smaller
           spectral radius even with an adaptive bias, since the problem requires large
           ASE and bias can only reduce ASE. This experiment demonstrates the

                                       <<FIGURE>>

           Figure 6: Thek-delay STM capacity of each ESN for delays 1,...,40 computed
           using the test signal. The results are averaged over 100 different realizations of
           each ESN type with the specifications given in the text for differentWandWin
           matrices. The STM capacities of R-ESN, U-ESN, ASE-ESN, and BASE-ESN are
           13.09, 13.55, 16.70, and 16.90, respectively.


           suitability of maximizing ASE in tasks that require a substantial memory
           length.

             5.2 Binary Parity Check.
             
             The effect of the adaptive bias was marginal
           in the previous experiment since the nature of the problem required large
           ASE values. However, there are tasks in which the optimal solutions re-
           quire smaller ASE values and smaller spectral radius. Those are the tasks
           where the adaptive bias becomes a crucial design parameter in our design
           methodology.
             Consider an ESN with 100 internal units and a single input unit. ESN is
           drivenbyabinaryinputsignal,u(n),thatassumesthevalues0or1.Thegoal
           is to train an ESN to generate them-bit parity corresponding to lastmbits
           received, wheremis 3,...,8. Similar to the previous experiments, we used
           the R-ESN, ASE-ESN, and BASE-ESN topologies. R-ESN is a randomly
           connected ESN where the entries ofWmatrix are set to 0, 0.06,−0.06
           with probabilities 0.8, 0.1, 0.1, respectively. This corresponds to a sparse
           connectivity of 20% and a spectral radius of 0.3. ASE-ESN and BASE-ESN
           are designed with a spectral radius of 0.9. The input weights are set to 1 or -1
           with equal probability, and direct connections from the input to the output
           are allowed whereasWback is set to 0. The echo states are calculated using
           equation 2.1 for 1000 samples of the input signal, and the first 100 samples
           corresponding to the initial transient are eliminated.Then the output weight        

                                         <<FIGURE>>

           Figure 7: The number of wrong decisions made by each ESN form=3,...,8
           in the binary parity check problem. The results are averaged over 100 differ-
           ent realizations of R-ESN, ASE-ESN, and BASE-ESN for differentWandWin
           matrices with the specifications given in the text. The total numbers of wrong
           decisions form=3,...,8 of R-ESN, ASE-ESN, and BASE-ESN are 722, 862, and
           699. 

           matrix is calculated using equation 2.4. For ESN with adaptive bias, the bias
           is trained for each task. The binary decision is made by a threshold detector
           that compares the output of the ESN to 0.5. Figure 7 shows the number of
           wrong decisions (averaged over 100 different realizations) made by each
           ESN for <<m=3,...,8>>.
             The total numbers of wrong decisions for <<m=3,...,8>> of R-ESN, ASE-
           ESN, and BASE-ESN are 722, 862, and 699, respectively. ASE-ESN performs
           poorly since the nature of the problem requires a short time constant for
           fast response, but ASE-ESN has a large spectral radius. For 5-bit parity, the
           R-ESN has no wrong decisions, whereas ASE-ESN has 53 wrong decisions.
           BASE-ESN performs a lot better than ASE-ESN and slightly better than
           the R-ESN since the adaptive bias reduces the spectral radius effectively.
           Note that form=7 and 8, the ASE-ESN performs similar to the R-ESN,
           since the task requires access to longer input history, which compromises
           the need for fast response. Indeed, the bias in the BASE-ESN takes effect
           when there are errors (m>4) and when the task benefits from smaller
           spectral radius. The optimal bias values are approximately 3.2, 2.8, 2.6, and
           2.7 form=3, 4, 5, and 6, respectively. Form=7 or 8, there is a wide
           range of bias values that result in similar MSE values (between 0 and 3). In 
           summary, this experiment clearly demonstrates the power of the bias signal
           to configure the ESN reservoir according to the mapping task.

             5.3 System Identification.
             This section presents a function approxima-
           tion task where the aim is to identify a nonlinear dynamical system. The
           unknown system is defined by the difference equation

               <<y(n+1)=0.3y(n)+0.6y(n−1)+f(u(n))>>,

           where

                <<f(u)=0.6sin(πu)+0.3sin(3πu)+0.1sin(5πu)>>.

           The input to the system is chosen to be <<sin(2πn/25)>>.
             We used three different ESNs—R-ESN, ASE-ESN, and BASE-ESN—with
           30 internal units and a single input unit. TheWmatrix of each ESN is scaled
           suchthatithasaspectralradiusof0.95.R-ESNisarandomlyconnectedESN
           where the entries ofWmatrix are set to 0, 0.35,−0.35 with probabilities 0.8,
           0.1, 0.1, respectively. In each ESN, the input weights are set to 1 or−1 with
           equal probability, and direct connections from the input to the output are
           allowed,whereasWback issetto0.Theoptimaloutputweightsarecalculated
           using equation 2.4. The MSE values (averaged over 100 realizations) for R-
           ESN and ASE-ESN are 1.23x10 −5 and 1.83x10 −6 , respectively. The addition
           of the adaptive bias to the ASE-ESN reduces the MSE value from 1.83x10^−6
           to 3.27x10^−9 .

           6 Discussion

           The great appeal of echo state networks (ESNs) and liquid state machine
           (LSM) is their ability to construct arbitrary mappings of signals with rich
           and time-varying temporal structures without requiring adaptation of the
           free parameters of the recurrent layer. The echo state condition allows the
           recurrent connections to be fixed with training limited to the linear output
           layer. However, the literature did not elucidate on how to properly choose
           the recurrent parameters for system identification applications. Here, we
           provide an alternate framework that interprets the echo states as a set
           of functional bases formed by fixed nonlinear combinations of the input.
           The linear readout at the output stage simply computes the projection of
           the desired output space onto this representation space. We further in-
           troduce an information-theoretic criterion, ASE, to better understand and
           evaluate the capability of a given ESN to construct such a representation
           layer. The average entropy of the distribution of the echo states quantifies
           thevolumespannedbythebases.Assuch,thisvolumeshouldbethelargest
           to achieve the smallest correlation among the bases and be able to cope with     
           arbitrary mappings. However, not all function approximation problems re-
           quire the same memory depth, which is coupled to the spectral radius. The
           effective spectral radius of an ESN can be optimized for the given problem
           with the help of an external bias signal that is adapted using the joint input-
           output space information. The interesting property of this method when
           applied to ESN built from sigmoidal nonlinearities is that it allows the fine
           tuning of the system dynamics for a given problem with a single external
           adaptive bias input and without changing internal system parameters. In
           our opinion, the combination of the largest possible ASE and the adapta-
           tion of the spectral radius by the bias produces the most parsimonious pole
           location of the linearized ESN when no knowledge about the mapping is
           available to optimally locate the bass functionals. Moreover, the bias can be
           easily trained with either a line search method or a gradient-based method
           since it is one-dimensional. We have illustrated experimentally that the de-
           sign of the ESN using the maximization of ASE with the adaptation of the
           spectral radius by the bias has provided consistently better performance
           across tasks that require different memory depths. This means that these
           two parameters’ design methodology is preferred to the spectral radius
           criterion proposed by Jaeger, and it is still easily incorporated in the ESN
           design.

             Experiments demonstrate that the ASE for ESN with uniform linearized
           poles is maximized when the spectral radius of the recurrent weight matrix
           approaches one (instability). It is interesting to relate this observation with
           the computational properties found in dynamical systems “at the edge of
           chaos” (Packard, 1988; Langton, 1990; Mitchell, Hraber, & Crutchfield, 1993;
           Bertschinger & Natschlager, 2004). Langton stated that when cellular au-¨
           tomata rules are evolved to perform a complex computation, evolution will
           tend to select rules with “critical” parameter values, which correlate with
           a phase transition between ordered and chaotic regimes. Recently, similar
           conclusions were suggested for LSMs (Bertschinger & Natschlager, 2004).¨
           Langton’s interpretation of edge of chaos was questioned by Mitchell et al.
           (1993). Here, we provide a system-theoretic view and explain the computa-
           tional behavior with the diversity of dynamics achieved with linearizations
           that have poles close to the unit circle. According to our results, the spectral
           radiusoftheoptimalESNinfunctionapproximationisproblemdependent,
           and in general it is impossible to forecast the computational performance
           as the system approaches instability (the spectral radius of the recurrent
           weight matrix approaches one). However, allowing the system to modu-
           late the spectral radius by either the output or internal biasing may allow
           a system close to instability to solve various problems requiring different
           spectral radii.

             Our emphasis here is mostly on ESNs without output feedback connec-
           tions. However, the proposed design methodology can also be applied to
           ESNs with output feedback. Both feedforward and feedback connections
           contribute to specify the bases to create the projection space. At the same
           time, there are applications where the output feedback contributes to the
           system dynamics in a different fashion. For example, it has been shown that
           a fixed weight (fully trained) RNN with output feedback can implement a
           family of functions (meta-learners) (Prokhorov, Feldkamp, & Tyukin, 1992).
           In meta-learning, the role of output feedback in the network is to bias the
           system to different regions of dynamics, providing multiple input-output
           mappings required (Santiago & Lendaris, 2004). However, results could not
           be replicated with ESNs (Prokhorov, 2005). We believe that more work has
           to be done on output feedback in the context of ESNs but also suspect that
           the echo state condition may be a restriction on the system dynamics for
           this type of problem.

             There are many interesting issues to be researched in this exciting new
           area. Besides an evaluation tool, ASE may also be utilized to train the ESN’s
           representation layer in an unsupervised fashion. In fact, we can easily adapt
           withtheSIG(stochasticinformationgradient)describedinErdogmus,Hild,
           and Principe (2003): extra weights linking the outputs of recurrent states to
           maximize output entropy. Output entropy maximization is a well-known
           metric to create independent components (Bell & Sejnowski, 1995), and
           here it means that the echo states will become as independent as possible.
           This would circumvent the linearization of the dynamical system to set the
           recurrent weights and would fine-tune continuously in an unsupervised
           manner the parameters of the ESN among different inputs. However, it
           goes against the idea of a fixed ESN reservoir.

             The reservoir of recurrent PEs can be thought of as a new form of a time-
           to-space mapping. Unlike the delay line that forms an embedding (Takens,
           1981), this mapping may have the advantage of filtering noise and produce
           representations with better SNRs to the peaks of the input, which is very
           appealing for signal processing and seems to be used in biology. However,
           further theoretical work is necessary in order to understand the embedding
           capabilities of ESNs. One of the disadvantages of the ESN correlated basis
           is in the design of the readout. Gradient-based algorithms will be very
           slow to converge (due to the large eigenvalue spread of modes), and even
           if recursive methods are used, their stability may be compromised by the
           condition number of the matrix. However, our recent results incorporating
           anL1 norm penalty in the LMS (Rao et al., 2005) show great promise of
           solving this problem.

             Finally we would like to briefly comment on the implications of these
           models to neurobiology and computational neuroscience. The work by
           Pouget and Sejnowski (1997) has shown that the available physiological
           data are consistent with the hypothesis that the response of a single neuron
           in the parietal cortex serves as a basis function generated by the sensory
           input in a nonlinear fashion. In other words, the neurons transform the
           sensory input into a format (representation space) such that the subsequent
           computation is simplified. Then, whenever a motor command (output of
           the biological system) needs to be generated, this simple computation to
           read out the neuronal activity is done. There is an intriguing similarity
           betweentheinterpretationoftheneuronalactivitybyPougetandSejnowski
           and our interpretation of echo states in ESN. We believe that similar ideas
           can be applied to improve the design of microcircuit implementations of
           LSMs. First, the framework of functional space interpretation (bases and
           projections) is also applicable to microcircuits. Second, the ASE measure
           may be directly utilized for LSM states because the states are normally low-
           pass-filtered before the readout. However, the control of ASE by changing
           the liquid dynamics is unclear. Perhaps global control of thresholds or bias
           current will be able to accomplish bias control as in ESN with sigmoid
           PEs.


           Acknowledgments

           This work was partially supported by NSFECS-0422718, NSFCNS-0540304,
           and ONR N00014-1-1-0405.


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<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>
                         Bayesian Compression for Deep Learning

                        Christos Louizos          Karen Ullrich               Max Welling
                     University of Amsterdam    University of Amsterdam    University of Amsterdam
                     TNO Intelligent Imaging     k.ullrich@uva.nl               CIFAR 
                      c.louizos@uva.nl                                       m.welling@uva.nl


                                             Abstract

                       Compression and computational efficiency in deep learning have become a problem
                       of great significance. In this work, we argue that the most principled and effective
                       way to attack this problem is by adopting a Bayesian point of view, where through
                       sparsity inducing priors we prune large parts of the network. We introduce two
                       novelties in this paper: 1) we use hierarchical priors to prune nodes instead of
                       individual weights, and 2) we use the posterior uncertainties to determine the
                       optimal fixed point precision to encode the weights. Both factors significantly
                       contribute to achieving the state of the art in terms of compression rates, while
                       still staying competitive with methods designed to optimize for speed or energy
                       efficiency.


                 1 Introduction

                 While deep neural networks have become extremely successful in in a wide range of applications,
                 often exceeding human performance, they remain difficult to apply in many real world scenarios. For
                 instance, making billions of predictions per day comes with substantial energy costs given the energy
                 consumption of common Graphical Processing Units (GPUs). Also, real-time predictions are often
                 about a factor100away in terms of speed from what deep NNs can deliver, and sending NNs with
                 millions of parameters through band limited channels is still impractical. As a result, running them on
                 hardware limited devices such as smart phones, robots or cars requires substantial improvements on
                 all of these issues. For all those reasons, compression and efficiency have become a topic of interest
                 in the deep learning community.
                 While all of these issues are certainly related, compression and performance optimizing procedures
                 might not always be aligned. As an illustration, consider the convolutional layers of Alexnet, which
                 account for only 4% of the parameters but 91% of the computation [68]. Compressing these layers
                 will not contribute much to the overall memory footprint.
                 There is a variety of approaches to address these problem settings. However, most methods have
                 the common strategy of reducing both the neural network structure and the effective fixed point
                 precision for each weight. A justification for the former is the finding that NNs suffer from significant
                 parameter redundancy [14]. Methods in this line of thought are network pruning, where unnecessary
                 connections are being removed [40,24,21], or student-teacher learning where a large network is
                 used to train a significantly smaller network [5, 27].
                 From a Bayesian perspective network pruning and reducing bit precision for the weights is aligned
                 with achieving high accuracy, because Bayesian methods search for the optimal model structure
                 (which leads to pruning with sparsity inducing priors), and reward uncertain posteriors over parameters
                 through the bits back argument [28] (which leads to removing insignificant bits). This relation is
                 made explicit in the MDL principle [20] which is known to be related to Bayesian inference.

                 In this paper we will use the variational Bayesian approximation for Bayesian inference which has
                 also been explicitly interpreted in terms of model compression [28]. By employing sparsity inducing
                 priors for hidden units (and not individual weights) we can prune neurons including all their ingoing
                 and outgoing weights. This avoids more complicated and inefficient coding schemes needed for
                 pruning or vector quantizing individual weights. As an additional Bayesian bonus we can use the
                 variational posterior uncertainty to assess which bits are significant and remove the ones which
                 fluctuate too much under approximate posterior sampling. From this we derive the optimal fixed
                 point precision per layer, which is still practical on chip.

                 2 Variational Bayes and Minimum Description Length

                 A fundamental theorem in information theory is the minimum description length (MDL) principle [20].
                 It relates to compression directly in that it defines the best hypothesis to be the one that communicates
                 the sum of the model (complexity costLC ) and the data misfit (error costLE ) with the minimum
                 number of bits [59,60]. It is well understood that variational inference can be reinterpreted from an
                 MDL point of view [56,72,28,30,19]. More specifically, assume that we are presented with a dataset QD that consists from N input-output pairs <<FORMULA>>. Let <<FORMULA>>
                 be a parametric model, e.g. a deep neural network, that maps inputs x to their corresponding outputs
                 y using parameters w governed by a prior distribution <<p(w)>>. In this scenario, we wish to approximate
                 the intractable posterior distribution <<p(w|D) =p(D|w)p(w)=p(D)>> with a fixed form approximate
                 posterior <<q(w)>> by optimizing the variational parameters  according to:

                             <<FORMULA>> 

                 where <<H()>> denotes the entropy and <<L()>> is known as the evidence-lower-bound (ELBO) or negative
                 variational free energy. As indicated in eq.1, <<L()>> naturally decomposes into a minimum cost for
                 communicating the targets <<FORMULA>> under the assumption that the sender and receiver agreed on a n=1 prior <<p(w)>> and that the receiver knows the inputs <<FORMULA>> and form of the parametric model. n=1
                 By using sparsity inducing priors for groups of weights that feed into a neuron the Bayesian mecha-
                 nism will start pruning hidden units that are not strictly necessary for prediction and thus achieving
                 compression. But there is also a second mechanism by which Bayes can help us compress. By
                 explicitly entertaining noisy weight encodings through <<q(w)>> we can benefit from the bits-back
                 argument [28,30] due to the entropy term; this is in contrast to infinitely precise weights that lead to
                 <<FORMULA>>. Nevertheless in practice, the data misfit termLE is intractable for neural network
                 models under a noisy weight encoding, so as a solution Monte Carlo integration is usually employed.
                 Continuous q(w) allow for the reparametrization trick [36,58]. Here, we replace sampling from
                 q(w) by a deterministic function of the variational parameters  and random samples from some
                 noise variables:

                            <<FORMULA>>;        (2)

                 where <<w=f(;)>>. By applying this trick, we obtain unbiased stochastic gradients of the ELBO
                 with respect to the variational parameters, thus resulting in a standard optimization problem that is
                 fit for stochastic gradient ascent. The efficiency of the gradient estimator resulting from eq. 2 can be
                 further improved for neural networks by utilizing local reparametrizations [37] (which we will use in
                 our experiments); they provide variance reduction in an efficient way by locally marginalizing the
                 weights at each layer and instead sampling the distribution of the pre-activations.

                 3 Related Work

                 One of the earliest ideas and most direct approaches to tackle efficiency is pruning. Originally
                 introduced by [40], pruning has recently been demonstrated to be applicable to modern architectures
                 [25,21]. It had been demonstrated that an overwhelming amount of up to 99,5% of parameters
                 can be pruned in common architectures. There have been quite a few encouraging results obtained
                 by (empirical) Bayesian approaches that employ weight pruning [19,7,52,70,51]. Nevertheless,

                    2 In practice this term is a large constant determined by the weight precision.

                 weight pruning is in general inefficient for compression since the matrix format of the weights is not
                 taken into consideration, therefore the Compressed Sparse Column (CSC) format has to be employed.
                 Moreover, note that in conventional CNNs most flops are used by the convolution operation. Inspired
                 by this observation, several authors proposed pruning schemes that take these considerations into
                 account [73, 74] or even go as far as efficiency aware architectures to begin with [32, 15, 31]. From
                 the Bayesian viewpoint, similar pruning schemes have been explored at [47, 53, 39, 34].
                 Given optimal architecture, NNs can further be compressed by quantization. More precisely, there
                 are two common techniques. First, the set of accessible weights can be reduced drastically. As an
                 extreme example, [13,48,57,76] and [11] trained NN to use only binary or tertiary weights with
                 floating point gradients. This approach however is in need of significantly more parameters than
                 their ordinary counterparts. Work by [18] explores various techniques beyond binary quantization:
                 k-means quantization, product quantization and residual quantization. Later studies extent this set to
                 optimal fixed point [44] and hashing quantization [10]. [25] apply k-means clustering and consequent
                 center training. From a practical point of view, however, all these are fairly unpractical during
                 test time. For the computation of each feature map in a net, the original weight matrix must be
                 reconstructed from the indexes in the matrix and a codebook that contains all the original weights.
                 This is an expensive operation and this is why some studies propose a different approach than set
                 quantization. Precision quantization simply reduces the bit size per weight. This has a great advantage
                 over set quantization at inference time since feature maps can simply be computed with less precision
                 weights. Several studies show that this has little to no effect on network accuracy when using 16bit
                 weights [49,22,12,71,9]. Somewhat orthogonal to the above discussion but certainly relevant are
                 approaches that customize the implementation of CNNs for hardware limited devices[31, 4, 62].



                 4 Bayesian compression with scale mixtures of normals


                 Consider the following prior over a parameter w where its scale z is governed by a distribution <<p(z)>>:


                                       <<FORMULA>>;                    (3)


                 with z2 serving as the variance of the zero-mean normal distribution over w. By treating the scales
                 of w as random variables we can recover marginal prior distributions over the parameters that have
                 heavier tails and more mass at zero; this subsequently biases the posterior distribution over w to
                 be sparse. This family of distributions is known as scale-mixtures of normals [6,2] and it is quite
                 general, as a lot of well known sparsity inducing distributions are special cases.
                 One example of the aforementioned framework is the spike-and-slab distribution [50], the golden
                 standard for sparse Bayesian inference. Under the spike-and-slab, the mixing density of the scales is a
                 Bernoulli distribution, thus the marginal <<p(w)>> has a delta “spike” at zero and a continuous “slab” over
                 the real line. Unfortunately, this prior leads to a computationally expensive inference since we have
                 to explore a space of2M models, whereMis the number of the model parameters. Dropout [29,67],
                 one of the most popular regularization techniques for neural networks, can be interpreted as positing a
                 spike and slab distribution over the weights where the variance of the “slab” is zero [17,45]. Another
                 example is the Laplace distribution which arises by considering <<FORMULA>>. The mode of
                 the posterior distribution under a Laplace prior is known as the Lasso [69] estimator and has been
                 previously used for sparsifying neural networks at [73,61]. While computationally simple, the
                 Lasso estimator is prone to “shrinking" large signals [8] and only provides point estimates about
                 the parameters. As a result it does not provide uncertainty estimates, it can potentially overfit and,
                 according to the bits-back argument, is inefficient for compression.
                 For these reasons, in this paper we will tackle the problem of compression and efficiency in neural
                 networks by adopting a Bayesian treatment and inferring an approximate posterior distribution over
                 the parameters under a scale mixture prior. We will consider two choices for the prior over the scales
                 p(z); the hyperparameter free log-uniform prior [16,37] and the half-Cauchy prior, which results into
                 a horseshoe [8] distribution. Both of these distributions correspond to a continuous relaxation of the
                 spike-and-slab prior and we provide a brief discussion on their shrinkage properties at Appendix C.

                 4.1 Reparametrizing variational dropout for group sparsity

                 One potential choice for p(z) is the improper log-uniform prior [37] <<FORMULA>>. It turns out that
                 we can recover the log-uniform prior over the weightswif we marginalize over the scales z: 
                 
                                              <<FORMULA>>                (4)
                 
                 This alternative parametrization of the log uniform prior is known in the statistics literature as the
                 normal-Jeffreys prior and has been introduced by [16]. This formulation allows to “couple" the
                 scales of weights that belong to the same group (e.g. neuron or feature map), by simply sharing the
                 corresponding scale variablezin the joint prior 3 :

                                              <<FORMULA>>;                  (5)
   
                 where W is the weight matrix of a fully connected neural network layer with A being the dimen-
                 sionality of the input and B the dimensionality of the output. Now consider performing variational
                 inference with a joint approximate posterior parametrized as follows:

                                             <<FORMULA>>;                  (6) 
                                       
                 where _i is the dropout rate [67,37,51] of the given group. As explained at [37,51], the multiplicative
                 parametrization of the approximate posterior over z suffers from high variance gradients; therefore
                 we will follow [51] and re-parametrize it in terms of <<FORMULA>>, hence optimize w.r.t._2 . 
                 The <<FORMULA>> lower bound under this prior and approximate posterior becomes:

                                              <<FORMULA>>                    (7)

                 Under this particular variational posterior parametrization the negative KL-divergence from the
                 conditional prior <<p(W|z)>> to the approximate posterior <<q(W|z)>> is independent of z:

                                                                        <<FORMULA>>       (8)

                 This independence can be better understood if we consider a non-centered parametrization of the
                 prior [55]. More specifically, consider reparametrizing the weights asw~ij =wij ; this will then result zi
                 into <<p(W|z)p(z) =p(W~)p(z)>>, where <<FORMULA>>. Now if <<FORMULA>> and <<W= diag(z)>>
                 we perform variational inference under the p(W~)p(z)prior with an approximate posterior that has Q the form of <<FORMULA>>, with <<FORMULA>>, then we see that we ij arrive at the same expressions for the negative KL-divergence from the prior to the approximate
                 posterior. Finally, the negative KL-divergence from the normal-Jeffreys scale prior p(z) to the
                 Gaussian variational posterior q depends only on the “implied” dropout rate, <<FORMULA>>, and zi z takes the following form [51]:       

                                               <<FORMULA>>;                  (9)
                                          
                 where <<FORMULA>> are the sigmoid and softplus functions respectively 4 and k1 = 0:63576,k2 =
                 1:87320,k3 = 1:48695. We can now prune entire groups of parameters by simply specifying a thresh-
                 old for the variational dropout rate of the corresponding group, e.g.<<FORMULA>>. It should be mentioned that this prior parametrization readily allows for a more flexible marginal pos-
                 terior over the weights as we now have a compound distribution, <<FORMULA>>; this
                 is in contrast to the original parametrization and the Gaussian approximations employed by [37,51].
                 Strictly speaking the result of eq. 4 only holds when each weight has its own scale and not when that scale is
                 shared across multiple weights. Nevertheless, in practice we obtain a prior that behaves in a similar way, i.e. it
                 biases the variational posterior to be sparse.

                                                <<FORMULA>>

                 Furthermore, this approach generalizes the low variance additive parametrization of variational
                 dropout proposed for weight sparsity at [51] to group sparsity (which was left as an open question
                 at [51]) in a principled way.
                 At test time, in order to have a single feedforward pass we replace the distribution overWat each
                 layer with a single weight matrix, the masked variational posterior mean:

                                                 <<FORMULA>>;                         (10)

                 where m is a binary mask determined according to the group variational dropout rate andMW are
                 the means ofq (W~). We further use the variational posterior marginal variances 5 for this particular
                 posterior approximation:              
                 
                                                <<FORMULA>>;                           (11)

                 to assess the bit precision of each weight in the weight matrix. More specifically, we employed the
                 mean variance across the weight matrixW^ to compute the unit round off necessary to represent the
                 weights. This method will give us the amount significant bits, and by adding 3 exponent and 1 sign
                 bits we arrive at the final bit precision for the entire weight matrixW^6 . We provide more details at
                 Appendix B.

                 4.2 Group horseshoe with half-Cauchy scale priors

                 Another choice for p(z) is a proper half-Cauchy distribution: <<FORMULA>>; it
                 induces a horseshoe prior [8] distribution over the weights, which is a well known sparsity inducing
                 prior in the statistics literature. More formally, the prior hierarchy over the weights is expressed as
                 (in a non-centered parametrization):

                                                  <<FORMULA>>;                           (12)

                 where0 is the free parameter that can be tuned for specific desiderata. The idea behind the horseshoe
                 is that of the “global-local" shrinkage; the global scale variablespulls all of the variables towards
                 zero whereas the heavy tailed local variableszi can compensate and allow for some weights to escape.
                 Instead of directly working with the half-Cauchy priors we will employ a decomposition of the
                 half-Cauchy that relies upon (inverse) gamma distributions [54] as this will allow us to compute
                 the negative KL-divergence from the scale priorp(z)to an approximate log-normal scale posterior
                 q (z)in closed form (the derivation is given in Appendix D). More specifically, we have that the
                 half-Cauchy prior can be expressed in a non-centered parametrization as:

                                                    <<FORMULA>>;                       (13)

                 where <<IG(;);G(;)>> correspond to the inverse Gamma and Gamma distributions in the scale
                 parametrization, and z follows a half-Cauchy distribution with scale k. Therefore we will re-express
                 the whole hierarchy as:

                                                  <<FORMULA>>;                           (14)

                 It should be mentioned that the improper log-uniform prior is the limiting case of the horseshoe prior
                 when the shapes of the (inverse) Gamma hyperpriors on <<FORMULA>> go to zero [8]. In fact, several well
                 known shrinkage priors can be expressed in this form by altering the shapes of the (inverse) Gamma
                 hyperpriors [3]. For the variational posterior we will employ the following mean field approximation:

                                                <<FORMULA>>.

                Notice that the fact that we are using mean-field variational approximations (which we chose for simplicity)
                 can potentially underestimate the variance, thus lead to higher bit precisions for the weights. We leave the
                 exploration of more involved posteriors for future work.

                  Where <<LN(;)>> is a log-normal distribution. It should be mentioned that a similar form of non-
                 centered variational inference for the horseshoe has been also successfully employed for undirected
                 models at [q     33]. Notice that we can also apply local reparametrizations [37] when we are sampling
                   <<FORMULA>>
                    i i and sa sb by exploiting properties of the log-normal distribution 7 and thus forming the
                 implied:

                                                    <<FORMULA>>                           (17)
                    
                 As a threshold rule for group pruning we will use the negative log-mode 8 of the local log-normal r.v.
                 <<FORMULA>> , i.e. prune when <<FORMULA>>, with <<FORMULA>>. This ignores <<FORMULA>> and <<FORMULA>>, but nonetheless we found <<FORMULA>> dependencies among the zi elements induced by the common scale
                 that it works well in practice. Similarly with the group normal-Jeffreys prior, we will replace the
                 distribution overWat each layer with the masked variational posterior mean during test time:

                                                       <<FORMULA>>;                        (19)

                 wheremis a binary mask determined according to the aforementioned threshold,MW are the means
                 ofq(W~)and;2 are the means and variances of the local log-normals over <<FORMULA>>. Furthermore,
                 similarly to the group normal-Jeffreys approach, we will use the variational posterior marginal
                 variances:
                                                      <<FORMULA>>;                           (20)

                 to compute the final bit precision for the entire weight matrix W.

                 5 Experiments

                 We validated the compression and speed-up capabilities of our models on the well-known architectures
                 of LeNet-300-100 [41], LeNet-5-Caffe 9 on MNIST [42] and, similarly with [51], VGG [63]10 on
                 CIFAR 10 [38]. The groups of parameters were constructed by coupling the scale variables for each
                 filter for the convolutional layers and for each input neuron for the fully connected layers. We provide
                 the algorithms that describe the forward pass using local reparametrizations for fully connected
                 and convolutional layers with each of the employed approximate posteriors at appendix F. For the
                 horseshoe prior we set the scale 0 of the global half-Cauchy prior to a reasonably small value, e.g.
                 0 = 1e5. This further increases the prior mass at zero, which is essential for sparse estimation
                 and compression. We also found that constraining the standard deviations as described at [46] and
                 “warm-up" [65] helps in avoiding bad local optima of the variational objective. Further details about
                 the experimental setup can be found at Appendix A. Determining the threshold for pruning can be
                 easily done with manual inspection as usually there are two well separated clusters (signal and noise).
                 We provide a sample visualization at Appendix E.

                 5.1 Architecture learning & bit precisions

                 We will first demonstrate the group sparsity capabilities of our methods by illustrating the learned
                 architectures at Table 1, along with the inferred bit precision per layer. As we can observe, our
                 methods infer significantly smaller architectures for the LeNet-300-100 and LeNet-5-Caffe, compared
                 to Sparse Variational Dropout, Generalized Dropout and Group Lasso. Interestingly, we observe
                 that for the VGG network almost all of big 512 feature map layers are drastically reduced to around
                 10 feature maps whereas the initial layers are mostly kept intact. Furthermore, all of the Bayesian
                 methods considered require far fewer than the standard 32 bits per-layer to represent the weights,
                 sometimes even allowing for 5 bit precisions.

                    The product of log-normal r.v.s is another log-normal and a power of a log-normal r.v. is another log-normal.
                    Empirically, it slightly better separates the scales compared to the negative log-mean <<FORMULA>>. 
                    https://github.com/BVLC/caffe/tree/master/examples/mnist
                    The adapted CIFAR 10 version described athttp://torch.ch/blog/2015/07/30/cifar.html.

                 Table 1: Learned architectures with Sparse VD [51], Generalized Dropout (GD) [66] and Group
                 Lasso (GL) [73]. Bayesian Compression (BC) with group normal-Jeffreys (BC-GNJ) and group
                 horseshoe (BC-GHS) priors correspond to the proposed models. We show the amount of neurons left
                 after pruning along with the average bit precisions for the weights at each layer.

                                        <<TABLE>>

                 5.2 Compression Rates

                 For the actual compression task we compare our method to current work in three different scenarios:
                 (i) compression achieved only by pruning, here, for non-group methods we use the CSC format
                 to store parameters; (ii) compression based on the former but with reduced bit precision per layer
                 (only for the weights); and (iii) the maximum compression rate as proposed by [25]. We believe

                 Table 2: Compression results for our methods. “DC” corresponds to Deep Compression method
                 introduced at [25], “DNS” to the method of [21] and “SWS” to the Soft-Weight Sharing of [70].
                 Numbers marked with * are best case guesses.

                            <<TABLE>>

                 these to be relevant scenarios because (i) can be applied with already existing frameworks such as
                 Tensorflow [1], (ii) is a practical scheme given upcoming GPUs and frameworks will be designed to
                 work with low and mixed precision arithmetics [43,23]. For (iii), we perform k-means clustering on
                 the weights with k=32 and consequently store a weight index that points to a codebook of available
                 weights. Note that the latter achieves highest compression rate but it is however fairly unpractical at
                 test time since the original matrix needs to be restored for each layer. As we can observe at Table 2,
                 our methods are competitive with the state-of-the art for LeNet-300-100 while offering significantly
                 better compression rates on the LeNet-5-Caffe architecture, without any loss in accuracy. Do note
                 that group sparsity and weight sparsity can be combined so as to further prune some weights when a
                 particular group is not removed, thus we can potentially further boost compression performance at
                 e.g. LeNet-300-100. For the VGG network we observe that training from a random initialization
                 yielded consistently less accuracy (around 1%-2% less) compared to initializing the means of the
                 approximate posterior from a pretrained network, similarly with [51], thus we only report the latter
                 results 11 . After initialization we trained the VGG network regularly for 200 epochs using Adam with
                 the default hyperparameters. We observe a small drop in accuracy for the final models when using
                 the deterministic version of the network for prediction, but nevertheless averaging across multiple
                 samples restores the original accuracy. Note, that in general we can maintain the original accuracy on
                 VGG without sampling by simply finetuning with a small learning rate, as done at [51]. This will
                 still induce (less) sparsity but unfortunately it does not lead to good compression as the bit precision
                 remains very high due to not appropriately increasing the marginal variances of the weights.

                 5.3 Speed and energy consumption

                 We demonstrate that our method is competitive with [73], denoted as GL, a method that explicitly
                 prunes convolutional kernels to reduce compute time. We measure the time and energy consumption
                 of one forward pass of a mini-batch with batch size 8192 through LeNet-5-Caffe. We average over10 4
                 forward passes and all experiments were run with Tensorflow 1.0.1, cuda 8.0 and respective cuDNN.
                 We apply 16 CPUs run in parallel (CPU) or a Titan X (GPU). Note that we only use the pruned
                 architecture as lower bit precision would further increase the speed-up but is not implementable in
                 any common framework. Further, all methods we compare to in the latter experiments would barely
                 show an improvement at all since they do not learn to prune groups but only parameters. In figure 1
                 we present our results. As to be expected the largest effect on the speed up is caused by GPU usage.
                 However, both our models and best competing models reach a speed up factor of around 8x. We
                 can further save about 3x energy costs by applying our architecture instead of the original one on a
                 GPU. For larger networks the speed-up is even higher: for the VGG experiments with batch size 256
                 we have a speed-up factor of 51x.

                                                <<FIGURE>>

                 Figure 1:Left:Avg. Time a batch of 8192 samples takes to pass through LeNet-5-Caffe. Numbers on
                 top of the bars represent speed-up factor relative to the CPU implementation of the original network.
                 Right:Energy consumption of the GPU of the same process (when run on GPU).

                 6 Conclusion

                 We introduced Bayesian compression, a way to tackle efficiency and compression in deep neural
                 networks in a unified and principled way. Our proposed methods allow for theoretically principled
                 compression of neural networks, improved energy efficiency with reduced computation while naturally
                 learning the bit precisions for each weight. This serves as a strong argument in favor of Bayesian
                 methods for neural networks, when we are concerned with compression and speed up.

                   11 We also tried to finetune the same network with Sparse VD, but unfortunately it increased the error
                 considerably (around 3% extra error), therefore we do not report those results.

                                                  8                   Acknowledgments
                   We would like to thank Dmitry Molchanov, Dmitry Vetrov, Klamer Schutte and Dennis Koelma for
                   valuable discussions and feedback. This research was supported by TNO, NWO and Google.


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                   Appendix

                   A. Detailed experimental setup

                   We implemented our methods in Tensorflow [1] and optimized the variational parameters using
                   Adam [35] with the default hyperparameters. The means of the conditional Gaussian <<q(W|z)>>


                     Table 3: Floating point formats Bits per Exponent 
                     
                                    <<TABLE>>

                 were initialized with the scheme proposed at [26], whereas the log of the standard deviations were
                 initialized by sampling from N(9;1e4). The parameters of q(z) were initialized such that the
                 overall mean of zise 1 and the overall variance is very low (1e^8); this ensures that all of the
                 groups are active during the initial training iterations.
                 As for the standard deviation constraints; for the LeNet-300-100 architecture we constrained the
                 standard deviation of the first layer to be 0:2 whereas for the LeNet-5-Caffe we constrained
                 the standard deviation of the first layer to be 0:5. The remaining standard deviations were left
                 unconstrained. For the VGG network we constrained the standard deviations of the 64 and 128
                 feature map layers to be 0:1, the standard deviations of the 256 feature map layers to be0:2
                 and left the rest of the standard deviations unconstrained. We also found beneficial the incorporation
                 of “warm-up” [65], i.e we annealed the negative KL-divergence from the prior to the approximate
                 posterior with a linear schedule for the first 100 epochs. We initialized the means of the approximate
                 posterior by the weights and biases obtained from a VGG network trained with batch normalization
                 and dropout on CIFAR 10. For our method we disabled batch-normalization during training.
                 As for preprocessing the data; for MNIST the only preprocessing we did was to rescale the digits to
                 lie at the [-1,1] range and for CIFAR 10 we used the preprocessed dataset provided by [75].
                 Furthermore, do note that by pruning a given filter at a particular convolutional layer we can also
                 prune the parameters corresponding to that feature map for the next layer. This similarly holds for
                 fully connected layers; if we drop a given input neuron then the weights corresponding to that node
                 from the previous layer can also be pruned.

                 B. Standards for Floating-Point Arithmetic

                 Floating points values eventually need to be represented in a binary basis in a computer. The most
                 common standard today is the IEEE 754-2008 convention [64]. It definesx-bit base-2 formats,
                 officially referred to as binaryx, withx2 f16;32;64;128g. The formats are also widely known as
                 half, single, double and quadruple precision floats, respectively and used in almost all programming
                 languages as a standard. The format considers 3 kinds of bits: one sign bit,wexponent bits andp
                 precision bits.

                                            <<FIGURE>>

                             Figure 2: A symbolic representation of the binaryxformat [64].


                 The Sign bit determines the sign of the number to be represented. The exponentEis anw-bit signed
                 integer, e.g. for single precisionw= 8and thusE2[127;128]. In practice, exponents range from
                 is smaller since the first and the last number are reserved for special numbers. The true significand or
                 mantissa includes t bits on the right of the binary point. There is an implicit leading bit with value
                 one. A values is consequently decomposed as follows

                                    <<FORMULA>>                          (21)
                                                
                                    <<FORMULA>>                          (22)

                 In table 3, we summarize common and less common floating point formats.

                 There is however the possibility to design a self defined format. There are 3 important quantities
                 when choosing the right specification: overflow, underflow and unit round off also known as machine
                 precision. Each one can be computed knowing the number of exponent and significant bits. in
                 our work for example we consider a format that uses significantly less exponent bits since network
                 parameters usually vary between [-10,10]. We set the unit round off equal to the precision and thus
                 can compute the significant bits necessary to represent a specific weight.
                 Beyond designing a tailored floating point format for deep learning, recent work also explored the
                 possibility of deep learning with mixed formats [43,23]. For example, imagine the activations having
                 high precision while weights can be low precision.

                 C. Shrinkage properties of the normal-Jeffreys and horseshoe priors

                            <<FIGURE>>

                 Figure 3: Comparison of the behavior of the log-uniform / normal-Jeffreys (NJ) prior and the
                 horseshoe (HS) prior (wheres= 1). Both priors behave similarly at zero but the normal-Jeffreys has
                 an extremely heavy tail (thus making it non-normalizable).

                 In this section we will provide some insights about the behavior of each of the priors we employ by
                 following the excellent analysis of [8]; we can perform a change of variables and express the scale
                 mixture distribution of eq.3 in the main paper in terms of a shrinkage coefficient,

                                                     <<FORMULA>>                  (23) 

                 It is easy to observe that eq. 23 corresponds to a continuous relaxation of the spike-and-slab prior:
                 when <<= 0>> we have that <<FORMULA>>, i.e. no shrinkage/regularization forw, when
                 <<= 1>> we have that <<FORMULA>>, i.e.wis exactly zero, and when <<=1>> we have that <<FORMULA>>. Now by examining the implied prior on the shrinkage coefficient  for both
                 the log-uniform and the horseshoe priors we can better study their behavior. As it is explained at                                                        
                 the half-Cauchy prior onzcorresponds to a beta prior on the shrinkage coefficient, <<FORMULA>>,
                 whereas the normal-Jeffreys / log-uniform prior onzcorresponds <<top() =B(;)>> with <<FORMULA>>.
                 The densities of both of these distributions can be seen at Figure 3b. As we can observe, the log-
                 uniform prior posits a distribution that concentrates almost all of its mass at either0or1,
                 essentially either pruning the parameter or keeping it close to the maximum likelihood estimate due
                 <<FORMULA>>. In contrast the horseshoe prior maintains enough probability mass for
                 the in-between values of  and thus can, potentially, offer better regularization and generalization.

                 D. Negative KL-divergences for log-normal approximating posteriors

                 Le <<FORMULA>> be a log-normal approximating posterior. Here we will derive the negative
                 KL-divergences toq(z)from inverse gamma, gamma and half-normal distributions.
                 Letp(z)be an inverse gamma distribution, i.e. <<p(z) =IG(;)>>. The negative KL-divergence can
                 be expressed as follows:
                     
                              <<FORMULA>>         (24)


                The second term is the entropy of the log-normal distribution which has the following form:

                                    <<FORMULA>>         (25)

                 The first term is the negative cross-entropy of the log-normal approximate posterior from the inverse-
                 Gamma prior:
                                  <<FORMULA>>        (26)

                                  <<FORMULA>>        (27)

                 Since the natural logarithm of a log-normal distribution <<FORMULA>> follows a normal distribution
                 <<FORMULA>> we have that <<FORMULA>>. Furthermore we have that <<FORMULA>> then <<FORMULA>>, therefore
                 <<FORMULA>>. Putting everything together we have that: 

                                  <<FORMULA>>         (28) 

                 Therefore the negative KL-divergence is:

                                          <<FORMULA>>                  (29)

                 Now let p(z) be a Gamma prior, i.e. <<p(z) =G(;)>>. We have that the negative cross-entropy
                 changes to:
                                  <<FORMULA>>        (30)

                                <<FORMULA>>      (31)
                                                            
                                <<FORMULA>>        (32)2

                 Therefore the negative KL-divergence is:

                                          <<FORMULA>>                   (33)

                 Now, by employing the aforementioned we can express the negative KL-divergence from
                <<FORMULA>> to <<FORMULA>> as follows:

                                              <<FORMULA>>

                 with the KL-divergence for the weight distribution <<q (W~)>> given by eq.8 in the main paper.

                            E. Visualizations

                                        <<FIGURE>>

                 Figure 4: Distribution of the thresholds for the Sparse Variational Dropout 4a, Bayesian Compression
                 with group normal-Jeffreys (BC-GNJ) 4b and group Horseshoe (BC-GHS) 4c priors for the three
                 layer LeNet-300-100 architecture. It is easily observed that there are usually two well separable
                 groups with BC-GNJ and BC-GHS, thus making the choice for the threshold easy. Smaller values
                 indicate signal whereas larger values indicate noise (i.e. useless groups).

                                        <<FIGURE>>

                 Figure 5: Distribution of the bit precisions for the Sparse Variational Dropout 5a, Bayesian Com-
                 pression with group normal-Jeffreys (BC-GNJ) 5b and group Horseshoe (BC-GHS) 5c priors for the
                 three layer LeNet-300-100 architecture. All of the methods usually require far fewer than 32bits for
                 the weights.

                 F. Algorithms for the feedforward pass

                 Algorithms 1, 2, 3, 4 describe the forward pass using local reparametrizations for fully connected and
                 convolutional layers with the approximate posteriors for the Bayesian Compression (BC) with group
                 normal-Jeffreys (BC-GNJ) and group Horseshoe (BC-GHS) priors employed at the experiments. For
                 the fully connected layers we coupled the scales for each input neuron whereas for the convolutional
                 we couple the scales for each output feature map.Mw ;w are the means and variances of each layer,
                 His a minibatch of activations of sizeK. For the first layer we have thatH=XwhereXis the
                 minibatch of inputs. For the convolutional layersNf are the number of convolutional filters,is the
                 convolution operator and we assume the [batch, height, width, feature maps] convention.

                   Algorithm 1 Fully connected BC-GNJ layer h. 
                   
                            <<ALGORITHM>>
                   
                   Algorithm 2Convolutional BC-GNJ layerh.
                
                            <<ALGORITHM>>

                 Algorithm 3 Fully connected BC-GHS layerh. 
                 
                            <<ALGORITHM>>
                 
                 Algorithm 4Convolutional BC-GHS layerh.

                            <<ALGORITHM>>           

<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>
Channel Pruning for Accelerating Very Deep Neural Networks 
Yihui He*  Xiangyu Zhang  Jian Sun  
Xifian Jiaotong University  Megvii Inc.  Megvii Inc.  
Xifian, 710049, China  Beijing, 100190, China  Beijing, 100190, China  
heyihui@stu.xjtu.edu.cn  zhangxiangyu@megvii.com  sunjian@megvii.com  

Abstract 
In this paper, we introduce a new channel pruning method to accelerate very deep convolutional neural net.works. Given a trained CNN model, we propose an it.erative two-step algorithm to effectively prune each layer, by a LASSO regression based channel selection and least square reconstruction. We further generalize this algorithm to multi-layer and multi-branch cases. Our method re.duces the accumulated error and enhance the compatibility with various architectures. Our pruned VGG-16 achieves the state-of-the-art results by 5. speed-up along with only 0.3% increase of error. More importantly, our method is able to accelerate modern networks like ResNet, exception and suffers only 1.4%, 1.0% accuracy loss under 2. speed.up respectively, which is significant. 
1. Introduction 
Recent CNN acceleration works fall into three categories: optimized implementation (e.g., FFT [47]), quantization (e.g., BinaryNet [8]), and structured simplification that convert a CNN into compact one [22]. This work focuses on the last one. 
Structured simplification mainly involves: tensor factorization [22], sparse connection [17], and channel pruning [48]. Tensor factorization factorizes a convolutional layer into several efficient ones (Fig. 1(c)). However, feature map width (number of channels) could not be reduced, which makes it difficult to decompose 1 . 1 convolutional layer favored by modern networks (e.g., GoogleNet [45], ResNet [18], Xception [7]). This type of method also intro.duces extra computation overhead. Sparse connection deactivates connections between neurons or channels (Fig. 1(b)). Though it is able to achieves high theoretical speed-up ratio, the sparse convolutional layers have an fiirregularfi shape which is not implementation friendly. In contrast, channel pruning directly reduces feature map width, which shrinks 

<<FIGURE>>

Figure 1. Structured simplification methods that accelerate CNNs: 
(a) a network with 3 conv layers. (b) sparse connection deactivates some connections between channels. (c) tensor factorization factorizes a convolutional layer into several pieces. (d) channel pruning reduces number of channels in each layer (focus of this paper). 
a network into thinner one, as shown in Fig. 1(d). It is efficient on both CPU and GPU because no special implementation is required. 
Pruning channels is simple but challenging because re.moving channels in one layer might dramatically change the input of the following layer. Recently, training-based channel pruning works [1, 48] have focused on imposing sparse constrain on weights during training, which could adaptively determine hyper-parameters. However, training from scratch is very costly and results for very deep CNNs on ImageNet have been rarely reported. Inference-time at.tempts [31, 3] have focused on analysis of the importance of individual weight. The reported speed-up ratio is very limited. 
In this paper, we propose a new inference-time approach for channel pruning, utilizing redundancy inter channels. Inspired by tensor factorization improvement by feature maps reconstruction [52], instead of analyzing filter weights [22, 31], we fully exploits redundancy within feature maps. Specifically, given a trained CNN model, pruning each layer is achieved by minimizing reconstruction error on its output feature maps, as showed in Fig. 2. We solve this mini.

<<FIGURE>>

Figure 2. Channel pruning for accelerating a convolutional layer. We aim to reduce the width of feature map B, while minimizing the reconstruction error on feature map C. Our optimization algorithm (Sec. 3.1) performs within the dotted box, which does not involve nonlinearity. This figure illustrates the situation that two channels are pruned for feature map B. Thus corresponding channels of filters W can be removed. Furthermore, even though not directly optimized by our algorithm, the corresponding filters in the previous layer can also be removed (marked by dotted filters). c, n: number of channels for feature maps B and C, kh . kw : kernel size. 
minimization problem by two alternative steps: channels selection and feature map reconstruction. In one step, we figure out the most representative channels, and prune redundant ones, based on LASSO regression. In the other step, we reconstruct the outputs with remaining channels with linear least squares. We alternatively take two steps. Further, we approximate the network layer-by-layer, with accumulated error accounted. We also discuss methodologies to prune multi-branch networks (e.g., ResNet [18], exception [7]). 
For VGG-16, we achieve 4. acceleration, with only 1.0% increase of top-5 error. Combined with tensor factorization, we reach 5. acceleration but merely suffer 0.3% increase of error, which outperforms previous state-of-the.arts. We further speed up ResNet-50 and Xception-50 by 2. with only 1.4%, 1.0% accuracy loss respectively. 

2. Related Work

There has been a significant amount of work on accelerating CNNs. Many of them fall into three categories: optimized implementation [4], quantization [40], and structured simplification [22]. 
Optimized implementation based methods [35, 47, 27, 4] accelerate convolution, with special convolution algorithms like FFT [47]. Quantization [8, 40] reduces floating point computational complexity. 
Sparse connection eliminates connections between neurons [17, 32, 29, 15, 14]. [51] prunes connections based on weights magnitude. [16] could accelerate fully connected layers up to 50.. However, in practice, the actual speed-up maybe very related to implementation. 
Tensor factorization [22, 28, 13, 24] decompose weights into several pieces. [50, 10, 12] accelerate fully connected layers with truncated SVD. [52] factorize a layer into 3 . 3 and 1 . 1 combination, driven by feature map redundancy. 
Channel pruning removes redundant channels on feature maps. There are several training-based approaches. [1, 48] regularize networks to improve accuracy. Channel-wise SSL [48] reaches high compression ratio for first few conv layers of LeNet [30] and AlexNet [26]. However, training-based approaches are more costly, and the effectiveness for very deep networks on large datasets is rarely exploited. 
Inference-time channel pruning is challenging, as re.ported by previous works [2, 39]. Some works [44, 34, 19] focus on model size compression, which mainly operate the fully connected layers. Data-free approaches [31, 3] results for speed-up ratio (e.g., 5.) have not been reported, and requires long retraining procedure. [3] select channels via over 100 random trials, however it need long time to eval.ate each trial on a deep network, which makes it infeasible to work on very deep models and large datasets. [31] is even worse than naive solution from our observation sometimes (Sec. 4.1.1). 

3. Approach 

In this section, we first propose a channel pruning algorithm for a single layer, then generalize this approach to multiple layers or the whole model. Furthermore, we dis.cuss variants of our approach for multi-branch networks. 

3.1. Formulation 

Fig. 2 illustrates our channel pruning algorithm for a sin.gle convolutional layer. We aim to reduce the width of feature map B, while maintaining outputs in feature map 
C. Once channels are pruned, we can remove correspond.ing channels of the filters that take these channels as in.put. Also, filters that produce these channels can also be removed. It is clear that channel pruning involves two key points. The first is channel selection, since we need to select most representative channels to maintain as much information. The second is reconstruction. We need to reconstruct the following feature maps using the selected channels. 
Motivated by this, we propose an iterative two-step algorithm. In one step, we aim to select most representative channels. Since an exhaustive search is infeasible even for tiny networks, we come up with a LASSO regression based method to figure out representative channels and prune redundant ones. In the other step, we reconstruct the outputs with remaining channels with linear least squares. We alter.natively take two steps. 
Formally, to prune a feature map with c channels, we consider applying n.c.kh .kw convolutional filters W on <<FORMULA>> input volumes X sampled from this feature map, which produces N . n output matrix Y. Here, N is the number of samples, n is the number of output channels, and kh,kw are the kernel size. For simple representation, bias term is not included in our formulation. To prune the 
.. 
input channels from c to desired <<FORMULA>>, while minimizing reconstruction error, we formulate our problem as follow: 

<<FORMULA>>       (1)

F is Frobenius norm. <<FORMULA>> matrix sliced from ith channel of input volumes X_i, i =1, ..., c. W_i is n . filter weights sliced from ith channel of W. is coefficient vector of length c for channel selection, and .i is ith entry of . Notice that, if .i =0, X_i will be no longer useful, which could be safely pruned from feature map. W_i could also be removed. Optimization Solving this minimization problem in Eqn. 1 is NP-hard. Therefore, we relax the l_0 to l_1 regularization: 

<<FORMULA>>       (2)

. is a penalty coefficient. By increasing l, there will be more zero terms in and one can get higher speed-up ratio. We also add a constrain .i WiF =1 to this formulation, which avoids trivial solution. 
Now we solve this problem in two folds. First, we fix W, solve for channel selection. Second, we fix , solve W to reconstruct error. 
(i) The subproblem of . In this case, W is fixed. We solve for channel selection. This problem can be solved by LASSO regression [46, 5], which is widely used for model selection. 

<<FORMULA>>       (3) 
.
Here Zi =XiWi (size N .n). We will ignore ith channels if .i =0. 
(ii) The subproblem of W. In this case, is fixed. We utilize the selected channels to minimize reconstruction error. We can find optimized solution by least squares: 

<<FORMULA>>. (4)

Here <<FORMULA>> (size N.). W is n reshaped W, <<FORMULA>>. After obtained result W, it is reshaped back to W. Then we assign <<FORMULA>>. Constrain <<FORMULA>> satisfies.
We alternatively optimize (i) and (ii). In the beginning, W is initialized from the trained model, <<FORMULA>>, namely no penalty, and <<k = c>>. We gradually increase <<FORMULA>> For each
change of <<FORMULA>>, we iterate these two steps until k is stable. 

After <<FORMULA>> satisfies, we obtain the final solution W from <<FORMULA>> In practice, we found that the two steps iteration is time consuming. So we apply (i) multiple times, 

<<FORMULA>>

until <<FORMULA>> satisfies. Then apply (ii) just once, to obtain 

<<FORMULA>>

the final result. From our observation, this result is comparable with two steps iterations. Therefore, in the following experiments, we adopt this approach for efficiency. 
Discussion: Some recent works [48, 1, 17] (though train.
ing based) also introduce .1-norm or LASSO. However, we must emphasis that we use different formulations. Many of them introduced sparsity regularization into training loss, instead of explicitly solving LASSO. Other work [1] solved LASSO, while feature maps or data were not considered during optimization. Because of these differences, our ap.proach could be applied at inference time. 

3.2. Whole Model Pruning 
Inspired by [52], we apply our approach layer by layer sequentially. For each layer, we obtain input volumes from the current input feature map, and output volumes from the output feature map of the un-pruned model. This could be formalized as: 

<<FORMULA>> (5)

Different from Eqn. 1, Y is replaced by Y . , which is from feature map of the original model. Therefore, the accumulated error could be accounted during sequential pruning. 

3.3. Pruning Multi.Branch Networks 
The whole model pruning discussed above is enough for single-branch networks like LeNet [30], AlexNet [26] and VGG Nets [43]. However, it is insufficient for multi-branch networks like GoogLeNet [45] and ResNet [18]. We mainly focus on pruning the widely used residual structure (e.g., ResNet [18], Xception [7]). Given a residual block shown in Fig. 3 (left), the input bifurcates into shortcut and residual branch. On the residual branch, there are several convolutional layers (e.g., 3 convolutional layers which have spatial size of 1 . 1, 3 . 3, 1 . 1, Fig. 3, left). Other layers except the first and last layer can be pruned as is described previously. For the first layer, the challenge is that the large input feature map width (for ResNet, 4 times of its output) can it be easily pruned, since it is shared with shortcut. For the last layer, accumulated error from the shortcut is hard to be recovered, since there is no parameter on the shortcut. To address these challenges, we propose several variants of our approach as follows. 

<<FIGURE>>

Figure 3. Illustration of multi-branch enhancement for residual block. Left: original residual block. Right: pruned residual block with enhancement, cx denotes the feature map width. Input channels of the first convolutional layer are sampled, so that the large input feature map width could be reduced. As for the last layer, rather than approximate Y2 , we try to approximate <<Y1+Y2>> directly (Sec. 3.3 Last layer of residual branch). 
Last layer of residual branch: Shown in Fig. 3, the output layer of a residual block consists of two inputs: feature map Y1 and Y2 from the shortcut and residual branch. We aim to recover Y1 +Y2 for this block. Here, Y1, Y2 are the original feature maps before pruning. Y2 could be approximated as in Eqn. 1. However, shortcut branch is parameter-free, then Y1 could not be recovered directly. To compensate this error, the optimization goal of the last layer is changed from Y2 to Y1 .Y . +Y2, which does not change 

<<FORMULA>>

our optimization. Here, Y . is the current feature map after

<<FORMULA>>

previous layers pruned. When pruning, volumes should be sampled correspondingly from these two branches. 
First layer of residual branch: Illustrated in Fig. 3(left), the input feature map of the residual block could not be pruned, since it is also shared with the short.cut branch. In this condition, we could perform feature map sampling before the first convolution to save computation. We still apply our algorithm as Eqn. 1. Differently, we sample the selected channels on the shared feature maps to construct a new input for the later convolution, shown in Fig. 3(right). Computational cost for this operation could be ignored. More importantly, after introducing feature map sampling, the convolution is still irregular. 
Filter-wise pruning is another option for the first con.volution on the residual branch. Since the input channels of parameter-free shortcut branch could not be pruned, we apply our Eqn. 1 to each filter independently (each fil.ter chooses its own representative input channels). Under single layer acceleration, filter-wise pruning is more accurate than our original one. From our experiments, it im.proves 0.5% top-5 accuracy for 2. ResNet-50 (applied on the first layer of each residual branch) without fine-tuning. However, after fine-tuning, there is no noticeable improvement. In addition, it outputs irregular convolutional layers, which need special library implementation support. We do not adopt it in the following experiments. 

4. Experiment 

We evaluation our approach for the popular VGG Nets [43], ResNet [18], Xception [7] on ImageNet [9], CIFAR.10 [25] and PASCAL VOC 2007 [11]. 
For Batch Normalization [21], we first merge it into convolutional weights, which do not affect the outputs of the networks. So that each convolutional layer is followed by ReLU [36]. We use Caffe [23] for deep network evaluation, and scikit-learn [38] for solvers implementation. For channel pruning, we found that it is enough to extract 5000 images, and 10 samples per image. On ImageNet, we evaluate the top-5 accuracy with single view. Images are re.sized such that the shorter side is 256. The testing is on center crop of 224 . 224 pixels. We could gain more per.formance with fine-tuning. We use a batch size of 128 and 
.5
learning rate 1e^-4. We fine-tune our pruned models for 10 epochs. The augmentation for fine-tuning is random crop of 224 . 224 and mirror. 

4.1. Experiments with VGG.16 

VGG-16 [43] is a 16 layers single path convolutional neural network, with 13 convolutional layers. It is widely used in recognition, detection and segmentation, etc. Single view top-5 accuracy for VGG-16 is 89.9%1. 

4.1.1 Single Layer Pruning 

In this subsection, we evaluate single layer acceleration performance using our algorithm in Sec. 3.1. For better under.standing, we compare our algorithm with two naive chan.nel selection strategies. first k selects the first k channels. max response selects channels based on corresponding filters that have high absolute weights sum [31]. For fair com.parison, we obtain the feature map indexes selected by each of them, then perform reconstruction (Sec. 3.1 (ii)). We hope that this could demonstrate the importance of channel selection. Performance is measured by increase of error af.ter a certain layer is pruned without fine-tuning, shown in Fig. 4. 
As expected, error increases as speed-up ratio increases. Our approach is consistently better than other approaches in different convolutional layers under different speed-up ra.tio. Unexpectedly, sometimes max response is even worse than first k. We argue that max response ignores correlations between different filters. Filters with large absolute weight may have strong correlation. Thus selection based on filter weights is less meaningful. Correlation on feature maps is worth exploiting. We can find that channel selection http://www.vlfeat.org/matconvnet/pretrained/ 

<<FIGURE>>

Figure 4. Single layer performance analysis under different speed-up ratios (without fine-tuning), measured by increase of error. To verify the importance of channel selection referred in Sec. 3.1, we considered two naive baselines. first k selects the first k feature maps. max response selects channels based on absolute sum of corresponding weights filter [31]. Our approach is consistently better (smaller is better). 

<<TABLE>>

Table 1. Accelerating the VGG-16 model [43] using a speedup ratio of 2., 4., or 5. (smaller is better). 
affects reconstruction error a lot. Therefore, it is important for channel pruning. 
Also notice that channel pruning gradually becomes hard, from shallower to deeper layers. It indicates that shallower layers have much more redundancy, which is consistent with [52]. We could prune more aggressively on shallower layers in whole model acceleration. 


4.1.2 Whole Model Pruning 
Shown in Table 1, whole model acceleration results under 2., 4., 5. are demonstrated. We adopt whole model pruning proposed in Sec. 3.2. Guided by single layer experiments above, we pruning more aggressive for shallower layers. Remaining channels ratios for shallow lay.ers (conv 1_x to conv 3_x) and deep layers (conv4_x) is 1:1.5. conv 5_x are not pruned, since they only con.tribute 9% computation in total and are not redundant. 
After fine-tuning, we could reach 2. speed-up without losing accuracy. Under 4., we only suffers 1.0% drops. Consistent with single layer analysis, our approach outperforms previous channel pruning approach (Li et al. [31]) by large margin. This is because we fully exploits channel redundancy within feature maps. Compared with tensor factorization algorithms, our approach is better than Jaderberg et al. [22], without fine-tuning. Though worse than Asym. [52], our combined model outperforms its combined Asym. 3D (Table 2). This may indicate that channel pruning is more challenging than tensor factorization, since removing channels in one layer might dramatically change the input of the following layer. However, channel pruning keeps the original model architecture, do not introduce additional layers, and the absolute speed-up ratio on GPU is much higher (Table 3). 
Since our approach exploits a new cardinality, we further combine our channel pruning with spatial factorization [22] and channel factorization [52]. Demonstrated in Table 2, 

<<TABLE>>

Table 2. Performance of combined methods on the VGG-16 model 

[43] using a speed-up ratio of 4. or 5.. Our 3C solution outperforms previous approaches (smaller is better). 
our 3 cardinalities acceleration (spatial, channel factorization, and channel pruning, denoted by 3C) outperforms previous state-of-the-arts. Asym. 3D [52] (spatial and chan.nel factorization), factorizes a convolutional layer to three parts: <<FORMULA>>. 
We apply spatial factorization, channel factorization, and our channel pruning together sequentially layer-by-layer. We fine-tune the accelerated models for 20 epochs, since they are 3 times deeper than the original ones. After fine-tuning, our 4. model suffers no degradation. Clearly, a combination of different acceleration techniques is better than any single one. This indicates that a model is redundant in each cardinality. 


4.1.3 Comparisons of Absolute Performance 
We further evaluate absolute performance of acceleration on GPU. Results in Table 3 are obtained under Caffe [23], CUDA 8 [37] and cuDNN5 [6], with a mini-batch of 32 on a GPU (GeForce GTX TITAN X). Results are averaged from 50 runs. Tensor factorization approaches decompose weights into too many pieces, which heavily increase over.head. They could not gain much absolute speed-up. Though our approach also encountered performance decadence, it generalizes better on GPU than other approaches. Our re.sults for tensor factorization differ from previous research [52, 22], maybe because current library and hardware prefer single large convolution instead of several small ones. 

4.1.4 Comparisons with Training from Scratch 
Though training a compact model from scratch is time-consuming (usually 120 epochs), it worths comparing our approach and from scratch counterparts. To be fair, we evaluated both from scratch counterpart, and normal setting net.work that has the same computational complexity and same architecture. 
Shown in Table 4, we observed that it is difficult for from scratch counterparts to reach competitive accuracy. our model outperforms from scratch one. Our approach successfully picks out informative channels and constructs highly compact models. We can safely draw the conclusion that the same model is difficult to be obtained from scratch. This coincides with architecture design researches [20, 1] that the model could be easier to train if there are more channels in shallower layers. However, channel prun.ing favors shallower layers. 
For from scratch (uniformed), the filters in each layers is reduced by half (eg. reduce conv1_1 from 64 to 32). We can observe that normal setting networks of the same complexity couldn't reach same accuracy either. This consolidates our idea that there is much redundancy in networks while training. However, redundancy can be opt out at inference-time. This maybe an advantage of inference-time acceleration approaches over training-based approaches. 
Notice that there is a 0.6% gap between the from scratch model and uniformed one, which indicates that there is room for model exploration. Adopting our approach is much faster than training a model from scratch, even for a thin.ner one. Further researches could alleviate our approach to do thin model exploring. 

4.1.5 Acceleration for Detection 
VGG-16 is popular among object detection tasks [42, 41, 33]. We evaluate transfer learning ability of our 2./4. pruned VGG-16, for Faster R-CNN [42] object detections. PASCAL VOC 2007 object detection benchmark [11] contains 5k trainable images and 5k test images. The performance is evaluated by mean Average Precision (mAP). In our experiments, we first perform channel pruning for VGG-16 on the ImageNet. Then we use the pruned model as the pre-trained model for Faster R-CNN. 
The actual running time of Faster R-CNN is 220ms / im.age. The convolutional layers contributes about 64%. We got actual time of 94ms for 4. acceleration. From Table 5, we observe 0.4% mAP drops of our 2. model, which is not harmful for practice consideration. 

4.2. Experiments with Residual Architecture Nets 
For Multi-path networks [45, 18, 7], we further explore the popular ResNet [18] and latest Xception [7], on Ima.geNet and CIFAR-10. Pruning residual architecture nets is more challenging. These networks are designed for both efficiency and high accuracy. Tensor factorization algorithms [52, 22] have difficult to accelerate these model. Spatially, 1 . 1 convolution is favored, which could hardly be factorized. 

4.2.1 ResNet Pruning 
ResNet complexity uniformly drops on each residual block. Guided by single layer experiments (Sec. 4.1.1), we still prefer reducing shallower layers heavier than deeper ones. 
Following similar setting as Filter pruning [31], we keep 70% channels for sensitive residual blocks (res5 and blocks close to the position where spatial size 

<<TABLE>>

Table 3. GPU acceleration comparison. We measure forward-pass time per image. Our approach generalizes well on GPU (smaller is better). 

<<TABLE>>

Table 4. Comparisons with training from scratch, under 4. acceleration. Our fine-tuned model outperforms scratch trained counterparts (smaller is better). 

<<TABLE>>

Table 5.Acceleration for Faster R-CNN detection. 
  
<<TABLE>>

Table 6. 2. acceleration for ResNet-50 on ImageNet, the base.line network is top-5 accuracy is 92.2% (one view). We improve performance with multi-branch enhancement (Sec. 3.3, smaller is better). 
change, e.g. res3a,res3d). As for other blocks, we keep 30% channels. With multi-branch enhancement, we prune branch 2a more aggressively within each residual block. The remaining channels ratios for branch 2a,branch 2b,branch 2c is 2:4:3 (e.g., Given 30%, we keep 40%, 80%, 60% respectively). 
We evaluate performance of multi-branch variants of our approach (Sec. 3.3). From Table 6, we improve 4.0% with our multi-branch enhancement. This is because we accounted the accumulated error from shortcut connection which could broadcast to every layer after it. And the large input feature map width at the entry of each residual block is well reduced by our feature map sampling. 
 
<<TABLE>>

Table 7. Comparisons for Xception-50, under 2. acceleration ra.tio. The baseline network is top-5 accuracy is 92.8%. Our approach outperforms previous approaches. Most structured simplification methods are not effective on Xception architecture (smaller is better). 


4.2.2 Xception Pruning 
Since computational complexity becomes important in model design, separable convolution has been payed much attention [49, 7]. Xception [7] is already spatially optimized and tensor factorization on 1 . 1 convolutional layer is destructive. Thanks to our approach, it could still be accelerated with graceful degradation. For the ease of comparison, we adopt Xception convolution on ResNet-50, denoted by Xception-50. Based on ResNet-50, we swap all convolutional layers with spatial conv blocks. To keep the same computational complexity, we increase the input channels of all branch2b layers by 2.. The baseline Xception.50 has a top-5 accuracy of 92.8% and complexity of 4450 MFLOPs. 
We apply multi-branch variants of our approach as de.scribed in Sec. 3.3, and adopt the same pruning ratio setting as ResNet in previous section. Maybe because of Xcep.tion block is unstable, Batch Normalization layers must be maintained during pruning. Otherwise it becomes nontrivial to fine-tune the pruned model. 
Shown in Table 7, after fine-tuning, we only suffer 1.0% increase of error under 2.. Filter pruning [31] could also apply on Xception, though it is designed for small speed.up ratio. Without fine-tuning, top-5 error is 100%. After training 20 epochs which is like training from scratch, in.creased error reach 4.3%. Our results for Xception-50 are not as graceful as results for VGG-16, since modern net.works tend to have less redundancy by design. 

<<TABLE>>

Table 8. 2. speed-up comparisons for ResNet-56 on CIFAR-10, the baseline accuracy is 92.8% (one view). We outperforms previous approaches and scratch trained counterpart (smaller is better). 


4.2.3 Experiments on CIFAR-10 
Even though our approach is designed for large datasets, it could generalize well on small datasets. We perform experiments on CIFAR-10 dataset [25], which is favored by many acceleration researches. It consists of 50k images for training and 10k for testing in 10 classes. 
We reproduce ResNet-56, which has accuracy of 92.8% (Serve as a reference, the official ResNet-56 [18] has ac.curacy of 93.0%). For 2. acceleration, we follow similar setting as Sec. 4.2.1 (keep the final stage unchanged, where the spatial size is 8 . 8). Shown in Table 8, our approach is competitive with scratch trained one, without fine-tuning, under 2. speed-up. After fine-tuning, our result is significantly better than Filter pruning [31] and scratch trained one. 

5. Conclusion 
To conclude, current deep CNNs are accurate with high inference costs. In this paper, we have presented an inference-time channel pruning method for very deep net.works. The reduced CNNs are inference efficient networks while maintaining accuracy, and only require off-the-shelf libraries. Compelling speed-ups and accuracy are demonstrated for both VGG Net and ResNet-like networks on Im.ageNet, CIFAR-10 and PASCAL VOC. 
In the future, we plan to involve our approaches into training time, instead of inference time only, which may also accelerate training procedure. 

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<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>
                                    Convex Neural Networks

                      Yoshua Bengio, Nicolas Le Roux, Pascal Vincent, Olivier Delalleau, Patrice Marcotte
                                         Dept. IRO, Universite de Montr´      eal´
                             P.O. Box 6128, Downtown Branch, Montreal, H3C 3J7, Qc, Canada
                              fbengioy,lerouxni,vincentp,delallea,marcotteg@iro.umontreal.ca

                                                Abstract
                           Convexity has recently received a lot of attention in the machine learning
                           community, and the lack of convexity has been seen as a major disad-
                           vantage of many learning algorithms, such as multi-layer artificial neural
                           networks. We show that training multi-layer neural networks in which the
                           number of hidden units is learned can be viewed as a convex optimization
                           problem. This problem involves an infinite number of variables, but can be
                           solved by incrementally inserting a hidden unit at a time, each time finding
                           a linear classifier that minimizes a weighted sum of errors.

                     1 Introduction
                     The objective of this paper is not to present yet another learning algorithm, but rather to point
                     to a previously unnoticed relation between multi-layer neural networks (NNs),Boosting (Fre-
                     und and Schapire, 1997) and convex optimization. Its main contributions concern the mathe-
                     matical analysis of an algorithm that is similar to previously proposed incremental NNs, with
                     L1 regularization on the output weights. This analysis helps to understand the underlying
                     convex optimization problem that one is trying to solve.
                     This paper was motivated by the unproven conjecture (based on anecdotal experience) that
                     when the number of hidden units is “large”, the resulting average error is rather insensitive to
                     the random initialization of the NN parameters. One way to justify this assertion is that to re-
                     ally stay stuck in a local minimum, one must have second derivatives positive simultaneously
                     in all directions. When the number of hidden units is large, it seems implausible for none of
                     them to offer a descent direction. Although this paper does not prove or disprove the above
                     conjecture, in trying to do so we found an interesting characterization of the optimization
                     problem for NNs as a convex program if the output loss function is convex in the NN out-
                     put and if the output layer weights are regularized by a convex penalty. More specifically,
                     if the regularization is the L1 norm of the output layer weights, then we show that a “rea-
                     sonable” solution exists, involving a finite number of hidden units (no more than the number
                     of examples, and in practice typically much less). We present a theoretical algorithm that
                     is reminiscent of Column Generation (Chvatal, 1983), in which hidden neurons are inserted ´
                     one at a time. Each insertion requires solving a weighted classification problem, very much
                     like in Boosting (Freund and Schapire, 1997) and in particular Gradient Boosting (Mason
                     et al., 2000; Friedman, 2001).
                     Neural Networks, Gradient Boosting, and Column Generation
                     Denote x~2Rd+1 the extension of vector x2Rd with one element with value 1. What
                     we call “Neural Network” (NN) here is a predictor for supervised learning of the form 
                     <<FORMULA>> where x is an input vector, <<h_i(x)>> is obtained from a linear dis-
                     criminant function hi <<FORMULA>> with e.g. <<s(a) = sign(a)>>, or <<s(a) = tanh(a)>> or
                     <<s(a) =  1>>. A learning algorithm must specify how to select m, the <<FORMULA>>                                        
                     i ’s and the vi ’s.                 

                     The classical solution (Rumelhart, Hinton and Williams, 1986) involves (a) selecting a loss
                     function Q(^y;y)that specifies how to penalize for mismatches between y^(x)and the ob-
                     served y’s (target output or target class), (b) optionally selecting a regularization penalty that
                     favors “small” parameters, and (c) choosing a method to approximately minimize the sum of
                     the losses on the training data D=f(x1 ;y 1 );:::;(xn ;y n )gplus the regularization penalty.
                     Note that in this formulation, an output non-linearity can still be used, by inserting it in the
                     loss function Q. Examples of such loss functions are the quadratic loss jjy^yjj 2 , the hinge
                     loss <<FORMULA>> (used in SVMs), the cross-entropy loss <<FORMULA>>
                     (used in logistic regression), and the exponential loss <<FORMULA>> (used in Boosting).
                     Gradient Boosting has been introduced in (Friedman, 2001) and (Mason et al., 2000) as a
                     non-parametric greedy-stagewise supervised learning algorithm in which one adds a function
                     at a time to the current solution <<y^(x)>>, in a steepest-descent fashion, to form an additive model
                     as above but with the functions hi typically taken in other kinds of sets of functions, such as
                     those obtained with decision trees. In a stagewise approach, when the (m+1)-th basis <<FORMULA>> is added, 
                     only <<w_m+1>> is optimized (by a line search), like in matching pursuit algorithms. Such
                     a greedy-stagewise approach is also at the basis of Boosting algorithms (Freund and Schapire,
                     1997), which is usually applied using decision trees as bases and Qthe exponential loss.
                     It may be difficult to minimize exactly for wm+1 and hm+1 when the previous bases and
                     weights are fixed, so (Friedman, 2001) proposes to “follow the gradient” in function space,
                     i.e., look for a base learner hm+1 that is best correlated with the gradient of the average
                     loss on the <<FORMULA>> (that would be the residue <<FORMULA>> in the case of the square loss). The
                     algorithm analyzed here also involves maximizing the correlation between Q0 (the derivative
                     of Q with respect to its first argument, evaluated on the training predictions) and the next
                     basis hm+1 . However, we follow a “stepwise”, less greedy, approach, in which all the output
                     weights are optimized at each step, in order to obtain convergence guarantees.
                     Our approach adapts the Column Generation principle (Chvatal, 1983), a decomposition´
                     technique initially proposed for solving linear programs with many variables and few con-
                     straints. In this framework, active variables, or “columns”, are only generated as they are
                     required to decrease the objective. In several implementations, the column-generation sub-
                     problem is frequently a combinatorial problem for which efficient algorithms are available.
                     In our case, the subproblem corresponds to determining an “optimal” linear classifier.

                     2 Core Ideas
                     Informally, consider the set Hof all possible hidden unit functions (i.e., of all possible hidden
                     unit weight vectors vi ). Imagine a NN that has all the elements in this set as hidden units. We
                     might want to impose precision limitations on those weights to obtain either a countable or
                     even a finite set. For such a NN, we only need to learn the output weights. If we end up with
                     a finite number of non-zero output weights, we will have at the end an ordinary feedforward
                     NN. This can be achieved by using a regularization penalty on the output weights that yields
                     sparse solutions, such as the L1 penalty. If in addition the loss function is convex in the output
                     layer weights (which is the case of squared error, hinge loss, -tube regression loss, and
                     logistic or softmax cross-entropy), then it is easy to show that the overall training criterion
                     is convex in the parameters (which are now only the output weights). The only problem is
                     that there are as many variables in this convex program as there are elements in the set H,
                     which may be very large (possibly infinite). However, we find that with L1 regularization,
                     a finite solution is obtained, and that such a solution can be obtained by greedily inserting
                     one hidden unit at a time. Furthermore, it is theoretically possible to check that the global
                     optimum has been reached.

                     Definition 2.1.Let Hbe a set of functions from an input space X to R. Elements of H
                     can be understood as “hidden units” in a NN. Let Wbe the Hilbert space of functions from
                     Hto R, with an inner product denoted by <<FORMULA>>. An element of W can be
                     understood as the output weights vector in a neural network. Let <<h(x):H -> R>> the function
                     that maps any element <<h_i>> of <<H to h_i(x)>>. <<h(x)>> can be understood as the vector of activations                     
                     of hidden units when input x is observed. Let w2 W represent a parameter(the output
                     weights). The NN prediction is denoted <<FORMULA>>. Let <<Q:R -> RxR>> be a
                     cost function convex in its first argument that takes a scalar prediction y^(x)and a scalar
                     target value y and returns a scalar cost. This is the cost to be minimized on example pair
                     (x;y). Let <<FORMULA>> be the training set. Let <<FORMULA>> be a convex
                     regularization functional that penalizes for the choice of more “complex” parameters (e.g.,
                     <<FORMULA>> according to a 1-norm in W, if His countable). We define the convex NN
                     criterion C(H;Q;;D;w)with parameter was follows: 

                                  <<FORMULA>>          (1)
 
                     The following is a trivial lemma, but it is conceptually very important as it is the basis for the
                     rest of the analysis in this paper.

                     Lemma 2.2.The convex NN cost <<FORMULA>> is a convex function of w.
                     Proof. <<FORMULA>> is convex in w and <<>> is convex in w, by the above construction. C
                     is additive in <<FORMULA>> and additive in . Hence C is convex in w.
                     Note that there are no constraints in this convex optimization program, so that at the global
                     minimum all the partial derivatives of C with respect to elements of w cancel.
                     Let jHj be the cardinality of the set H. If it is not finite, it is not obvious that an optimal
                     solution can be achieved in finitely many iterations.

                     Lemma 2.2 says that training NNs from a very large class (with one or more hidden layer)
                     can be seen as convex optimization problems, usually in a very high dimensional space,as
                     long as we allow the number of hidden units to be selected by the learning algorithm.
                     By choosing a regularizer that promotes sparse solutions, we obtain a solution that has a
                     finite number of “active” hidden units (non-zero entries in the output weights vector w).
                     This assertion is proven below, in theorem 3.1, for the case of the hinge loss.
                     However, even if the solution involves a finite number of active hidden units, the convex
                     optimization problem could still be computationally intractable because of the large number
                     of variables involved. One approach to this problem is to apply the principles already suc-
                     cessfully embedded in Gradient Boosting, but more specifically in Column Generation (an
                     optimization technique for very large scale linear programs), i.e., add one hidden unit at a
                     time in an incremental fashion. The important ingredient here is a way to know that we
                     have reached the global optimum, thus not requiring to actually visit all the possible
                     hidden units.We show that this can be achieved as long as we can solve the sub-problem
                     of finding a linear classifier that minimizes the weighted sum of classification errors. This
                     can be done exactly only on low dimensional data sets but can be well approached using
                     weighted linear SVMs, weighted logistic regression, or Perceptron-type algorithms.
                     Another idea (not followed up here) would be to consider first a smaller set H1 , for which
                     the convex problem can be solved in polynomial time, and whose solution can theoretically
                     be selected as initialization for minimizing the criterion <<FORMULA>>, with <<FORMULA>>,
                     and where H2 may have infinite cardinality (countable or not). In this way we could show
                     that we can find a solution whose cost satisfies <<FORMULA>>,
                     i.e., is at least as good as the solution of a more restricted convex optimization problem. The
                     second minimization can be performed with a local descent algorithm, without the necessity
                     to guarantee that the global optimum will be found.

                     3 Finite Number of Hidden Neurons
                     In this section we consider the special case with <<FORMULA>> the hinge loss,
                     and <<L1>> regularization, and we show that the global optimum of the convex cost involves at
                     most n+ 1 hidden neurons, using an approach already exploited in (Ratsch, Demiriz and¨
                     Bennett, 2002) for L1-loss regression Boosting with L1 regularization of output weights.                                                    Xn
                     The training criterion is <<FORMULA>>. Let us rewrite t=1 this cost function as the 
                     constrained optimization problem: 
                     
                                          <<FORMULA>>      (C1)
                  
                                          <<FORMULA>>      (C2)

                     Using a standard technique, the above program can be recast as a linear program. Defin-
                     ing <<FORMULA>> the vector of Lagrangian multipliers for the constraints C1 , its dual
                     problem (P)takes the form (in the case of a finite number Jof base learners): 
                     
                                          <<FORMULA>>
                          
                     In the case of a finite number Jof base learners, <<FORMULA>>. If
                     the number of hidden units is uncountable, then Iis a closed bounded interval of R.
                     Such an optimization problem satisfies all the conditions needed for using Theorem 4.2
                     from (Hettich and Kortanek, 1993). Indeed:
                     <<FORMULA>> it is compact (as a closed bounded interval of <<FORMULA>> is a concave function 
                     it is even a linear function);
                     <<FORMULA>> is convex in <<>> (it is actually linear in <<>>);
                     <<FORMULA>> (therefore finite) ( (P)is the largest value of F satisfying the constraints);
                      for every set of n+1 points <<FORMULA>>, there exists ~such that <<FORMULA>> for
                     <<FORMULA>> (one can take <<FORMULA>> since K>0).

                     Then, from Theorem 4.2 from (Hettich and Kortanek, 1993), the following theorem holds:
                     Theorem 3.1.The solution of (P) can be attained with constraints C0 and only n+1 constraints C0 
                     (i.e., there exists a subset of n+1 constraints C0 giving rise to the same maximum 1                               
                     as when using the whole set of constraints). Therefore, the primal problem associated is the
                     minimization of the cost function of a NN with n+1 hidden neurons.

                     4 Incremental Convex NN Algorithm
                     In this section we present a stepwise algorithm to optimize a NN, and show that there is a cri-
                     terion that allows to verify whether the global optimum has been reached. This is a specializa-
                     tion of minimizing <<FORMULA>>, with <<FORMULA>> 1 and <<FORMULA>>
                     is the set of soft or hard linear classifiers (depending on choice of s()).

                                        Algorithm ConvexNN( D, Q, , s)

                                                <<ALGORITHM>>
                     
                     Theorem 4.1.AlgorithmConvexNN Pstops when it reaches the global optimum of

                                      <<FORMULA>>.

                     Proof.Let wbe the output weights vector when the algorithm stops. Because the set of
                     hidden units Hwe consider is such that when his in H, h is also in H, we can assume
                     all weights to be non-negative. By contradiction, if w0 6=wis the global optimum, with
                     <<C(w_0) < C(w)>>, then, since Cis convex in the output weights, for any 2(0;1) , we have
                     <<FORMULA>>. For
                      small enough, we can assume all weights in w that are strictly positive to be also strictly
                     positive in w . Let us denote by Ip the set of strictly positive weights in w (and w), by 
                     Iz the set of weights set to zero in w but to a non-zero value in w , and by k the difference
                     w;k wk in the weight of hidden unit hk between wand w . We can assume j < 0 for
                     j2Iz , because instead of setting a small positive weight to hj , one can decrease the weight
                     of hj by the same amount, which will give either the same cost, or possibly a lower one
                     when the weight of <<FORMULA>> is positive. With o() denoting a quantity such that  o()!0
                     when !0, the difference  (w) =XC(w )C(w)can now be written:

                                       <<FORMULA>>

                     since for i2Ip , thanks to step (7) of the algorithm, we have @C (w) = 0 . Thus the @w
                     inequality <<FORMULA>> rewrites into  <<FORMULA>>
                     which, when !0, yields (note that <<FORMULA>> does not depend on !  since j is linear in ):

                                      <<FORMULA>>             (2)

                     i being the optimal classifier chosen in step (5a) or (5c), all hidden units <<FORMULA>> verify <<FORMULA>>

                                       <<FORMULA>>

                     <<FORMULA>> , contradicting eq. 2.

                     (Mason et al., 2000) prove a related global convergence result for the AnyBoost algorithm,
                     a non-parametric Boosting algorithm that is also similar to Gradient Boosting (Friedman,
                     2001). Again, this requires solving as a sub-problem an exact minimization to find a function
                     hi 2 H that is maximally correlated with the gradient Q0 on the output. We now show a
                     simple procedure to select a hyperplane with the best weighted classification error.
                     Exact Minimization                     
                     In step (5a) we are required to find a linear classifier that minimizes the weighted sum of
                     classification errors. Unfortunately, this is an NP-hard problem (w.r.t. d, see theorem 4
                     in (Marcotte and Savard, 1992)). However, an exact solution can be easily found in O(n3 )
                     computations for d= 2 inputs.

                     Proposition 4.2.Finding a linear classifier that minimizes the weighted sum of classification
                     error can be achieved in O(n3 )steps when the input dimension is d= 2 .
                     Proof.We want to <<FORMULA>> with respect to u and b, the c’s being
                     in <<FORMULA>> Consider u fixed and sort the xi ’s according to their dot product with u and denote r
                     the function which maps ito r(i) such that xr(i) is in i-th position in the sort. Depending on P       
                     the value of b, we will have n+1 possible sums, respectively <<FORMULA>>,
                     <<FORMULA>>. It is obvious that those sums only depend on the order of the products <<FORMULA>>,
                     <<FORMULA>>. When u varies smoothly on the unit circle, as the dot product is a continuous
                     function of its arguments, the changes in the order of the dot products will occur only when
                     there is a pair (i,j) such that <<FORMULA>>. Therefore, there are at most as many order
                     changes as there are pairs of different points, i.e., <<FORMULA>>. In the case of d=2, we
                     can enumerate all the different angles for which there is a change, namely a1 ;:::;a z with
                     <<FORMULA>>. We then need to test at least one <<FORMULA>> for each interval a2                                                    i <
                     <<FORMULA>>, and also one u for <<FORMULA>>, which makes a total of <<FORMULA>> possibilities. 2
                     It is possible to generalize this result in higher dimensions, and as shown in (Marcotte and
                     Savard, 1992), one can achieve <<O(log(n)nd)>> time.

                     Algorithm 1 Optimal linear classifier search
                    
                                       <<ALGORITHM>>

                     Approximate Minimization

                     For data in higher dimensions, the exact minimization scheme to find the optimal linear
                     classifier is not practical. Therefore it is interesting to consider approximate schemes for
                     obtaining a linear classifier with weighted costs. Popular schemes for doing so are the linear
                     SVM (i.e., linear classifier with hinge loss), the logistic regression classifier, and variants of
                     the Perceptron algorithm. In that case, step (5c) of the algorithm is not an exact minimization,
                     and one cannot guarantee that the global optimum will be reached. However, it might be
                     reasonable to believe that finding a linear classifier by minimizing a weighted hinge loss
                     should yield solutions close to the exact minimization. Unfortunately, this is not generally
                     true, as we have found out on a simple toy data set described below. On the other hand,
                     if in step (7) one performs an optimization not only of the output weights wj (ji) but
                     also of the corresponding weight vectors vj , then the algorithm finds a solution close to the
                     global optimum (we could only verify this on 2-D data sets, where the exact solution can be
                     computed easily). It means that at the end of each stage, one first performs a few training
                     iterations of the whole NN (for the hidden units ji) with an ordinary gradient descent
                     mechanism (we used conjugate gradients but stochastic gradient descent would work too),
                     optimizing the wj ’s and the vj ’s, and then one fixes the vj ’s and obtains the optimal wj ’s for
                     these vj ’s (using a convex optimization procedure). In our experiments we used a quadratic                     
                     Q, for which the optimization of the output weights can be done with a neural network, using
                     the outputs of the hidden layer as inputs.

                     Let us consider now a bit more carefully what it means to tune the v_j’s in step (7). Indeed,
                     changing the weight vector vj of a selected hidden neuron to decrease the cost is equivalent
                     to a change in the output weights w’s. More precisely, consider the step in which the
                     value of vj becomes v0 . This is equivalent to the following operation on the w’s, when wj                                            j is the corresponding output weight value: the output weight associated with the value vj of
                     a hidden neuron is set to 0, and the output weight associated with the value v0 of a hidden j 
                     neuron is set to wj . This corresponds to an exchange between two variables in the convex
                     program. We are justified to take any such step as long as it allows us to decrease the cost
                     C(w). The fact that we are simultaneously making such exchanges on all the hidden units
                     when we tune the vj ’s allows us to move faster towards the global optimum.
                     Extension to multiple outputs
                     The multiple outputs case is more involved than the single-output case because it is not P 
                     enough to check the condition <<FORMULA>>. Consider a new hidden neuron whose output is
                     hi when the input is xi . Let us also denote <<FORMULA>> the vector of output weights
                     between the new hidden neuron and the <<FORMULA>> output neurons. The gradient with respect to j
                     is <<FORMULA>> with <<FORMULA>> the value of the j-th output neuron with input <<FORMULA>>. 
                     This means that if, for a given j, we have <<FORMULA>>, moving Pj away from 0 can
                     only increase the cost. Therefore, the right quantity to consider is <<FORMULA>>.
                     We must therefore find <<FORMULA>>. As before, this sub-problem is not + convex, but it is not 
                     as obvious how to approximate it by a convex problem. The stopping P criterion becomes: if there is no j 
                     such that <<FORMULA>>, then all weights must remain equal to 0 and a global minimum is reached.

                     Experimental Results
                     We performed experiments on the 2-D double moon toy dataset (as used in (Delalleau, Ben-
                     gio and Le Roux, 2005)), to be able to compare with the exact version of the algorithm. In
                     these experiments, <<FORMULA>>. The set-up is the following:

                     Select a new linear classifier, either (a) the optimal one or (b) an approximate using logistic
                     regression.
                     Optimize the output weights using a convex optimizer.
                     In case (b), tune both input and output weights by conjugate gradient descent on Cand
                     finally re-optimize the output weights using LASSO regression.
                     Optionally, remove neurons whose output weight has been set to 0.
                     Using the approximate algorithm yielded for 100 training examples an average penalized
                     ( = 1 ) squared error of 17.11 (over 10 runs), an average test classification error of 3.68%
                     and an average number of neurons of 5.5 . The exact algorithm yielded a penalized squared
                     error of 8.09, an average test classification error of 5.3%, and required 3 hidden neurons. A
                     penalty of = 1 was nearly optimal for the exact algorithm whereas a smaller penalty further
                     improved the test classification error of the approximate algorithm. Besides, when running
                     the approximate algorithm for a long time, it converges to a solution whose quadratic error is
                     extremely close to the one of the exact algorithm.

                     5 Conclusion
                     We have shown that training a NN can be seen as a convex optimization problem, and have
                     analyzed an algorithm that can exactly or approximately solve this problem. We have shown
                     that the solution with the hinge loss involved a number of non-zero weights bounded by
                     the number of examples, and much smaller in practice. We have shown that there exists a
                     stopping criterion to verify if the global optimum has been reached, but it involves solving a
                     sub-learning problem involving a linear classifier with weighted errors, which can be computationally 
                     hard if the exact solution is sought, but can be easily implemented for toy data
                     sets (in low dimension), for comparing exact and approximate solutions.
                     The above experimental results are in agreement with our initial conjecture: when there are
                     many hidden units we are much less likely to stall in the optimization procedure, because
                     there are many more ways to descend on the convex cost C(w). They also suggest, based
                     on experiments in which we can compare with the exact sub-problem minimization, that
                     applying Algorithm ConvexNN with an approximate minimization for adding each hidden
                     unit while continuing to tune the previous hidden unit s tends to lead to fast convergence
                     to the global minimum. What can get us stuck in a “local minimum” (in the traditional sense,
                     i.e., of optimizing w’s and v’s together) is simply the inability to find a new hidden unit
                     weight vector that can improve the total cost (fit and regularization term) even if there
                     exists one.

                     Note that as a side-effect of the results presented here, we have a simple way to train P neural
                     networks with hard-threshold hidden units, since increasing <<FORMULA>> can be either achieved 
                     exactly (at great price) or approximately (e.g. by using a cross-entropy
                     or hinge loss on the corresponding linear classifier).

                     Acknowledgments

                     The authors thank the following for support: NSERC, MITACS, and the Canada Research
                     Chairs. They are also grateful for the feedback and stimulating exchanges with Sam Roweis,
                     Nathan Srebro, and Aaron Courville.

                     References

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                        Operations Research (Theory), 36:517–545.
                     Mason, L., Baxter, J., Bartlett, P. L., and Frean, M. (2000). Boosting algorithms as gradient descent.
                        InAdvances in Neural Information Processing Systems 12, pages 512–518.
                     Ratsch, G., Demiriz, A., and Bennett, K. P. (2002). Sparse regression ensembles in infinite and finite¨
                        hypothesis spaces.Machine Learning.
                     Rumelhart, D., Hinton, G., and Williams, R. (1986). Learning representations by back-propagating
                        errors.Nature, 323:533–536
<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>                  
                  DEEP COMPRESSION: COMPRESSING DEEP NEURAL
                  NETWORKS WITH PRUNING , T RAINED QUANTIZATION
                 AND HUFFMAN CODING


                  Song Han
                  Stanford University, Stanford, CA 94305, USA
                  songhan@stanford.edu

                  Huizi Mao
                  Tsinghua University, Beijing, 100084, China
                  mhz12@mails.tsinghua.edu.cn

                  William J. Dally
                  Stanford University, Stanford, CA 94305, USA
                  NVIDIA, Santa Clara, CA 95050, USA
                  dally@stanford.edu



                                              ABSTRACT

                       Neural networks are both computationally intensive and memory intensive, making
                       them difficult to deploy on embedded systems with limited hardware resources. To
                       address this limitation, we introduce “deep compression”, a three stage pipeline:
                       pruning, trained quantization and Huffman coding, that work together to reduce
                       the storage requirement of neural networks by 35% to 49% without affecting their
                       accuracy. Our method first prunes the network by learning only the important
                       connections. Next, we quantize the weights to enforce weight sharing, finally, we
                       apply Huffman coding. After the first two steps we retrain the network to fine
                       tune the remaining connections and the quantized centroids. Pruning, reduces the
                       number of connections by 9% to 13%; Quantization then reduces the number of
                       bits that represent each connection from 32 to 5. On the ImageNet dataset, our
                       method reduced the storage required by AlexNet by 35%, from 240MB to 6.9MB,
                       without loss of accuracy. Our method reduced the size of VGG-16 by 49% from
                       552MB to 11.3MB, again with no loss of accuracy. This allows fitting the model
                       into on-chip SRAM cache rather than off-chip DRAM memory. Our compression
                       method also facilitates the use of complex neural networks in mobile applications
                       where application size and download bandwidth are constrained. Benchmarked on
                       CPU, GPU and mobile GPU, compressed network has 3% to 4% layerwise speedup
                       and 3% to 7% better energy efficiency.


                  1 INTRODUCTION

                 Deep neural networks have evolved to the state-of-the-art technique for computer vision tasks
                 (Krizhevsky et al., 2012)(Simonyan & Zisserman, 2014). Though these neural networks are very
                 powerful, the large number of weights consumes considerable storage and memory bandwidth. For
                 example, the AlexNet Caffemodel is over 200MB, and the VGG-16 Caffemodel is over 500MB
                 (BVLC). This makes it difficult to deploy deep neural networks on mobile system.
                 First, for many mobile-first companies such as Baidu and Facebook, various apps are updated via
                 different app stores, and they are very sensitive to the size of the binary files. For example, App
                 Store has the restriction “apps above 100 MB will not download until you connect to Wi-Fi”. As a
                 result, a feature that increases the binary size by 100MB will receive much more scrutiny than one
                 that increases it by 10MB. Although having deep neural networks running on mobile has many great

                                                 <<FIGURE>>

                 Figure 1: The three stage compression pipeline: pruning, quantization and Huffman coding. Pruning
                 reduces the number of weights by10%, while quantization further improves the compression rate:
                 between27%and31%. Huffman coding gives more compression: between35%and49%. The
                 compression rate already included the meta-data for sparse representation. The compression scheme
                 doesn’t incur any accuracy loss.

                 features such as better privacy, less network bandwidth and real time processing, the large storage
                 overhead prevents deep neural networks from being incorporated into mobile apps.
                 The second issue is energy consumption. Running large neural networks require a lot of memory
                 bandwidth to fetch the weights and a lot of computation to do dot products— which in turn consumes
                 considerable energy. Mobile devices are battery constrained, making power hungry applications such
                 as deep neural networks hard to deploy.
                 Energy consumption is dominated by memory access. Under 45nm CMOS technology, a 32 bit
                 floating point add consumes 0.9PJ, a 32bit SRAM cache access takes 5PJ, while a 32bit DRAM
                 memory access takes 640PJ, which is 3 orders of magnitude of an add operation. Large networks
                 do not fit in on-chip storage and hence require the more costly DRAM accesses. Running a 1 billion
                 connection neural network, for example, at 20fps would require (20Hz)(1G)(640PJ) = 12.8W just
                 for DRAM access - well beyond the power envelope of a typical mobile device.
                 Our goal is to reduce the storage and energy required to run inference on such large networks so they
                 can be deployed on mobile devices. To achieve this goal, we present “deep compression”: a three-
                 stage pipeline (Figure 1) to reduce the storage required by neural network in a manner that preserves
                 the original accuracy. First, we prune the networking by removing the redundant connections, keeping
                 only the most informative connections. Next, the weights are quantized so that multiple connections
                 share the same weight, thus only the codebook (effective weights) and the indices need to be stored.
                 Finally, we apply Huffman coding to take advantage of the biased distribution of effective weights.
                 Our main insight is that, pruning and trained quantization are able to compress the network without
                 interfering each other, thus lead to surprisingly high compression rate. It makes the required storage
                 so small (a few megabytes) that all weights can be cached on chip instead of going to off-chip DRAM
                 which is energy consuming. Based on “deep compression”, the EIE hardware accelerator Han et al.
                 (2016) was later proposed that works on the compressed model, achieving significant speedup and
                 energy efficiency improvement.

                  2 NETWORK PRUNING

                 Network pruning has been widely studied to compress CNN models. In early work, network pruning
                 proved to be a valid way to reduce the network complexity and over-fitting (LeCun et al., 1989;
                 Hanson & Pratt, 1989; Hassibi et al., 1993; Strom, 1997). Recently Han et al. (2015) pruned state- ¨
                 of-the-art CNN models with no loss of accuracy. We build on top of that approach. As shown on
                 the left side of Figure 1, we start by learning the connectivity via normal network training. Next, we
                 prune the small-weight connections: all connections with weights below a threshold are removed
                 from the network. Finally, we retrain the network to learn the final weights for the remaining sparse
                 connections. Pruning reduced the number of parameters by9%and13%for AlexNet and VGG-16
                 model.

                                                <<FIGURE>>

                 Figure 2: Representing the matrix sparsity with relative index. Padding filler zero to prevent overflow.

                                                <<FIGURE>>

                     Figure 3: Weight sharing by scalar quantization (top) and centroids fine-tuning (bottom).


                 We store the sparse structure that results from pruning using compressed sparse row (CSR) or
                 compressed sparse column (CSC) format, which requires2a+n+1numbers, where a is the number
                 of non-zero elements and n is the number of rows or columns.
                 To compress further, we store the index difference instead of the absolute position, and encode this
                 difference in 8 bits for conv layer and 5 bits for fc layer. When we need an index difference larger
                 than the bound, we the zero padding solution shown in Figure 2: in case when the difference exceeds
                 8, the largest 3-bit (as an example) unsigned number, we add a filler zero.

                  3 TRAINED QUANTIZATION AND WEIGHT SHARING

                 Network quantization and weight sharing further compresses the pruned network by reducing the
                 number of bits required to represent each weight. We limit the number of effective weights we need to
                 store by having multiple connections share the same weight, and then fine-tune those shared weights.
                 Weight sharing is illustrated in Figure 3. Suppose we have a layer that has 4 input neurons and 4
                 output neurons, the weight is a 4x4 matrix. On the top left is the 4x4 weight matrix, and on the
                 bottom left is the 4x4 gradient matrix. The weights are quantized to 4 bins (denoted with 4 colors),
                 all the weights in the same bin share the same value, thus for each weight, we then need to store only
                 a small index into a table of shared weights. During update, all the gradients are grouped by the color
                 and summed together, multiplied by the learning rate and subtracted from the shared centroids from
                 last iteration. For pruned AlexNet, we are able to quantize to 8-bits (256 shared weights) for each
                 CONV layers, and 5-bits (32 shared weights) for each FC layer without any loss of accuracy.
                 To calculate the compression rate, given k clusters, we only need log_2(k) bits to encode the index. In
                 general, for a network with n connections and each connection is represented with b bits, constraining
                 the connections to have only k shared weights will result in a compression rate of:

                                                  <<FORMULA>>                                   (1)

                 For example, Figure 3 shows the weights of a single layer neural network with four input units and
                 four output units. There are4%4 = 16weights originally but there are only4shared weights: similar
                 weights are grouped together to share the same value. Originally we need to store 16 weights each

                                                    <<FIGURE>>                 

                 Figure 4: Left: Three different methods for centroids initialization. Right: Distribution of weights
                 (blue) and distribution of codebook before (green cross) and after fine-tuning (red dot).


                 has 32 bits, now we need to store only 4 effective weights (blue, green, red and orange), each has 32
                 bits, together with 16 2-bit indices giving a compression rate of <<FORMULA>>

                  3.1 WEIGHT SHARING

                 We use k-means clustering to identify the shared weights for each layer of a trained network, so that
                 all the weights that fall into the same cluster will share the same weight. Weights are not shared across
                 layers. We partition n original weights <<FORMULA>> into k clusters <<FORMULA>>,
                 n%k, so as to minimize the within-cluster sum of squares (WCSS):

                                               <<FORMULA>>                      (2)

                 Different from HashNet (Chen et al., 2015) where weight sharing is determined by a hash function
                 before the networks sees any training data, our method determines weight sharing after a network is
                 fully trained, so that the shared weights approximate the original network.

                  3.2 INITIALIZATION OF SHARED WEIGHTS

                 Centroid initialization impacts the quality of clustering and thus affects the network’s prediction
                 accuracy. We examine three initialization methods: Forgy(random), density-based, and linear
                 initialization. In Figure 4 we plotted the original weights’ distribution of conv3 layer in AlexNet
                 (CDF in blue, PDF in red). The weights forms a bimodal distribution after network pruning. On the
                 bottom it plots the effective weights (centroids) with 3 different initialization methods (shown in blue,
                 red and yellow). In this example, there are 13 clusters.
                 Forgy(random) initialization randomly chooses k observations from the data set and uses these as
                 the initial centroids. The initialized centroids are shown in yellow. Since there are two peaks in the
                 bimodal distribution, Forgy method tend to concentrate around those two peaks.
                 Density-based initialization linearly spaces the CDF of the weights in the y-axis, then finds the
                 horizontal intersection with the CDF, and finally finds the vertical intersection on the x-axis, which
                 becomes a centroid, as shown in blue dots. This method makes the centroids denser around the two
                 peaks, but more scatted than the Forgy method.
                 Linear initialization linearly spaces the centroids between the [min, max] of the original weights.
                 This initialization method is invariant to the distribution of the weights and is the most scattered
                 compared with the former two methods.
                 Larger weights play a more important role than smaller weights (Han et al., 2015), but there are fewer
                 of these large weights. Thus for both Forgy initialization and density-based initialization, very few
                 centroids have large absolute value which results in poor representation of these few large weights.
                 Linear initialization does not suffer from this problem. The experiment section compares the accuracy

                                                  <<FIGURE>>

                      Figure 5: Distribution for weight (Left) and index (Right). The distribution is biased.
                 of different initialization methods after clustering and fine-tuning, showing that linear initialization
                 works best.

                  3.3 FEED-FORWARD AND BACK-PROPAGATION

                 The centroids of the one-dimensional k-means clustering are the shared weights. There is one level
                 of indirection during feed forward phase and back-propagation phase looking up the weight table.
                 An index into the shared weight table is stored for each connection. During back-propagation, the
                 gradient for each shared weight is calculated and used to update the shared weight. This procedure is
                 shown in Figure 3.
                 We denote the loss byL, the weight in the ith column and jth row by Wij, the centroid index of
                 element Wij by Iij, the kth centroid of the layer by Ck. By using the indicator function <<1(.)>>, the
                 gradient of the centroids is calculated as:

                                                   <<FORMULA>>               (3)
 
                  4 HUFFMAN CODING

                 A Huffman code is an optimal prefix code commonly used for lossless data compression(Van Leeuwen,
                 1976). It uses variable-length codewords to encode source symbols. The table is derived from the
                 occurrence probability for each symbol. More common symbols are represented with fewer bits.
                 Figure 5 shows the probability distribution of quantized weights and the sparse matrix index of the
                 last fully connected layer in AlexNet. Both distributions are biased: most of the quantized weights are
                 distributed around the two peaks; the sparse matrix index difference are rarely above 20. Experiments
                 show that Huffman coding these non-uniformly distributed values saves 20% to 30% of network
                 storage.

                  5 EXPERIMENTS

                 We pruned, quantized, and Huffman encoded four networks: two on MNIST and two on ImageNet
                 data-sets. The network parameters and accuracy- 1 before and after pruning are shown in Table 1. The
                 compression pipeline saves network storage by 35% to 49% across different networks without loss
                 of accuracy. The total size of AlexNet decreased from 240MB to 6.9MB, which is small enough to
                 be put into on-chip SRAM, eliminating the need to store the model in energy-consuming DRAM
                 memory.

                 Training is performed with the Caffe framework (Jia et al., 2014). Pruning is implemented by adding
                 a mask to the blobs to mask out the update of the pruned connections. Quantization and weight
                 sharing are implemented by maintaining a codebook structure that stores the shared weight, and
                 group-by-index after calculating the gradient of each layer. Each shared weight is updated with all
                 the gradients that fall into that bucket. Huffman coding doesn’t require training and is implemented
                 offline after all the fine-tuning is finished.

                  5.1 LE NET-300-100 AND LE NET-5 ON MNIST

                 We first experimented on MNIST dataset with LeNet-300-100 and LeNet-5 network (LeCun et al.,
                 1998). LeNet-300-100 is a fully connected network with two hidden layers, with 300 and 100
                    1 Reference model is from Caffe model zoo, accuracy is measured without data augmentation

                 Table 1: The compression pipeline can save35%to49%parameter storage with no loss of accuracy.

                                                <<TABLE>>

                 Table 2: Compression statistics for LeNet-300-100. P: pruning, Q:quantization, H:Huffman coding.

                                                <<TABLE>>

                 Table 3: Compression statistics for LeNet-5. P: pruning, Q:quantization, H:Huffman coding.

                                                <<TABLE>>

                 neurons each, which achieves 1.6% error rate on Mnist. LeNet-5 is a convolutional network that
                 has two convolutional layers and two fully connected layers, which achieves 0.8% error rate on
                 Mnist. Table 2 and table 3 show the statistics of the compression pipeline. The compression rate
                 includes the overhead of the codebook and sparse indexes. Most of the saving comes from pruning
                 and quantization (compressed 32%), while Huffman coding gives a marginal gain (compressed 40%)

                  5.2 ALEX NET ON IMAGE NET

                 We further examine the performance of Deep Compression on the ImageNet ILSVRC-2012 dataset,
                 which has 1.2M training examples and 50k validation examples. We use the AlexNet Caffe model as
                 the reference model, which has 61 million parameters and achieved a top-1 accuracy of 57.2% and a
                 top-5 accuracy of 80.3%. Table 4 shows that AlexNet can be compressed to2:88%of its original size
                 without impacting accuracy. There are 256 shared weights in each CONV layer, which are encoded
                 with 8 bits, and 32 shared weights in each FC layer, which are encoded with only 5 bits. The relative
                 sparse index is encoded with 4 bits. Huffman coding compressed additional 22%, resulting in 35%
                 compression in total.

                  5.3 VGG-16 ON IMAGE NET

                 With promising results on AlexNet, we also looked at a larger, more recent network, VGG-16 (Si-
                 monyan & Zisserman, 2014), on the same ILSVRC-2012 dataset. VGG-16 has far more convolutional
                 layers but still only three fully-connected layers. Following a similar methodology, we aggressively
                 compressed both convolutional and fully-connected layers to realize a significant reduction in the
                 number of effective weights, shown in Table5.
                 The VGG16 network as a whole has been compressed by49%. Weights in the CONV layers are
                 represented with 8 bits, and FC layers use 5 bits, which does not impact the accuracy. The two largest
                 fully-connected layers can each be pruned to less than 1.6% of their original size. This reduction

                    Table 4: Compression statistics for AlexNet. P: pruning, Q: quantization, H:Huffman coding.

                                                   <<TABLE>>

                    Table 5: Compression statistics for VGG-16. P: pruning, Q:quantization, H:Huffman coding.

                                                    <<TABLE>>

                 is critical for real time image processing, where there is little reuse of these layers across images
                 (unlike batch processing). This is also critical for fast object detection algorithms where one CONV
                 pass is used by many FC passes. The reduced layers will fit in an on-chip SRAM and have modest
                 bandwidth requirements. Without the reduction, the bandwidth requirements are prohibitive.

                  6 DISCUSSIONS

                  6.1 PRUNING AND QUANTIZATION WORKING TOGETHER

                 Figure 6 shows the accuracy at different compression rates for pruning and quantization together
                 or individually. When working individually, as shown in the purple and yellow lines, accuracy of
                 pruned network begins to drop significantly when compressed below 8% of its original size; accuracy
                 of quantized network also begins to drop significantly when compressed below 8% of its original
                 size. But when combined, as shown in the red line, the network can be compressed to 3% of original
                 size with no loss of accuracy. On the far right side compared the result of SVD, which is inexpensive
                 but has a poor compression rate.
                 The three plots in Figure 7 show how accuracy drops with fewer bits per connection for CONV layers
                 (left), FC layers (middle) and all layers (right). Each plot reports both top-1 and top-5 accuracy.
                 Dashed lines only applied quantization but without pruning; solid lines did both quantization and
                 pruning. There is very little difference between the two. This shows that pruning works well with
                 quantization.
                 Quantization works well on pruned network because unpruned AlexNet has 60 million weights to
                 quantize, while pruned AlexNet has only 6.7 million weights to quantize. Given the same amount of
                 centroids, the latter has less error.

                                              <<FIGURE>>

                 Figure 6: Accuracy v.s. compression rate under different compression methods. Pruning and
                 quantization works best when combined.

                                              <<FIGURE>>

                 Figure 7: Pruning doesn’t hurt quantization. Dashed: quantization on unpruned network. Solid:
                 quantization on pruned network; Accuracy begins to drop at the same number of quantization bits
                 whether or not the network has been pruned. Although pruning made the number of parameters less,
                 quantization still works well, or even better(3 bits case on the left figure) as in the unpruned network.

                                                <<FIGURE>>

                 Figure 8: Accuracy of different initialization methods. Left: top-1 accuracy. Right: top-5 accuracy.
                 Linear initialization gives best result.

                 The first two plots in Figure 7 show that CONV layers require more bits of precision than FC layers.
                 For CONV layers, accuracy drops significantly below 4 bits, while FC layer is more robust: not until
                 2 bits did the accuracy drop significantly.


                  6.2 CENTROID INITIALIZATION

                 Figure 8 compares the accuracy of the three different initialization methods with respect to top-1
                 accuracy (Left) and top-5 accuracy (Right). The network is quantized to2%8bits as shown on
                 x-axis. Linear initialization outperforms the density initialization and random initialization in all
                 cases except at 3 bits.
                 The initial centroids of linear initialization spread equally across the x-axis, from the min value to the
                 max value. That helps to maintain the large weights as the large weights play a more important role
                 than smaller ones, which is also shown in network pruning Han et al. (2015). Neither random nor
                 density-based initialization retains large centroids. With these initialization methods, large weights are
                 clustered to the small centroids because there are few large weights. In contrast, linear initialization
                 allows large weights a better chance to form a large centroid.

                                            <<FIGURE>>

                 Figure 9: Compared with the original network, pruned network layer achieved 3% speedup on CPU,
                 3.5% on GPU and 4.2% on mobile GPU on average. Batch size = 1 targeting real time processing.
                 Performance number normalized to CPU.

                                            <<FIGURE>>

                 Figure 10: Compared with the original network, pruned network layer takes 7% less energy on CPU,
                 3.3% less on GPU and 4.2% less on mobile GPU on average. Batch size = 1 targeting real time
                 processing. Energy number normalized to CPU.

                  6.3 SPEEDUP AND ENERGY EFFICIENCY

                 Deep Compression is targeting extremely latency-focused applications running on mobile, which
                 requires real-time inference, such as pedestrian detection on an embedded processor inside an
                 autonomous vehicle. Waiting for a batch to assemble significantly adds latency. So when bench-
                 marking the performance and energy efficiency, we consider the case when batch size = 1. The cases
                 of batching are given in Appendix A.
                 Fully connected layer dominates the model size (more than90%) and got compressed the most by
                 Deep Compression (96%weights pruned in VGG-16). In state-of-the-art object detection algorithms
                 such as fast R-CNN (Girshick, 2015), up to 38% computation time is consumed on FC layers on
                 uncompressed model. So it’s interesting to benchmark on FC layers, to see the effect of Deep
                 Compression on performance and energy. Thus we setup our benchmark on FC6, FC7, FC8 layers of
                 AlexNet and VGG-16. In the non-batched case, the activation matrix is a vector with just one column,
                 so the computation boils down to dense / sparse matrix-vector multiplication for original / pruned
                 model, respectively. Since current BLAS library on CPU and GPU doesn’t support indirect look-up
                 and relative indexing, we didn’t benchmark the quantized model.
                 We compare three different off-the-shelf hardware: the NVIDIA GeForce GTX Titan X and the Intel
                 Core i7 5930K as desktop processors (same package as NVIDIA Digits Dev Box) and NVIDIA Tegra
                 K1 as mobile processor. To run the benchmark on GPU, we used cuBLAS GEMV for the original
                 dense layer. For the pruned sparse layer, we stored the sparse matrix in in CSR format, and used
                 cuSPARSE CSRMV kernel, which is optimized for sparse matrix-vector multiplication on GPU. To
                 run the benchmark on CPU, we used MKL CBLAS GEMV for the original dense model and MKL
                 SPBLAS CSRMV for the pruned sparse model.

                 To compare power consumption between different systems, it is important to measure power at a
                 consistent manner (NVIDIA, b). For our analysis, we are comparing pre-regulation power of the
                 entire application processor (AP) / SOC and DRAM combined. On CPU, the benchmark is running on
                 single socket with a single Haswell-E class Core i7-5930K processor. CPU socket and DRAM power
                 are as reported by the pcm-power utility provided by Intel. For GPU, we used nvidia-smi
                 utility to report the power of Titan X. For mobile GPU, we use a Jetson TK1 development board and
                 measured the total power consumption with a power-meter. We assume 15% AC to DC conversion
                 loss,85% regulator efficiency and 15% power consumed by peripheral components (NVIDIA, a) to
                 report the AP+DRAM power for Tegra K1.

                 Table 6: Accuracy of AlexNet with different aggressiveness of weight sharing and quantization. 8/5
                 bit quantization has no loss of accuracy; 8/4 bit quantization, which is more hardware friendly, has
                 negligible loss of accuracy of 0.01%; To be really aggressive, 4/2 bit quantization resulted in 1.99%
                 and 2.60% loss of accuracy.

                                                <<TABLE>>

                 The ratio of memory access over computation characteristic with and without batching is different.
                 When the input activations are batched to a matrix the computation becomes matrix-matrix multipli-
                 cation, where locality can be improved by blocking. Matrix could be blocked to fit in caches and
                 reused efficiently. In this case, the amount of memory access isO(n2 ), and that of computation is
                 O(n3 ), the ratio between memory access and computation is in the order of1=n.
                 In real time processing when batching is not allowed, the input activation is a single vector and the
                 computation is matrix-vector multiplication. In this case, the amount of memory access isO(n2 ), and
                 the computation isO(n2 ), memory access and computation are of the same magnitude (as opposed
                 to1=n). That indicates MV is more memory-bounded than MM. So reducing the memory footprint
                 is critical for the non-batching case.

                 Figure 9 illustrates the speedup of pruning on different hardware. There are 6 columns for each
                 benchmark, showing the computation time of CPU / GPU / TK1 on dense / pruned network. Time is
                 normalized to CPU. When batch size = 1, pruned network layer obtained 3% to 4% speedup over the
                 dense network on average because it has smaller memory footprint and alleviates the data transferring
                 overhead, especially for large matrices that are unable to fit into the caches. For example VGG16’s
                 FC6 layer, the largest layer in our experiment, contains 400MB data, which is far from the capacity of L3 cache.

                 In those latency-tolerating applications, batching improves memory locality, where weights could
                 be blocked and reused in matrix-matrix multiplication. In this scenario, pruned network no longer
                 shows its advantage. We give detailed timing results in Appendix A.

                 Figure 10 illustrates the energy efficiency of pruning on different hardware. We multiply power
                 consumption with computation time to get energy consumption, then normalized to CPU to get
                 energy efficiency. When batch size = 1, pruned network layer consumes 3% to 7% less energy over
                 the dense network on average. Reported by nvidia-smi, GPU utilization is 99% for both dense
                 and sparse cases.

                  6.4 RATIO OF WEIGHTS, INDEX AND CODEBOOK

                 Pruning makes the weight matrix sparse, so extra space is needed to store the indexes of non-zero
                 elements. Quantization adds storage for a codebook. The experiment section has already included
                 these two factors. Figure 11 shows the breakdown of three different components when quantizing
                 four networks. Since on average both the weights and the sparse indexes are encoded with 5 bits,
                 their storage is roughly half and half. The overhead of codebook is very small and often negligible.

                                                <<FIGURE>>

                                Figure 11: Storage ratio of weight, index and codebook.

                 Table 7: Comparison with other compression methods on AlexNet. (Collins & Kohli, 2014) reduced
                 the parameters by 4% and with inferior accuracy. Deep Fried Conv nets(Yang et al., 2014) worked
                 on fully connected layers and reduced the parameters by less than 4%. SVD save parameters but
                 suffers from large accuracy loss as much as 2%. Network pruning (Han et al., 2015) reduced the
                 parameters by 9%, not including index overhead. On other networks similar to AlexNet, (Denton
                 et al., 2014) exploited linear structure of conv nets and compressed the network by 2.4% to 13.4%
                 layer wise, with 0.9% accuracy loss on compressing a single layer. (Gong et al., 2014) experimented
                 with vector quantization and compressed the network by 16% to 24%, incurring 1% accuracy loss.

                                                     <<TABLE>>

                  7 RELATED WORK

                 Neural networks are typically over-parametrized, and there is significant redundancy for deep learning
                 models(Denil et al., 2013). This results in a waste of both computation and memory usage. There
                 have been various proposals to remove the redundancy: Vanhoucke et al. (2011) explored a fixed-
                 point implementation with 8-bit integer (vs 32-bit floating point) activations. Hwang & Sung
                 (2014) proposed an optimization method for the fixed-point network with ternary weights and 3-bit
                 activations. Anwar et al. (2015) quantized the neural network using L2 error minimization and
                 achieved better accuracy on MNIST and CIFAR-10 datasets.Denton et al. (2014) exploited the linear
                 structure of the neural network by finding an appropriate low-rank approximation of the parameters
                 and keeping the accuracy within 1% of the original model.
                 The empirical success in this paper is consistent with the theoretical study of random-like sparse
                 networks with +1/0/-1 weights (Arora et al., 2014), which have been proved to enjoy nice properties
                 (e.g. reversibility), and to allow a provably polynomial time algorithm for training.
                 Much work has been focused on binning the network parameters into buckets, and only the values in
                 the buckets need to be stored. HashedNets(Chen et al., 2015) reduce model sizes by using a hash
                 function to randomly group connection weights, so that all connections within the same hash bucket
                 share a single parameter value. In their method, the weight binning is pre-determined by the hash
                 function, instead of being learned through training, which doesn’t capture the nature of images. Gong
                 et al. (2014) compressed deep conv nets using vector quantization, which resulted in 1% accuracy
                 loss. Both methods studied only the fully connected layer, ignoring the convolutional layers.
                 There have been other attempts to reduce the number of parameters of neural networks by replacing
                 the fully connected layer with global average pooling. The Network in Network architecture(Lin et al.,
                 2013) and GoogLenet(Szegedy et al., 2014) achieves state-of-the-art results on several benchmarks by
                 adopting this idea. However, transfer learning, i.e. reusing features learned on the ImageNet dataset
                 and applying them to new tasks by only fine-tuning the fully connected layers, is more difficult with
                 this approach. This problem is noted by Szegedy et al. (2014) and motivates them to add a linear
                 layer on the top of their networks to enable transfer learning.
                 Network pruning has been used both to reduce network complexity and to reduce over-fitting. An
                 early approach to pruning was biased weight decay (Hanson & Pratt, 1989). Optimal Brain Damage
                 (LeCun et al., 1989) and Optimal Brain Surgeon (Hassibi et al., 1993) prune networks to reduce
                 the number of connections based on the Hessian of the loss function and suggest that such pruning
                 is more accurate than magnitude-based pruning such as weight decay. A recent work (Han et al.,
                 2015) successfully pruned several state of the art large scale networks and showed that the number of
                 parameters could be reduce by an order of magnitude. There are also attempts to reduce the number
                 of activations for both compression and acceleration Van Nguyen et al. (2015).

                  8 FUTURE WORK

                 While thE pruned network has been benchmarked on various hardware, the quantized network with
                 weight sharing has not, because off-the-shelf cuSPARSE or MKL SPBLAS library does not support
                 indirect matrix entry lookup, nor is the relative index in CSC or CSR format supported. So the full
                 advantage of Deep Compression that fit the model in cache is not fully unveiled. A software solution
                 is to write customized GPU kernels that support this. A hardware solution is to build custom ASIC
                 architecture specialized to traverse the sparse and quantized network structure, which also supports
                 customized quantization bit width. We expect this architecture to have energy dominated by on-chip
                 SRAM access instead of off-chip DRAM access.

                  9 CONCLUSION

                 We have presented “Deep Compression” that compressed neural networks without affecting accuracy.
                 Our method operates by pruning the unimportant connections, quantizing the network using weight
                 sharing, and then applying Huffman coding. We highlight our experiments on AlexNet which
                 reduced the weight storage by 35% without loss of accuracy. We show similar results for VGG-16
                 and LeNet networks compressed by 49% and 39% without loss of accuracy. This leads to smaller
                 storage requirement of putting conv nets into mobile app. After Deep Compression the size of these
                 networks fit into on-chip SRAM cache (5pJ/access) rather than requiring off-chip DRAM memory
                 (640pJ/access). This potentially makes deep neural networks more energy efficient to run on mobile.
                 Our compression method also facilitates the use of complex neural networks in mobile applications
                 where application size and download bandwidth are constrained.

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                  A APPENDIX :DETAILED TIMING / POWER REPORTS OF DENSE & SPARSE
                     NETWORK LAYERS

                 Table 8: Average time on different layers. To avoid variance, we measured the time spent on each
                 layer for 4096 input samples, and averaged the time regarding each input sample. For GPU, the time
                 consumed bycudaMallocandcudaMemcpyis not counted. For batch size = 1,gemvis used;
                 For batch size = 64,gemmis used. For sparse case,csrmvandcsrmmis used, respectively.

                                              <<TABLE>>

                 Table 9: Power consumption of different layers. We measured the Titan X GPU power with
                 nvidia-smi, Core i7-5930k CPU power withpcm-powerand Tegra K1 mobile GPU power with
                 an external power meter (scaled to AP+DRAM, see paper discussion). During power measurement,
                 we repeated each computation multiple times in order to get stable numbers. On CPU, dense matrix
                 multiplications consume2xenergy than sparse ones because it is accelerated with multi-threading.

                                              <<TABLE>>
<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>
                  DEEP DOUBLE DESCENT: WHERE BIGGER MODELS AND MORE DATA HURT

                  Preetum Nakkiran    Gal Kaplun y       Yamini Bansal y     Tristan Yang
                  Harvard University    Harvard University    Harvard University    Harvard University

                  Boaz Barak         Ilya Sutskever
                  Harvard University    OpenAI


                                         ABSTRACT

                       We show that a variety of modern deep learning tasks exhibit a “double-descent”
                       phenomenon where, as we increase model size, performance first gets worse and
                       then gets better. Moreover, we show that double descent occurs not just as a
                       function of model size, but also as a function of the number of training epochs.
                       We unify the above phenomena by defining a new complexity measure we call
                       the effective model complexity and conjecture a generalized double descent with
                       respect to this measure. Furthermore, our notion of model complexity allows us to
                       identify certain regimes where increasing (even quadrupling) the number of train
                       samples actually hurts test performance.


                  1 INTRODUCTION

                                        <<FIGURE>>

                 Figure 1:Left:Train and test error as a function of model size, for ResNet18s of varying width
                  on CIFAR-10 with 15% label noise.Right:Test error, shown for varying train epochs. All models
                 trained using Adam for 4K epochs. The largest model (width64) corresponds to standard ResNet18.


                  The bias-variance trade-off is a fundamental concept in classical statistical learning theory (e.g.,
                  Hastie et al. (2005)). The idea is that models of higher complexity have lower bias but higher vari-
                  ance. According to this theory, once model complexity passes a certain threshold, models “overfit”
                  with the variance term dominating the test error, and hence from this point onward, increasing model
                  complexity will only decrease performance (i.e., increase test error). Hence conventional wisdom
                 in classical statistics is that, once we pass a certain threshold,“larger models are worse.”
                  However, modern neural networks exhibit no such phenomenon. Such networks have millions of
                  parameters, more than enough to fit even random labels (Zhang et al. (2016)), and yet they perform
                  much better on many tasks than smaller models. Indeed, conventional wisdom among practitioners
                  is that“larger models are better’’ (Krizhevsky et al. (2012), Huang et al. (2018), Szegedy et al.
                   
                              <<FIGURE>>

                  Figure 2:Left:Test error as a function of model size and train epochs. The horizontal line corre-
                  sponds to model-wise double descent–varying model size while training for as long as possible. The
                  vertical line corresponds to epoch-wise double descent, with test error undergoing double-descent
                  as train time increases.RightTrain error of the corresponding models. All models are Resnet18s
                  trained on CIFAR-10 with 15% label noise, data-augmentation, and Adam for up to 4K epochs.


                  (2015), Radford et al. (2019)). The effect of training time on test performance is also up for debate.
                  In some settings, “early stopping” improves test performance, while in other settings training neu-
                  ral networks to zero training error only improves performance. Finally, if there is one thing both
                  classical statisticians and deep learning practitioners agree on is“more data is always better”.
                 In this paper, we present empirical evidence that both reconcile and challenge some of the above
                 “conventional wisdoms.” We show that many deep learning settings have two different regimes.
                 In the under-parameterized regime, where the model complexity is small compared to the number
                  of samples, the test error as a function of model complexity follows the U-like behavior predicted
                  by the classical bias/variance tradeoff. However, once model complexity is sufficiently large to
                  interpolate i.e., achieve (close to) zero training error, then increasing complexity only decreases test
                 error, following the modern intuition of “bigger models are better”. Similar behavior was previously
                 observed in Opper (1995; 2001), Advani & Saxe (2017), Spigler et al. (2018), and Geiger et al.
                 (2019b). This phenomenon was first postulated in generality by Belkin et al. (2018) who named
                 it “double descent”, and demonstrated it for decision trees, random features, and 2-layer neural
                 networks with‘2 loss, on a variety of learning tasks including MNIST and CIFAR-10.


                  Main contributions. We show that double descent is a robust phenomenon that occurs in a variety
                  of tasks, architectures, and optimization methods (see Figure 1 and Section 5; our experiments are
                  summarized in Table A). Moreover, we propose a much more general notion of “double descent”
                  that goes beyond varying the number of parameters. We define the effective model complexity (EMC)
                 of a training procedure as the maximum number of samples on which it can achieve close to zero
                 training error. The EMC depends not just on the data distribution and the architecture of the classifier
                 but also on the training procedure—and in particular increasing training time will increase the EMC.
                 We hypothesize that for many natural models and learning algorithms, double descent occurs as a
                 function of the EMC. Indeed we observe “epoch-wise double descent” when we keep the model fixed
                 and increase the training time, with performance following a classical U-like curve in the underfitting
                 stage (when the EMC is smaller than the number of samples) and then improving with training time
                 once the EMC is sufficiently larger than the number of samples (see Figure 2). As a corollary, early
                 stopping only helps in the relatively narrow parameter regime of critically parameterized models.


                 Sample non-monotonicity. Finally, our results shed light on test performance as a function of
                 the number of train samples. Since the test error peaks around the point where EMC matches the
                 number of samples (the transition from the under- to over-parameterization), increasing the number
                 of samples has the effect of shifting this peak to the right. While in most settings increasing the
                 number of samples decreases error, this shifting effect can sometimes result in a setting wheremore
                 data is worse!For example, Figure 3 demonstrates cases in which increasing the number of samples
                 by a factor of4:5results in worse test performance.

                                                      Figure 3: Test loss (per-token perplexity) as a
                                                      function of Transformer model size (embed-
                                                      ding dimension d model) on language trans-
                     <<FIGURE>>                       lation (IWSLT‘14 German-to-English). The
                                                      curve for 18k samples is generally lower than
                                                      the one for 4k samples, but also shifted to
                                                      the right, since fitting 18k samples requires
                                                      a larger model. Thus, for some models, the
                                                      performance for 18k samples is worse than
                                                      for 4k samples.



                  2 OUR RESULTS

                  To state our hypothesis more precisely, we define the notion of effective model complexity. We define
                  a training procedure T to be any procedure that takes as input a set <<FORMULA>>
                  of labeled training samples and outputs a classifier <<T(S)>> mapping data to labels. We define the
                  effective model complexity of T (w.r.t. distributionD) to be the maximum number of samples non
                  which T achieves on average <<FORMULA>> training error.

                  Definition 1 (Effective Model Complexity)TheEffective Model Complexity(EMC) of a training
                  procedureT, with respect to distribution D and parameter <<FORMULA>>, is defined as:

                                <<FORMULA>>

                  whereError <<S(M)>> is the mean error of modelMon train samplesS.

                  Our main hypothesis can be informally stated as follows:

                  Hypothesis 1 (Generalized Double Descent hypothesis, informal)For any natural data distribu-
                  tion D, neural-network-based training procedureT, and small <<FORMULA>>, if we consider the task of
                  predicting labels based on n samples from D then:

                  Under-parametrized regime.If <<FORMULA>> is sufficiently smaller than n, any perturbation of T
                       that increases its effective complexity will decrease the test error.
                  Over-parameterized regime.If <<FORMULA>> is sufficiently larger than n, any perturbation of T
                       that increases its effective complexity will decrease the test error.
                  Critically parameterized regime.If <<FORMULA>>, then a perturbation of T that increases its
                       effective complexity might decrease or increase the test error.

                  Hypothesis 1 is informal in several ways. We do not have a principled way to choose the parameter
                  <<FORMULA>> (and currently heuristically use <<FORMULA>>). We also are yet to have a formal specification for
                  “sufficiently smaller” and “sufficiently larger”. Our experiments suggest that there is a critical
                  interval around the interpolation threshold when <<FORMULA>>: below and above this interval
                  increasing complexity helps performance, while within this interval it may hurt performance. The
                  width of the critical interval depends on both the distribution and the training procedure in ways we
                  do not yet completely understand.

                  We believe Hypothesis 1 sheds light on the interaction between optimization algorithms, model size,
                  and test performance and helps reconcile some of the competing intuitions about them. The main
                  result of this paper is an experimental validation of Hypothesis 1 under a variety of settings, where
                  we considered several natural choices of datasets, architectures, and optimization algorithms, and
                  we changed the “interpolation threshold” by varying the number of model parameters, the length of
                  training, the amount of label noise in the distribution, and the number of train samples.
                  Model-wise Double Descent.In Section 5, we study the test error of models of increasing size,
                  for a fixed large number of optimization steps. We show that “model-wise double-descent” occurs
                  for various modern datasets (CIFAR-10, CIFAR-100, IWSLT‘14 de-en, with varying amounts of
                  label noise), model architectures (CNNs, ResNets, Transformers), optimizers (SGD, Adam), number
                  of train samples, and training procedures (data-augmentation, and regularization). Moreover, the
                 peak in test error systematically occurs at the interpolation threshold. In particular, we demonstrate
                 realistic settings in which bigger models are worse.

                 Epoch-wise Double Descent.In Section 6, we study the test error of a fixed, large architecture over
                 the course of training. We demonstrate, in similar settings as above, a corresponding peak in test
                 performance when models are trained just long enough to reach <<FORMULA>> train error. The test error of a
                 large model first decreases (at the beginning of training), then increases (around the critical regime),
                 then decreases once more (at the end of training)—that is,training longer can correct overfitting.
                 Sample-wise Non-monotonicity.In Section 7, we study the test error of a fixed model and training
                 procedure, for varying number of train samples. Consistent with our generalized double-descent
                 hypothesis, we observe distinct test behavior in the “critical regime”, when the number of samples
                 is near the maximum that the model can fit. This often manifests as a long plateau region, in which
                 taking significantly more data might not help when training to completion (as is the case for CNNs on
                 CIFAR-10). Moreover, we show settings (Transformers on IWSLT‘14 en-de), where this manifests
                 as a peak—and for a fixed architecture and training procedure,more data actually hurts.
                 Remarks on Label Noise.We observe all forms of double descent most strongly in settings with
                 label noise in the train set (as is often the case when collecting train data in the real-world). How-
                 ever, we also show several realistic settings with a test-error peak even without label noise: ResNets
                 (Figure 4a) and CNNs (Figure 20) on CIFAR-100; Transformers on IWSLT‘14 (Figure 8). More-
                 over, all our experiments demonstrate distinctly different test behavior in the critical regime— often
                 manifesting as a “plateau” in the test error in the noiseless case which develops into a peak with
                 added label noise. See Section 8 for further discussion.

                  3 RELATED WORK

                 Model-wise double descent was first proposed as a general phenomenon by Belkin et al. (2018).
                 Similar behavior had been observed in Opper (1995; 2001), Advani & Saxe (2017), Spigler et al.
                 (2018), and Geiger et al. (2019b). Subsequently, there has been a large body of work studying the
                 double descent phenomenon. A growing list of papers that theoretically analyze it in the tractable
                 setting of linear least squares regression includes Belkin et al. (2019); Hastie et al. (2019); Bartlett
                 et al. (2019); Muthukumar et al. (2019); Bibas et al. (2019); Mitra (2019); Mei & Montanari (2019).
                 Moreover, Geiger et al. (2019a) provide preliminary results for model-wise double descent in con-
                 volutional networks trained on CIFAR-10. Our work differs from the above papers in two crucial
                 aspects: First, we extend the idea of double-descent beyond the number of parameters to incorpo-
                 rate the training procedure under a unified notion of “Effective Model Complexity”, leading to novel
                 insights like epoch-wise double descent and sample non-monotonicity. The notion that increasing
                 train time corresponds to increasing complexity was also presented in Nakkiran et al. (2019). Sec-
                 ond, we provide an extensive and rigorous demonstration of double-descent for modern practices
                 spanning a variety of architectures, datasets optimization procedures. An extended discussion of the
                 related work is provided in Appendix C.

                  4 EXPERIMENTAL SETUP

                 We briefly describe the experimental setup here; full details are in Appendix B1. We consider three
                 families of architectures: ResNets, standard CNNs, and Transformers.ResNets:We parameterize
                 a family of ResNet18s (He et al. (2016)) by scaling the width (number of filters) of convolutional
                 layers. Specifically, we use layer widths [k;2k;4k;8k] for varying k. The standard ResNet18
                 corresponds tok= 64. Standard CNNs:We consider a simple family of 5-layer CNNs, with
                 4 convolutional layers of widths [k;2k;4k;8k] for varying k, and a fully-connected layer. For
                 context, the CNN with width k=64, can reach over 90% test accuracy on CIFAR-10 with data-
                 augmentation.Transformers:We consider the 6 layer encoder-decoder from Vaswani et al. (2017),
                 as implemented by Ott et al. (2019). We scale the size of the network by modifying the embedding
                 dimension d model , and setting the width of the fully-connected layers proportionally (<<FORMULA>>).

                 The raw data from our experiments are available at:   https://gitlab.com/harvard-machine-learning/double-descent/tree/master

                 For ResNets and CNNs, we train with cross-entropy loss, and the following optimizers: (1) Adam
                 with learning-rate0:0001for 4K epochs; (2) SGD with learning rate/p1 for 500K gradient steps. T We train Transformers for 80K gradient steps, with 10% label smoothing and no drop-out.

                 Label Noise. In our experiments, label noise of probability prefers to training on a samples which
                  have the correct label with probability (<<FORMULA>>), and a uniformly random incorrect label otherwise
                  (label noise is sampled only once and not per epoch). Figure 1 plots test error on the noisy distribu-
                  tion, while the remaining figures plot test error with respect to the clean distribution (the two curves
                  are just linear rescaling of one another).

                  5 MODEL-WISE DOUBLE DESCENT

                                            <<FIGURE>>

                 Figure 4:Model-wise double descent for ResNet18s.Trained on CIFAR-100 and CIFAR-10, with
                 varying label noise. Optimized using Adam with LR0:0001for 4K epochs, and data-augmentation.

                 In this section, we study the test error of models of increasing size, when training to completion
                 (for a fixed large number of optimization steps). We demonstrate model-wise double descent across
                 different architectures, datasets, optimizers, and training procedures. The critical region exhibits
                 distinctly different test behavior around the interpolation point and there is often a peak in test error
                 that becomes more prominent in settings with label noise.
                 For the experiments in this section (Figures 4, 5, 6, 7, 8), notice that all modifications which increase
                 the interpolation threshold (such as adding label noise, using data augmentation, and increasing the
                 number of train samples) also correspondingly shift the peak in test error towards larger models.
                 Additional plots showing the early-stopping behavior of these models, and additional experiments
                 showing double descent in settings with no label noise (e.g. Figure 19) are in Appendix E.2. We
                 also observed model-wise double descent for adversarial training, with a prominent robust test error
                 peak even in settings without label noise. See Figure 26 in Appendix E.2.

                 Discussion. Fully understanding the mechanisms behind model-wise double descent in deep neu-
                 ral networks remains an important open question. However, an analog of model-wise double descent
                 occurs even for linear models. A recent stream of theoretical works analyzes this setting (Bartlett
                 et al. (2019); Muthukumar et al. (2019); Belkin et al. (2019); Mei & Montanari (2019); Hastie et al.
                 (2019)). We believe similar mechanisms may be at work in deep neural networks.
                 Informally, our intuition is that for model-sizes at the interpolation threshold, there is effectively
                 only one model that fits the train data and this interpolating model is very sensitive to noise in the

                         <<FIGURE>>

                 Figure 5: Effect of Data Augmentation. 5-layer CNNs on CIFAR10, with and without data-
                 augmentation. Data-augmentation shifts the interpolation threshold to the right, shifting the test
                 error peak accordingly. Optimized using SGD for 500K steps. See Figure 27 for larger models.

                          <<FIGURE>>                                  <<FIGURE>>

                   Figure 6:SGD vs. Adam.5-Layer CNNs       Figure 7: Noiseless settings.  5-layer
                   on CIFAR-10 with no label noise, and no       CNNs on CIFAR-100 with no label noise;
                   data augmentation. Optimized using SGD       note the peak in test error. Trained with
                   for 500K gradient steps, and Adam for 4K       SGD and no data augmentation. See Fig-
                   epochs.                             ure 20 for the early-stopping behavior of
                                                      these models.


                 train set and/or model mis-specification. That is, since the model is just barely able to fit the train
                 data, forcing it to fit even slightly-noisy or mis-specified labels will destroy its global structure, and
                 result in high test error. (See Figure 28 in the Appendix for an experiment demonstrating this noise
                 sensitivity, by showing that ensembling helps significantly in the critically-parameterized regime).
                 However for over-parameterized models, there are many interpolating models that fit the train set,
                 and SGD is able to find one that “memorizes” (or “absorbs”) the noise while still performing well
                 on the distribution.
                 The above intuition is theoretically justified for linear models. In general, this situation manifests
                 even without label noise for linear models (Mei & Montanari (2019)), and occurs whenever there

                                                      Figure 8:Transformers on language trans-
                                                      lation tasks:Multi-head-attention encoder-
                                                      decoder Transformer model trained for
                          <<FIGURE>>                  80k gradient steps with labeled smoothed
                                                      cross-entropy loss on IWSLT‘14 German-
                                                      to-English (160K sentences) and WMT‘14
                                                      English-to-French (subsampled to 200K sen-
                                                      tences) dataset. Test loss is measured as per-
                                                      token perplexity.





                 is model mis-specification between the structure of the true distribution and the model family. We
                  believe this intuition extends to deep learning as well, and it is consistent with our experiments.


                  6 EPOCH-WISE DOUBLE DESCENT

                  In this section, we demonstrate a novel form of double-descent with respect to training epochs,
                  which is consistent with our unified view of effective model complexity (EMC) and the generalized
                  double descent hypothesis. Increasing the train time increases the EMC—and thus a sufficiently
                  large model transitions from under- to over-parameterized over the course of training.

                                      <<FIGURE>>

                 Figure 9:Left:Training dynamics for models in three regimes. Models are ResNet18s on CIFAR10
                 with 20% label noise, trained using Adam with learning rate0:0001, and data augmentation.Right:
                 Test error over (Model size  Epochs). Three slices of this plot are shown on the left.


                 As illustrated in Figure 9, sufficiently large models can undergo a “double descent” behavior where
                 test error first decreases then increases near the interpolation threshold, and then decreases again. In
                 contrast, for “medium sized” models, for which training to completion will only barely reach 0
                 error, the test error as a function of training time will follow a classical U-like curve where it is
                 better to stop early. Models that are too small to reach the approximation threshold will remain in
                 the “under parameterized” regime where increasing train time monotonically decreases test error.
                 Our experiments (Figure 10) show that many settings of dataset and architecture exhibit epoch-wise
                 double descent, in the presence of label noise. Further, this phenomenon is robust across optimizer
                 variations and learning rate schedules (see additional experiments in Appendix E.1). As in model-
                 wise double descent, the test error peak is accentuated with label noise.
                 Conventional wisdom suggests that training is split into two phases: (1) In the first phase, the net-
                 work learns a function with a small generalization gap (2) In the second phase, the network starts
                 to over-fit the data leading to an increase in test error. Our experiments suggest that this is not the
                 complete picture—in some regimes, the test error decreases again and may achieve a lower value at
                 the end of training as compared to the first minimum (see Fig 10 for 10% label noise).

                               <<FIGURE>>

                 Figure 10:Epoch-wise double descent for ResNet18 and CNN (width=128). ResNets trained using
                 Adam with learning rate0:0001, and CNNs trained with SGD with inverse-square root learning rate.


                  7 SAMPLE-WISE NON-MONOTONICITY

                  In this section, we investigate the effect of varying the number of train samples, for a fixed model and
                  training procedure. Previously, in model-wise and epoch-wise double descent, we explored behavior
                  in the critical regime, where <<FORMULA>>, by varying the EMC. Here, we explore the critical
                  regime by varying the number of train samples n. By increasing n, the same training procedure T
                  can switch from being effectively over-parameterized to effectively under-parameterized.
                  We show that increasing the number of samples has two different effects on the test error vs. model
                  complexity graph. On the one hand, (as expected) increasing the number of samples shrinks the area
                  under the curve. On the other hand, increasing the number of samples also has the effect of “shifting
                  the curve to the right” and increasing the model complexity at which test error peaks.

                                          <<FIGURE>>

                                    Figure 11: Sample-wise non-monotonicity.


                  These twin effects are shown in Figure 11a. Note that there is a range of model sizes where the
                 effects “cancel out”—and having 4% more train samples does not help test performance when
                  training to completion. Outside the critically-parameterized regime, for sufficiently under- or over-
                  parameterized models, having more samples helps. This phenomenon is corroborated in Figure 12,
                  which shows test error as a function of both model and sample size, in the same setting as Figure 11a.

                                                  <<FIGURE>>

                 Figure 12:Left:Test Error as a function of model size and number of train samples, for 5-layer
                 CNNs on CIFAR-10 +20% noise. Note the ridge of high test error again lies along the interpolation
                 threshold. Right: Three slices of the left plot, showing the effect of more data for models of
                 different sizes. Note that, when training to completion, more data helps for small and large models,
                 but does not help for near-critically-parameterized models (green).

                 In some settings, these two effects combine to yield a regime of model sizes where more data actually
                 hurts test performance as in Figure 3 (see also Figure 11b). Note that this phenomenon is not unique
                 to DNNs: more data can hurt even for linear models (see Appendix D).

                  8 CONCLUSION AND DISCUSSION

                 We introduce a generalized double descent hypothesis: models and training procedures exhibit atyp-
                 ical behavior when their Effective Model Complexity is comparable to the number of train samples.
                 We provide extensive evidence for our hypothesis in modern deep learning settings, and show that
                 it is robust to choices of dataset, architecture, and training procedures. In particular, we demon-
                 strate “model-wise double descent” for modern deep networks and characterize the regime where
                 bigger models can perform worse. We also demonstrate “epoch-wise double descent,” which, to the
                 best of our knowledge, has not been previously proposed. Finally, we show that the double descent
                 phenomenon can lead to a regime where training on more data leads to worse test performance.
                 Preliminary results suggest that double descent also holds as we vary the amount of regularization
                 for a fixed model (see Figure 22).
                 We also believe our characterization of the critical regime provides a useful way of thinking for
                 practitioners—if a model and training procedure are just barely able to fit the train set, then small
                 changes to the model or training procedure may yield unexpected behavior (e.g. making the model
                 slightly larger or smaller, changing regularization, etc. may hurt test performance).

                 Early stopping. We note that many of the phenomena that we highlight often do not occur with
                 optimal early-stopping. However, this is consistent with our generalized double descent hypothesis:
                 if early stopping prevents models from reaching0train error then we would not expect to see double-
                  descent, since the EMC does not reach the number of train samples. Further, we show at least one

                 setting where model-wise double descent can still occur even with optimal early stopping (ResNets
                 on CIFAR-100 with no label noise, see Figure 19). We have not observed settings where more data
                 hurts when optimal early-stopping is used. However, we are not aware of reasons which preclude
                 this from occurring. We leave fully understanding the optimal early stopping behavior of double
                 descent as an important open question for future work.

                 Label Noise. In our experiments, we observe double descent most strongly in settings with label
                 noise. However, we believe this effect is not fundamentally about label noise, but rather about
                 model mis-specification. For example, consider a setting where the label noise is not truly random,
                 but rather pseudorandom (with respect to the family of classifiers being trained). In this setting,
                 the performance of the Bayes optimal classifier would not change (since the pseudorandom noise
                 is deterministic, and invertible), but we would observe an identical double descent as with truly
                 random label noise. Thus, we view adding label noise as merely a proxy for making distributions
                 “harder”— i.e. increasing the amount of model mis-specification.

                 Other Notions of Model Complexity. Our notion of Effective Model Complexity is related to
                 classical complexity notions such as Rademacher complexity, but differs in several crucial ways:
                 (1) EMC depends on the true labels of the data distribution, and (2) EMC depends on the training
                 procedure, not just the model architecture.
                 Other notions of model complexity which do not incorporate features (1) and (2) would not suffice
                 to characterize the location of the double-descent peak. Rademacher complexity, for example, is
                 determined by the ability of a model architecture to fit a randomly-labeled train set. But Rademacher
                 complexity and VC dimension are both insufficient to determine the model-wise double descent
                 peak location, since they do not depend on the distribution of labels— and our experiments show
                 that adding label noise shifts the location of the peak.
                 Moreover, both Rademacher complexity and VC dimension depend only on the model family and
                 data distribution, and not on the training procedure used to find models. Thus, they are not capable
                 of capturing train-time double-descent effects, such as “epoch-wise” double descent, and the effect
                 of data-augmentation on the peak location.

                  ACKNOWLEDGMENTS
                 We thank Mikhail Belkin for extremely useful discussions in the early stages of this work. We
                 thank Christopher Olah for suggesting the Model SizeEpoch visualization, which led to the
                 investigation of epoch-wise double descent, as well as for useful discussion and feedback. We also
                 thank Alec Radford, Jacob Steinhardt, and Vaishaal Shankar for helpful discussion and suggestions.
                 P.N. thanks OpenAI, the Simons Institute, and the Harvard Theory Group for a research environment
                 that enabled this kind of work.
                 We thank Dimitris Kalimeris, Benjamin L. Edelman, and Sharon Qian, and Aditya Ramesh for
                 comments on an early draft of this work.
                 This work supported in part by NSF grant CAREER CCF 1452961, BSF grant 2014389, NSF US-
                 ICCS proposal 1540428, a Google Research award, a Facebook research award, a Simons Investiga-
                 tor Award, a Simons Investigator Fellowship, and NSF Awards CCF 1715187, CCF 1565264, CCF
                 1301976, IIS 1409097, and CNS 1618026. Y.B. would like to thank the MIT-IBM Watson AI Lab
                 for contributing computational resources for experiments.


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                           A SUMMARY TABLE OF EXPERIMENTAL RESULTS

                                     <<TABLE>>


                  B APPENDIX: EXPERIMENTAL DETAILS

                  B.1 MODELS

                 We use the following families of architectures. The PyTorch Paszke et al. (2017)
                 specification of our ResNets and CNNs are available at https://gitlab.com/harvard-machine-learning/double-descent/tree/master.

                 ResNets. We define a family of ResNet18s of increasing size as follows. We follow the Preac-
                 tivation ResNet18 architecture of He et al. (2016), using 4 ResNet blocks, each consisting of two
                 BatchNorm-ReLU-Convolution layers. The layer widths for the 4 blocks are [k;2k;4k;8k] for
                 varyingk2Nand the strides are [1, 2, 2, 2]. The standard ResNet18 corresponds to k=64 con-
                  volutional channels in the first layer. The scaling of model size withkis shown in Figure 13b. Our
                  implementation is adapted from https://github.com/kuangliu/pytorch-cifar.

                  Standard CNNs. We consider a simple family of 5-layer CNNs, with four Conv-BatchNorm-
                  ReLU-MaxPool layers and a fully-connected output layer. We scale the four convolutional layer
                  widths as [k;2k;4k;8k]. The MaxPool is [1, 2, 2, 8]. For all the convolution layers, the kernel
                 size = 3, stride = 1 and padding=1. This architecture is based on the “backbone” architecture from
                 Page (2018). For k=64, this CNN has 1558026 parameters and can reach >90% test accuracy on
                 CIFAR-10 (Krizhevsky (2009)) with data-augmentation. The scaling of model size with k is shown
                 in Figure 13a.

                 Transformers. We consider the encoder-decoder Transformer model from Vaswani et al. (2017)
                  with 6 layers and 8 attention heads per layer, as implemented by fairseq Ott et al. (2019). We scale
                  the size of the network by modifying the embedding dimension (d model), and scale the width of the
                 fully-connected layers proportionally (dff = 4d model). We train with 10% label smoothing and no
                  drop-out, for 80 gradient steps.

                                         <<FIGURE>>            

                   Figure 13: Scaling of model size with our parameterization of width & embedding dimension.


                  B.2 IMAGE CLASSIFICATION: EXPERIMENTAL SETUP

                  We describe the details of training for CNNs and ResNets below.
                  Loss function:Unless stated otherwise, we use the cross-entropy loss for all the experiments.
                  Data-augmentation:  In experiments where data-augmentation was used, we apply
                  RandomCrop(32, padding=4)andRandomHorizontalFlip. In experiments with
                 added label noise, the label for all augmentations of a given training sample are given the same
                 label.
                 Regularization:No explicit regularization like weight decay or dropout was applied unless explic-
                  itly stated.
                  Initialization:We use the default initialization provided by PyTorch for all the layers.
                  Optimization:

                      Adam: Unless specified otherwise, learning rate was set at constant to 1e^4 and all other
                       parameters were set to their default PyTorch values.
                      SGD: Unless specified otherwise, learning rate schedule inverse-square root (defined be-
                       low) was used with initial learning rate <<FORMULA>> and updates every L=512 gradient steps.
                       No momentum was used.

                  We found our results are robust to various other natural choices of optimizers and learning rate
                  schedule. We used the above settings because (1) they optimize well, and (2) they do not require
                  experiment-specific hyperparameter tuning, and allow us to use the same optimization across many
                  experiments.
                  Batch size: All experiments use a batchsize of 128.
                  Learning rate schedule descriptions:

                      Inverse-square root (<<FORMULA>>): At gradient stept, the learning rate is set to <<FORMULA>>. We set learning-rate with respect to number of gradient steps, and not epochs, <<FORMULA>>
                       in order to allow comparison between experiments with varying train-set sizes.
                      Dynamic drop (<<FORMULA>>, drop, patience): Starts with an initial learning rate of 0 and drops by
                       a factor of ’drop’ if the training loss has remained constant or become worse for ’patience’
                       number of gradient steps.

                  B.3 NEURAL MACHINE TRANSLATION: EXPERIMENTAL SETUP

                 Here we describe the experimental setup for the neural machine translation experiments.
                 Training procedure.

                 In this setting, the distributionDconsists of triples

                                   <<FORMULA>>
                                   
                 where V_src and V_tgt are the source and target vocabularies, the stringxis a sentence in the source
                 language,yis its translation in the target language, andiis the index of the token to be predicted by
                  the model. We assume that <<FORMULA>> is distributed uniformly on <<FORMULA>>.
                  A standard probabilistic model defines an autoregressive factorization of the likelihood:

                                                <<FORMULA>>

                 Given a set of training samplesS, we define

                                             <<FORMULA>>

                  In practice,S is not constructed from independent samples from D, but rather by first sampling
                  <<(x,y)>> and then including all <<FORMULA>> in S.
                  For training transformers, we replicate the optimization procedure specified in Vaswani et al. (2017)
                  section 5.3, where the learning rate schedule consists of a “warmup” phase with linearly increasing
                  learning rate followed by a phase with inverse square-root decay. We preprocess the data using byte
                  pair encoding (BPE) as described in Sennrich et al. (2015). We use the implementation provided by
                  fairseq (https://github.com/pytorch/fairseq).

                  Datasets. The IWSLT’14 German to English dataset contains TED Talks as described in Cettolo
                  et al. (2012). The WMT’14 English to French dataset is taken from http://www.statmt.org/wmt14/translation-task.html.

                  B.4 PER-SECTION EXPERIMENTAL DETAILS

                  Here we provide full details for experiments in the body, when not otherwise provided.
                  Introduction: Experimental Details Figure 1: All models were trained using Adam with learning-
                  rate 0.0001 for 4K epochs. Plotting means and standard deviations for 5 trials, with random network
                  initialization.

                  Model-wise Double Descent: Experimental Details Figure 7: Plotting means and standard devia-
                  tions for 5 trials, with random network initialization.
                  Sample-wise Nonmonotonicity: Experimental DetailsFigure 11a: All models are trained with
                  SGD for 500K epochs, and data-augmentation. Bottom: Means and standard deviations from 5
                  trials with random initialization, and random subsampling of the train set.

                       C EXTENDED DISCUSSION OF RELATED WORK

                 Belkin et al. (2018): This paper proposed, in very general terms, that the apparent contradiction
                 between traditional notions of the bias-variance trade-off and empirically successful practices in
                 deep learning can be reconciled under a double-descent curve—as model complexity increases, the
                 test error follows the traditional “U-shaped curve”, but beyond the point of interpolation, the error
                 starts todecrease. This work provides empirical evidence for the double-descent curve with fully
                  connected networks trained on subsets of MNIST, CIFAR10, SVHN and TIMIT datasets. They use
                  thel2 loss for their experiments. They demonstrate that neural networks are not an aberration in this
                  regard—double-descent is a general phenomenon observed also in linear regression with random
                  features and random forests.

                  Theoretical works on linear least squares regression: A variety of papers have attempted to the-
                  oretically analyze this behavior in restricted settings, particularly the case of least squares regression
                  under various assumptions on the training data, feature spaces and regularization method.

                     1.Advani & Saxe (2017); Hastie et al. (2019) both consider the linear regression problem
                       stated above and analyze the generalization behavior in the asymptotic limit <<FORMULA>>
                       using random matrix theory. Hastie et al. (2019) highlight that when the model is mis-
                       specified, the minimum of training error can occur for over-parameterized models
                     2.Belkin et al. (2019) Linear least squares regression for two data models, where the input
                       data is sampled from a Gaussian and a Fourier series model for functions on a circle. They
                       provide a finite-sample analysis for these two cases
                     3.Bartlett et al. (2019) provides generalization bounds for the minimuml2 -norm interpolant
                       for Gaussian features
                     4.Muthukumar et al. (2019) characterize the fundamental limit of of any interpolating solu-
                       tion in the presence of noise and provide some interesting Fourier-theoretic interpretations.
                     5.Mei & Montanari (2019): This work provides asymptotic analysis for ridge regression over
                       random features

                 Similar double descent behavior was investigated in Opper (1995; 2001)
                 Geiger et al. (2019b) showed that deep fully connected networks trained on the MNIST dataset with
                 hinge loss exhibit a “jamming transition” when the number of parameters exceeds a threshold that
                 allows training to near-zero train loss. Geiger et al. (2019a) provide further experiments on CIFAR-
                 10 with a convolutional network. They also highlight interesting behavior with ensembling around
                 the critical regime, which is consistent with our informal intuitions in Section 5 and our experiments
                 in Figures 28, 29.
                 Advani & Saxe (2017); Geiger et al. (2019b;a) also point out that double-descent is not observed
                 when optimal early-stopping is used.

                     D RANDOM FEATURES: A CASE STUDY

                              <<FIGURE>>

                 Figure 14:Random Fourier Featureson the Fashion MNIST dataset. The setting is equivalent
                 to two-layer neural network witheix activation, with randomly-initialized first layer that is fixed
                 throughout training. The second layer is trained using gradient flow.


                 In this section, for completeness sake, we show that both the model- and sample-wise double de-
                 scent phenomena are not unique to deep neural networks—they exist even in the setting of Random
                 Fourier Features of Rahimi & Recht (2008). This setting is equivalent to a two-layer neural network
                 with <<FORMULA>> activation. The first layer is initialized with aN(0;1 )Gaussian distribution and then
                 fixed throughout training. The width (or embedding dimension) d dof the first layer parameterizes
                 the model size. The second layer is initialized with0s and trained with MSE loss.
                 Figure 14 shows the grid of Test Error as a function of both number of samplesnand model sized.
                 Note that in this settingEMC =d(the embedding dimension). As a result, as demonstrated in the
                 figure, the peak follows the path ofn=d. Both model-wise and sample-wise (see figure 15) double
                 descent phenomena are captured, by horizontally and vertically crossing the grid, respectively.

                                    <<FIGURE>>

                 Figure 15: Sample-wise double-descent slice for Random Fourier Features on the Fashion MNIST
                 dataset. In this figure the embedding dimension (number of random features) is 1000.

                            E APPENDIX: ADDITIONAL EXPERIMENTS


                  E.1 EPOCH-WISE DOUBLE DESCENT: ADDITIONAL RESULTS

                 Here, we provide a rigorous evaluation of epoch-wise double descent for a variety of optimizers and
                 learning rate schedules. We train ResNet18 on CIFAR-10 with data-augmentation and 20% label
                 noise with three different optimizers—Adam, SGD, SGD + Momentum (momentum set to 0.9) and
                 three different learning rate schedules—constant, inverse-square root, dynamic drop for differnet
                 values of initial learning rate. We observe that double-descent occurs reliably for all optimizers and
                 learning rate schedules and the peak of the double descent curve shifts with the interpolation point.

                                              <<FIGURE>>

                 Figure 16:Epoch-wise double descentfor ResNet18 trained with Adam and multiple learning rate
                  schedules

                  A practical recommendation resulting from epoch-wise double descent is that stopping the training
                  when the test error starts to increase may not always be the best strategy. In some cases, the test error
                  may decrease again after reaching a maximum, and the final value may be lower than the minimum
                  earlier in training.

                                      <<FIGURE>>

                 Figure 17:Epoch-wise double descentfor ResNet18 trained with SGD and multiple learning rate
                 schedules

                                                                <<FIGURE>>

                 Figure 18:Epoch-wise double descentfor ResNet18 trained with SGD+Momentum and multiple
                 learning rate schedules


                  E.2 MODEL-WISE DOUBLE DESCENT: ADDITIONAL RESULTS

                  E.2.1 CLEAN SETTINGS WITH MODEL-WISE DOUBLE DESCENT

                  <<FIGURE>>

                 Figure 19:Top:Train and test performance as a function of both model size and train epochs.
                  Bottom:Test error dynamics of the same model (ResNet18, on CIFAR-100 with no label noise,
                  data-augmentation and Adam optimizer trained for 4k epochs with learning rate 0.0001). Note that
                  even with optimal early stopping this setting exhibits double descent.

                                                  <<FIGURE>>

                 Figure 20:Top:Train and test performance as a function of both model size and train epochs.
                  Bottom:Test error dynamics of the same models. 5-Layer CNNs, CIFAR-100 with no label noise,
                 no data-augmentation Trained with SGD for 1e6 steps. Same experiment as Figure 7.


                  E.2.2 WEIGHT DECAY

                                          <<FIGURE>>

                 Figure 21:Left:Test error dynamics with weight decay of 5e-4 (bottom left) and without weight
                 decay (top left). Right:Test and train error andtest lossfor models with varying amounts of
                 weight decay. All models are 5-Layer CNNs on CIFAR-10 with 10% label noise, trained with
                 data-augmentation and SGD for 500K steps.

                 Here, we now study the effect of varying the level of regularization on test error. We train CIFAR10
                 with data-augmentation and 20% label noise on ResNet18 for weight decay coefficients <<FORMULA>> rang-
                 ing from 0 to 0.1. We train the networks using SGD + inverse-square root learning rate. Figure
                 below shows a picture qualitatively very similar to that observed for model-wise double descent
                 wherein ”model complexity” is now controlled by the regularization parameter. This confirms our
                 generalized double descent hypothesis along yet another axis of Effective Model Complexity.

                            <<FIGURE>>

                 Figure 22: Generalized double descent for weight decay. We found that using the same initial
                 learning rate for all weight decay values led to training instabilities. This resulted in some noise in
                 the Test Error (Weight DecayEpochs) plot shown above.


                  E.2.3 EARLY STOPPING DOES NOT EXHIBIT DOUBLE DESCENT

                                      <<FIGURE>>

                 Figure 23: Model-wise test error dynamics for a subsampled IWSLT‘14 dataset. Left: 4k samples,
                 Right: 18k samples. Note that with optimal early-stopping, more samples is always better.

                          <<FIGURE>>

                 Figure 24: Model-wise test error dynamics for a IWSLT‘14 de-en and subsampled WMT‘14 en-fr
                 datasets.Left: IWSLT‘14,Right: subsampled (200k samples) WMT‘14. Note that with optimal
                  early-stopping, the test error is much lower for this task.

                                          <<FIGURE>>

                 Figure 25:Top:Train and test performance as a function of both model size and train epochs.
                  Bottom:Test error dynamics of the same model (CNN, on CIFAR-10 with 10% label noise, data-paugmentation and SGD optimizer with learning rate/1= T).


                   E.2.4 TRAINING PROCEDURE

                                  <<FIGURE>>

                 Figure 26:Model-wise double descent for adversarial trainingResNet18s on CIFAR-10 (sub-
                  sampled to 25k train samples) with no label noise. We train for L2 robustness of radius <<FORMULA>> and
                  <<FORMULA>>, using 10-step PGD (Goodfellow et al. (2014); Madry et al. (2017)). Trained using SGD
                  (batch size 128) with learning rate0:1for 400 epochs, then0:01for 400 epochs.

                                              <<FIGURE>>

                                               Figure 27


                    E.3 ENSEMBLING

                                            <<FIGURE>> 

                 Figure 28:Effect of Ensembling (ResNets, 15% label noise). Test error of an ensemble of 5
                 models, compared to the base models. The ensembled classifier is determined by plurality vote over
                 the 5 base models. Note that emsembling helps most around the critical regime. All models are
                 ResNet18s trained on CIFAR-10 with 15% label noise, using Adam for 4K epochs (same setting
                 as Figure 1). Test error is measured against the original (not noisy) test set, and each model in the
                 ensemble is trained using a train set with independently-sampled 15% label noise.

                                                              <<FIGURE>>

                 Figure 29:Effect of Ensembling (CNNs, no label noise). Test error of an ensemble of 5 models,
                 compared to the base models. All models are 5-layer CNNs trained on CIFAR-10 with no label
                 noise, using SGD and no data augmentation. (same setting as Figure 7).
<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>
Deep Residual Learning for Image Recognition 
Kaiming He Xiangyu Zhang Shaoqing Ren Jian Sun Microsoft Research {kahe, v-xiangz, v-shren, jiansun}@microsoft.com 

Abstract

Deeper neural networks are more difficult to train. We present a residual learning framework to ease the training of networks that are substantially deeper than those used previously. We explicitly reformulate the layers as learn.ing residual functions with reference to the layer inputs, in.stead of learning unreferenced functions. We provide comprehensive empirical evidence showing that these residual networks are easier to optimize, and can gain accuracy from considerably increased depth. On the ImageNet dataset we evaluate residual nets with a depth of up to 152 layers 8. deeper than VGG nets [41] but still having lower complex.
ity. An ensemble of these residual nets achieves 3.57% error on the ImageNet test set. This result won the 1st place on the ILSVRC 2015 classification task. We also present analysis on CIFAR-10 with 100 and 1000 layers. 
The depth of representations is of central importance for many visual recognition tasks. Solely due to our extremely deep representations, we obtain a 28% relative improvement on the COCO object detection dataset. Deep residual nets are foundations of our submissions to ILSVRC & COCO 2015 competitions1 , where we also won the 1st places on the tasks of ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation. 

1. Introduction 

Deep convolutional neural networks [22, 21] have led to a series of breakthroughs for image classification [21, 50, 40]. Deep networks naturally integrate low/mid/high.level features [50] and classifiers in an end-to-end multi.layer fashion, and the levels of features can be enriched by the number of stacked layers (depth). Recent evidence [41, 44] reveals that network depth is of crucial importance, and the leading results [41, 44, 13, 16] on the challenging ImageNet dataset [36] all exploit very deep [41] models, with a depth of sixteen [41] to thirty [16]. Many other non.trivial visual recognition tasks [8, 12, 7, 32, 27] have also

            <<FIGURE>>

Figure 1. Training error (left) and test error (right) on CIFAR-10 with 20-layer and 56-layer plain networks. The deeper network has higher training error, and thus test error. Similar phenomena on ImageNet is presented in Fig. 4. 

greatly benefited from very deep models. Driven by the significance of depth, a question arises: Is learning better networks as easy as stacking more layers? 
An obstacle to answering this question was the notorious problem of vanishing/exploding gradients [1, 9], which hamper convergence from the beginning. This problem, however, has been largely addressed by normalized initialization [23, 9, 37, 13] and intermediate normalization layers [16], which enable networks with tens of layers to start con.
verging for stochastic gradient descent (SGD) with back-propagation [22]. 
When deeper networks are able to start converging, a degradation problem has been exposed: with the network depth increasing, accuracy gets saturated (which might be unsurprising) and then degrades rapidly. Unexpectedly, such degradation is not caused by overfitting, and adding more layers to a suitably deep model leads to higher training error, as reported in [11, 42] and thoroughly verified by our experiments. Fig. 1 shows a typical example. 
The degradation (of training accuracy) indicates that not all systems are similarly easy to optimize. Let us consider a shallower architecture and its deeper counterpart that adds more layers onto it. There exists a solution by construction to the deeper model: the added layers are identity mapping, and the other layers are copied from the learned shallower model. The existence of this constructed solution indicates that a deeper model should produce no higher training error than its shallower counterpart. But experiments show that our current solvers on hand are unable to find solutions that are comparably good or better than the constructed solution (or unable to do so in feasible time). 

In this paper, we address the degradation problem by introducing a deep residual learning framework. In.stead of hoping each few stacked layers directly fit a desired underlying mapping, we explicitly let these lay.ers fit a residual mapping. Formally, denoting the desired underlying mapping as <<H(x)>>, we let the stacked nonlinear layers fit another mapping of <<FORMULA>>. The original mapping is recast into <<F(x)+x>>. We hypothesize that it is easier to optimize the residual mapping than to optimize the original, unreferenced mapping. To the extreme, if an identity mapping were optimal, it would be easier to push the residual to zero than to fit an identity mapping by a stack of nonlinear layers. 
The formulation of <<FORMULA>> can be realized by feedforward neural networks with shortcut connections (Fig. 2). Shortcut connections [2, 34, 49] are those skipping one or more layers. In our case, the shortcut connections simply perform identity mapping, and their outputs are added to the outputs of the stacked layers (Fig. 2). Identity short.cut connections add neither extra parameter nor computational complexity. The entire network can still be trained end-to-end by SGD with backpropagation, and can be easily implemented using common libraries (e.g., Caffe [19]) without modifying the solvers. 
We present comprehensive experiments on ImageNet 
[36] to show the degradation problem and evaluate our method. We show that: 1) Our extremely deep residual nets are easy to optimize, but the counterpart plain nets (that simply stack layers) exhibit higher training error when the depth increases; 2) Our deep residual nets can easily enjoy accuracy gains from greatly increased depth, producing results substantially better than previous networks. 
Similar phenomena are also shown on the CIFAR-10 set [20], suggesting that the optimization difficulties and the effects of our method are not just akin to a particular dataset. We present successfully trained models on this dataset with over 100 layers, and explore models with over 1000 layers. 
On the ImageNet classification dataset [36], we obtain excellent results by extremely deep residual nets. Our 152.layer residual net is the deepest network ever presented on ImageNet, while still having lower complexity than VGG nets [41]. Our ensemble has 3.57% top-5 error on the ImageNet test set, and won the 1st place in the ILSVRC 2015 classification competition. The extremely deep representations also have excellent generalization performance on other recognition tasks, and lead us to further win the 1st places on: ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation in ILSVRC & COCO 2015 competitions. This strong evidence shows that the residual learning principle is generic, and we expect that it is applicable in other vision and non-vision problems. 

2. Related Work Residual Representations. In image recognition, VLAD 

[18] is a representation that encodes by the residual vectors with respect to a dictionary, and Fisher Vector [30] can be formulated as a probabilistic version [18] of VLAD. Both of them are powerful shallow representations for image retrieval and classification [4, 48]. For vector quantization, encoding residual vectors [17] is shown to be more effective than encoding original vectors. 
In low-level vision and computer graphics, for solving Partial Differential Equations (PDEs), the widely used Multigrid method [3] reformulates the system as subproblems at multiple scales, where each subproblem is responsible for the residual solution between a coarser and a finer scale. An alternative to Multigrid is hierarchical basis pre.conditioning [45, 46], which relies on variables that represent residual vectors between two scales. It has been shown [3, 45, 46] that these solvers converge much faster than standard solvers that are unaware of the residual nature of the solutions. These methods suggest that a good reformulation or preconditioning can simplify the optimization. 
Shortcut Connections. Practices and theories that lead to shortcut connections [2, 34, 49] have been studied for a long time. An early practice of training multi-layer perceptrons (MLPs) is to add a linear layer connected from the network input to the output [34, 49]. In [44, 24], a few intermediate layers are directly connected to auxiliary classifiers for addressing vanishing/exploding gradients. The papers of [39, 38, 31, 47] propose methods for centering layer responses, gradients, and propagated errors, implemented by shortcut connections. In [44], an inception layer is composed of a shortcut branch and a few deeper branches. 
Concurrent with our work, highway networks [42, 43] present shortcut connections with gating functions [15]. These gates are data-dependent and have parameters, in contrast to our identity shortcuts that are parameter-free. When a gated shortcut is closed (approaching zero), the layers in highway networks represent non-residual functions. On the contrary, our formulation always learns residual functions; our identity shortcuts are never closed, and all information is always passed through, with additional residual functions to be learned. In addition, high.way networks have not demonstrated accuracy gains with extremely increased depth (e.g., over 100 layers). 

3. Deep Residual Learning 

3.1. Residual Learning 
Let us consider <<H(x)>> as an underlying mapping to be fit by a few stacked layers (not necessarily the entire net), with x denoting the inputs to the first of these layers. If one hypothesizes that multiple nonlinear layers can asymptotically approximate complicated functions, then it is equivalent to hypothesize that they can asymptotically approximate the residual functions, i.e., <<H(x) . x>> (assuming that the input and output are of the same dimensions). So rather than expect stacked layers to approximate <<H(x)>>, we explicitly let these layers approximate a residual function <<F(x) := H(x) . x>>. The original function thus becomes <<F(x)+x>>. Although both forms should be able to asymptotically approximate the desired functions (as hypothesized), the ease of learning might be different. 
This reformulation is motivated by the counterintuitive phenomena about the degradation problem (Fig. 1, left). As we discussed in the introduction, if the added layers can be constructed as identity mappings, a deeper model should have training error no greater than its shallower counterpart. The degradation problem suggests that the solvers might have difficulties in approximating identity mappings by multiple nonlinear layers. With the residual learning reformulation, if identity mappings are optimal, the solvers may simply drive the weights of the multiple nonlinear lay.ers toward zero to approach identity mappings. 
In real cases, it is unlikely that identity mappings are optimal, but our reformulation may help to precondition the problem. If the optimal function is closer to an identity mapping than to a zero mapping, it should be easier for the solver to find the perturbations with reference to an identity mapping, than to learn the function as a new one. We show by experiments (Fig. 7) that the learned residual functions in general have small responses, suggesting that identity map.pings provide reasonable preconditioning. 

3.2. Identity Mapping by Shortcuts 
We adopt residual learning to every few stacked layers. A building block is shown in Fig. 2. Formally, in this paper we consider a building block defined as: 

<<y = F(x, {Wi})+ x>>. (1) 

Here x and y are the input and output vectors of the lay.ers considered. The function <<F(x, {Wi})>> represents the residual mapping to be learned. For the example in Fig. 2 that has two layers, <<F = W_2.(W_1 . x)>> in which <<FORMULA>> denotes 
ReLU [29] and the biases are omitted for simplifying notations. The operation <<FORMULA>> is performed by a shortcut connection and element-wise addition. We adopt the second nonlinearity after the addition (i.e., <<FORMULA>>, see Fig. 2). 
The shortcut connections in Eqn.(1) introduce neither extra parameter nor computation complexity. This is not only attractive in practice but also important in our comparisons between plain and residual networks. We can fairly compare plain/residual networks that simultaneously have the same number of parameters, depth, width, and computational cost (except for the negligible element-wise addition). 
The dimensions of x and F must be equal in Eqn.(1). If this is not the case (e.g., when changing the input/output channels), we can perform a linear projection Ws by the shortcut connections to match the dimensions: 

<<FORMULA>>. (2) 

We can also use a square matrix <<W_s>> in Eqn.(1). But we will show by experiments that the identity mapping is sufficient for addressing the degradation problem and is economical, and thus Ws is only used when matching dimensions. 
The form of the residual function F is flexible. Experiments in this paper involve a function F that has two or three layers (Fig. 5), while more layers are possible. But if F has only a single layer, Eqn.(1) is similar to a linear layer: <<y = W1x + x>>, for which we have not observed advantages. 
We also note that although the above notations are about fully-connected layers for simplicity, they are applicable to convolutional layers. The function <<F(x, {Wi})>> can represent multiple convolutional layers. The element-wise addition is performed on two feature maps, channel by channel. 

3.3. Network Architectures 
We have tested various plain/residual nets, and have observed consistent phenomena. To provide instances for discussion, we describe two models for ImageNet as follows. 
Plain Network. Our plain baselines (Fig. 3, middle) are mainly inspired by the philosophy of VGG nets [41] (Fig. 3, left). The convolutional layers mostly have 3.3 filters and follow two simple design rules: (i) for the same output feature map size, the layers have the same number of filters; and (ii) if the feature map size is halved, the number of filters is doubled so as to preserve the time complexity per layer. We perform downsampling directly by convolutional layers that have a stride of 2. The network ends with a global average pooling layer and a 1000-way fully-connected layer with softmax. The total number of weighted layers is 34 in Fig. 3 (middle). 
It is worth noticing that our model has fewer filters and lower complexity than VGG nets [41] (Fig. 3, left). Our 34 layer baseline has 3.6 billion FLOPs (multiply-adds), which is only 18% of VGG-19 (19.6 billion FLOPs). 


Residual Network. Based on the above plain network, we insert shortcut connections (Fig. 3, right) which turn the network into its counterpart residual version. The identity shortcuts (Eqn.(1)) can be directly used when the input and output are of the same dimensions (solid line shortcuts in Fig. 3). When the dimensions increase (dotted line shortcuts in Fig. 3), we consider two options: (A) The shortcut still performs identity mapping, with extra zero entries padded for increasing dimensions. This option introduces no extra parameter; (B) The projection shortcut in Eqn.(2) is used to match dimensions (done by 1.1 convolutions). For both options, when the shortcuts go across feature maps of two sizes, they are performed with a stride of 2. 

3.4. Implementation 
Our implementation for ImageNet follows the practice in [21, 41]. The image is resized with its shorter side randomly sampled in [256, 480] for scale augmentation [41]. A 224.224 crop is randomly sampled from an image or its horizontal flip, with the per-pixel mean subtracted [21]. The standard color augmentation in [21] is used. We adopt batch normalization (BN) [16] right after each convolution and before activation, following [16]. We initialize the weights as in [13] and train all plain/residual nets from scratch. We use SGD with a mini-batch size of 256. The learning rate starts from 0.1 and is divided by 10 when the error plateaus, and the models are trained for up to 60 . 104 iterations. We use a weight decay of 0.0001 and a momentum of 0.9. We do not use dropout [14], following the practice in [16]. 
In testing, for comparison studies we adopt the standard 10-crop testing [21]. For best results, we adopt the fully 
convolutional form as in [41, 13], and average the scores at multiple scales (images are resized such that the shorter side is in {224, 256, 384, 480, 640}). 

4. Experiments 

4.1. ImageNet Classification 
We evaluate our method on the ImageNet 2012 classification dataset [36] that consists of 1000 classes. The models are trained on the 1.28 million training images, and evaluated on the 50k validation images. We also obtain a final result on the 100k test images, reported by the test server. We evaluate both top-1 and top-5 error rates. 
Plain Networks. We first evaluate 18-layer and 34-layer plain nets. The 34-layer plain net is in Fig. 3 (middle). The 18-layer plain net is of a similar form. See Table 1 for de.
tailed architectures. 
The results in Table 2 show that the deeper 34-layer plain net has higher validation error than the shallower 18-layer plain net. To reveal the reasons, in Fig. 4 (left) we com.pare their training/validation errors during the training procedure. We have observed the degradation problem 

                        <<TABLE>>

Table 1. Architectures for ImageNet. Building blocks are shown in brackets (see also Fig. 5), with the numbers of blocks stacked. Down-sampling is performed by conv3 1, conv4 1, and conv5 1 with a stride of 2. 

Figure 4. Training on ImageNet. Thin curves denote training error, and bold curves denote validation error of the center crops. Left: plain networks of 18 and 34 layers. Right: ResNets of 18 and 34 layers. In this plot, the residual networks have no extra parameter compared to their plain counterparts. 

                    <<FIGURE>>  

Table 2. Top-1 error (%, 10-crop testing) on ImageNet validation. Here the ResNets have no extra parameter compared to their plain counterparts. Fig. 4 shows the training procedures. 
34-layer plain net has higher training error throughout the whole training procedure, even though the solution space of the 18-layer plain network is a subspace of that of the 34-layer one. 
We argue that this optimization difficulty is unlikely to be caused by vanishing gradients. These plain networks are trained with BN [16], which ensures forward propagated signals to have non-zero variances. We also verify that the backward propagated gradients exhibit healthy norms with BN. So neither forward nor backward signals vanish. In fact, the 34-layer plain net is still able to achieve compet.itive accuracy (Table 3), suggesting that the solver works to some extent. We conjecture that the deep plain nets may have exponentially low convergence rates, which impact the reducing of the training error3. The reason for such opti.mization difficulties will be studied in the future. 
Residual Networks. Next we evaluate 18-layer and 34.layer residual nets (ResNets). The baseline architectures are the same as the above plain nets, expect that a shortcut connection is added to each pair of 3.3 filters as in Fig. 3 (right). In the first comparison (Table 2 and Fig. 4 right), we use identity mapping for all shortcuts and zero-padding for increasing dimensions (option A). So they have no extra parameter compared to the plain counterparts. 
We have three major observations from Table 2 and Fig. 4. First, the situation is reversed with residual learn.ing fi the 34-layer ResNet is better than the 18-layer ResNet (by 2.8%). More importantly, the 34-layer ResNet exhibits considerably lower training error and is generalizable to the validation data. This indicates that the degradation problem is well addressed in this setting and we manage to obtain accuracy gains from increased depth. 
Second, compared to its plain counterpart, the 34-layer 
3We have experimented with more training iterations (3.) and still ob.served the degradation problem, suggesting that this problem cannot be feasibly addressed by simply using more iterations. 

                    <<TABLE>>

Table 3. Error rates (%, 10-crop testing) on ImageNet validation. VGG-16 is based on our test. ResNet-50/101/152 are of option B that only uses projections for increasing dimensions. 

                    <<TABLE>>

Table 4. Error rates (%) of single-model results on the ImageNet validation set (except fi reported on the test set). 

                    <<TABLE>>

Table 5. Error rates (%) of ensembles. The top-5 error is on the test set of ImageNet and reported by the test server.

                    <<TABLE>>

ResNet reduces the top-1 error by 3.5% (Table 2), resulting from the successfully reduced training error (Fig. 4 right vs. left). This comparison verifies the effectiveness of residual learning on extremely deep systems. 
Last, we also note that the 18-layer plain/residual nets are comparably accurate (Table 2), but the 18-layer ResNet converges faster (Fig. 4 right vs. left). When the net is not overly deep (18 layers here), the current SGD solver is still able to find good solutions to the plain net. In this case, the ResNet eases the optimization by providing faster convergence at the early stage. 
Identity vs. Projection Shortcuts. We have shown that 

Figure 5. A deeper residual function F for ImageNet. Left: a building block (on 56.56 feature maps) as in Fig. 3 for ResNet.

        <<FIGURE>>

parameter-free, identity shortcuts help with training. Next we investigate projection shortcuts (Eqn.(2)). In Table 3 we compare three options: (A) zero-padding shortcuts are used for increasing dimensions, and all shortcuts are parameter-free (the same as Table 2 and Fig. 4 right); (B) projection shortcuts are used for increasing dimensions, and other shortcuts are identity; and (C) all shortcuts are projections. 
Table 3 shows that all three options are considerably bet.
ter than the plain counterpart. B is slightly better than A. We argue that this is because the zero-padded dimensions in A indeed have no residual learning. C is marginally better than B, and we attribute this to the extra parameters introduced by many (thirteen) projection shortcuts. But the small differences among A/B/C indicate that projection shortcuts are not essential for addressing the degradation problem. So we do not use option C in the rest of this paper, to reduce mem.ory/time complexity and model sizes. Identity shortcuts are particularly important for not increasing the complexity of the bottleneck architectures that are introduced below. 
Deeper Bottleneck Architectures. Next we describe our deeper nets for ImageNet. Because of concerns on the train.ing time that we can afford, we modify the building block as a bottleneck design4. For each residual function F, we use a stack of 3 layers instead of 2 (Fig. 5). The three layers are 1.1, 3.3, and 1.1 convolutions, where the 1.1 layers are responsible for reducing and then increasing (restoring) dimensions, leaving the 3.3 layer a bottleneck with smaller input/output dimensions. Fig. 5 shows an example, where both designs have similar time complexity. 
The parameter-free identity shortcuts are particularly important for the bottleneck architectures. If the identity short.cut in Fig. 5 (right) is replaced with projection, one can show that the time complexity and model size are doubled, as the shortcut is connected to the two high-dimensional ends. So identity shortcuts lead to more efficient models for the bottleneck designs. 
50-layer ResNet: We replace each 2-layer block in the 
4Deeper non-bottleneck ResNets (e.g., Fig. 5 left) also gain accuracy from increased depth (as shown on CIFAR-10), but are not as economical as the bottleneck ResNets. So the usage of bottleneck designs is mainly due to practical considerations. We further note that the degradation problem of plain nets is also witnessed for the bottleneck designs. 

34-layer net with this 3-layer bottleneck block, resulting in a 50-layer ResNet (Table 1). We use option B for increasing dimensions. This model has 3.8 billion FLOPs. 
101-layer and 152-layer ResNets: We construct 101.layer and 152-layer ResNets by using more 3-layer blocks (Table 1). Remarkably, although the depth is significantly increased, the 152-layer ResNet (11.3 billion FLOPs) still has lower complexity than VGG-16/19 nets (15.3/19.6 bil.lion FLOPs). 
The 50/101/152-layer ResNets are more accurate than the 34-layer ones by considerable margins (Table 3 and 4). We do not observe the degradation problem and thus enjoy significant accuracy gains from considerably increased depth. The benefits of depth are witnessed for all evaluation metrics (Table 3 and 4). 
Comparisons with State-of-the-art Methods. In Table 4 we compare with the previous best single-model results. Our baseline 34-layer ResNets have achieved very competitive accuracy. Our 152-layer ResNet has a single-model top-5 validation error of 4.49%. This single-model result outperforms all previous ensemble results (Table 5). We combine six models of different depth to form an ensemble (only with two 152-layer ones at the time of submitting). This leads to 3.57% top-5 error on the test set (Table 5). 
This entry won the 1st place in ILSVRC 2015. 

4.2. CIFAR-10 and Analysis 

We conducted more studies on the CIFAR-10 dataset [20], which consists of 50k training images and 10k test.ing images in 10 classes. We present experiments trained on the training set and evaluated on the test set. Our focus is on the behaviors of extremely deep networks, but not on pushing the state-of-the-art results, so we intentionally use simple architectures as follows. 
The plain/residual architectures follow the form in Fig. 3 (middle/right). The network inputs are 32.32 images, with the per-pixel mean subtracted. The first layer is 3.3 convolutions. Then we use a stack of 6n layers with 3.3 convolutions on the feature maps of sizes {32, 16, 8} respectively, with 2n layers for each feature map size. The numbers of filters are {16, 32, 64} respectively. The subsampling is per.formed by convolutions with a stride of 2. The network ends with a global average pooling, a 10-way fully-connected layer, and softmax. There are totally 6n+2 stacked weighted layers. The following table summarizes the architecture: 

<<TABLE>>

When shortcut connections are used, they are connected to the pairs of 3.3 layers (totally 3n shortcuts). On this dataset we use identity shortcuts in all cases (i.e., option A), 

<<TABLE>>

Table 6. Classification error on the CIFAR-10 test set. All meth.ods are with data augmentation. For ResNet-110, we run it 5 times and show best (mean std) as in [43]. 

so our residual models have exactly the same depth, width, and number of parameters as the plain counterparts. 
We use a weight decay of 0.0001 and momentum of 0.9, and adopt the weight initialization in [13] and BN [16] but with no dropout. These models are trained with a mini-batch size of 128 on two GPUs. We start with a learning rate of 0.1, divide it by 10 at 32k and 48k iterations, and terminate training at 64k iterations, which is determined on a 45k/5k train/val split. We follow the simple data augmen.tation in [24] for training: 4 pixels are padded on each side, and a 32.32 crop is randomly sampled from the padded image or its horizontal fiip. For testing, we only evaluate the single view of the original 32.32 image. 
We compare n = {3, 5, 7, 9}, leading to 20, 32, 44, and 56-layer networks. Fig. 6 (left) shows the behaviors of the plain nets. The deep plain nets suffer from increased depth, and exhibit higher training error when going deeper. This phenomenon is similar to that on ImageNet (Fig. 4, left) and on MNIST (see [42]), suggesting that such an optimization difficulty is a fundamental problem. 
Fig. 6 (middle) shows the behaviors of ResNets. Also similar to the ImageNet cases (Fig. 4, right), our ResNets manage to overcome the optimization difficulty and demon.strate accuracy gains when the depth increases. 
We further explore n = 18 that leads to a 110-layer ResNet. In this case, we find that the initial learning rate of 0.1 is slightly too large to start converging5. So we use 
0.01 to warm up the training until the training error is below 80% (about 400 iterations), and then go back to 0.1 and continue training. The rest of the learning schedule is as done previously. This 110-layer network converges well (Fig. 6, middle). It has fewer parameters than other deep and thin 
5With an initial learning rate of 0.1, it starts converging (<90% error) after several epochs, but still reaches similar accuracy. 

<<FIGURE>>

Figure 7. Standard deviations (std) of layer responses on CIFAR.
10. The responses are the outputs of each 3.3 layer, after BN and before nonlinearity. Top: the layers are shown in their original order. Bottom: the responses are ranked in descending order. 
networks such as FitNet [35] and Highway [42] (Table 6), yet is among the state-of-the-art results (6.43%, Table 6). 
Analysis of Layer Responses. Fig. 7 shows the standard deviations (std) of the layer responses. The responses are the outputs of each 3.3 layer, after BN and before other nonlinearity (ReLU/addition). For ResNets, this analysis reveals the response strength of the residual functions. Fig. 7 shows that ResNets have generally smaller responses than their plain counterparts. These results support our ba.sic motivation (Sec.3.1) that the residual functions might be generally closer to zero than the non-residual functions. We also notice that the deeper ResNet has smaller magnitudes of responses, as evidenced by the comparisons among ResNet-20, 56, and 110 in Fig. 7. When there are more layers, an individual layer of ResNets tends to modify the signal less. 
Exploring Over 1000 layers. We explore an aggressively deep model of over 1000 layers. We set n = 200 that leads to a 1202-layer network, which is trained as described above. Our method shows no optimization difficulty, and this 103-layer network is able to achieve training error <0.1% (Fig. 6, right). Its test error is still fairly good (7.93%, Table 6). 
But there are still open problems on such aggressively deep models. The testing result of this 1202-layer network is worse than that of our 110-layer network, although both 

<<TABLE>>

Table 7. Object detection mAP (%) on the PASCAL VOC 2007/2012 test sets using baseline Faster R-CNN. See also Ta.ble 10 and 11 for better results. 

<<TABLE>>

Table 8. Object detection mAP (%) on the COCO validation set using baseline Faster R-CNN. See also Table 9 for better results. 
have similar training error. We argue that this is because of overfitting. The 1202-layer network may be unnecessarily large (19.4M) for this small dataset. Strong regularization such as maxout [10] or dropout [14] is applied to obtain the best results ([10, 25, 24, 35]) on this dataset. In this paper, we use no maxout/dropout and just simply impose regularization via deep and thin architectures by design, without distracting from the focus on the difficulties of optimization. But combining with stronger regularization may im.prove results, which we will study in the future. 

4.3. Object Detection on PASCAL and MS COCO 
Our method has good generalization performance on other recognition tasks. Table 7 and 8 show the object detection baseline results on PASCAL VOC 2007 and 2012 
[5] and COCO [26]. We adopt Faster R-CNN [32] as the detection method. Here we are interested in the improvements of replacing VGG-16 [41] with ResNet-101. The detection implementation (see appendix) of using both models is the same, so the gains can only be attributed to better networks. Most remarkably, on the challenging COCO dataset we ob.tain a 6.0% increase in COCOfis standard metric (mAP@[.5, .95]), which is a 28% relative improvement. This gain is solely due to the learned representations. 
Based on deep residual nets, we won the 1st places in several tracks in ILSVRC & COCO 2015 competitions: Im.ageNet detection, ImageNet localization, COCO detection, and COCO segmentation. The details are in the appendix. 

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A. Object Detection Baselines 

In this section we introduce our detection method based on the baseline Faster R-CNN [32] system. The models are initialized by the ImageNet classification models, and then fine-tuned on the object detection data. We have experi.mented with ResNet-50/101 at the time of the ILSVRC & COCO 2015 detection competitions. 
Unlike VGG-16 used in [32], our ResNet has no hidden fc layers. We adopt the idea of fiNetworks on Conv feature maps (NoC) [33] to address this issue. We compute the full-image shared conv feature maps using those lay.ers whose strides on the image are no greater than 16 pixels (i.e., conv1, conv2 x, conv3 x, and conv4 x, totally 91 conv layers in ResNet-101; Table 1). We consider these layers as analogous to the 13 conv layers in VGG-16, and by doing so, both ResNet and VGG-16 have conv feature maps of the same total stride (16 pixels). These layers are shared by a region proposal network (RPN, generating 300 proposals) 
[32] and a Fast R-CNN detection network [7]. RoI pool.ing [7] is performed before conv5 1. On this RoI-pooled feature, all layers of conv5 x and up are adopted for each region, playing the roles of VGG-16fis fc layers. The final classification layer is replaced by two sibling layers (classi.fication and box regression [7]). 
For the usage of BN layers, after pre-training, we compute the BN statistics (means and variances) for each layer on the ImageNet training set. Then the BN layers are fixed during fine-tuning for object detection. As such, the BN layers become linear activations with constant offsets and scales, and BN statistics are not updated by fine-tuning. We fix the BN layers mainly for reducing memory consumption in Faster R-CNN training. 
PASCAL VOC 
Following [7, 32], for the PASCAL VOC 2007 test set, we use the 5k trainval images in VOC 2007 and 16k train-val images in VOC 2012 for training (fi07+12fi). For the PASCAL VOC 2012 test set, we use the 10k trainval+test images in VOC 2007 and 16k trainval images in VOC 2012 for training (fi07++12fi). The hyper-parameters for train.ing Faster R-CNN are the same as in [32]. Table 7 shows the results. ResNet-101 improves the mAP by >3% over VGG-16. This gain is solely because of the improved features learned by ResNet. 
MS COCO 
The MS COCO dataset [26] involves 80 object cate.gories. We evaluate the PASCAL VOC metric (mAP @ IoU = 0.5) and the standard COCO metric (mAP @ IoU = .5:.05:.95). We use the 80k images on the train set for train.ing and the 40k images on the val set for evaluation. Our detection system for COCO is similar to that for PASCAL VOC. We train the COCO models with an 8-GPU imple.mentation, and thus the RPN step has a mini-batch size of 8 images (i.e., 1 per GPU) and the Fast R-CNN step has a mini-batch size of 16 images. The RPN step and Fast R.CNN step are both trained for 240k iterations with a learn.ing rate of 0.001 and then for 80k iterations with 0.0001. 
Table 8 shows the results on the MS COCO validation set. ResNet-101 has a 6% increase of mAP@[.5, .95] over VGG-16, which is a 28% relative improvement, solely con.tributed by the features learned by the better network. Re.markably, the mAP@[.5, .95]fis absolute increase (6.0%) is nearly as big as mAP@.5fis (6.9%). This suggests that a deeper network can improve both recognition and localiza.tion. 
B. Object Detection Improvements 
For completeness, we report the improvements made for the competitions. These improvements are based on deep features and thus should benefit from residual learning. 
MS COCO 
Box refinement. Our box refinement partially follows the it.erative localization in [6]. In Faster R-CNN, the final output is a regressed box that is different from its proposal box. So for inference, we pool a new feature from the regressed box and obtain a new classification score and a new regressed box. We combine these 300 new predictions with the orig.inal 300 predictions. Non-maximum suppression (NMS) is applied on the union set of predicted boxes using an IoU threshold of 0.3 [8], followed by box voting [6]. Box re.finement improves mAP by about 2 points (Table 9). 
Global context. We combine global context in the Fast R-CNN step. Given the full-image conv feature map, we pool a feature by global Spatial Pyramid Pooling [12] (with a fisingle-levelfi pyramid) which can be implemented as fiRoIfi pooling using the entire imagefis bounding box as the RoI. This pooled feature is fed into the post-RoI layers to obtain a global context feature. This global feature is con.catenated with the original per-region feature, followed by the sibling classification and box regression layers. This new structure is trained end-to-end. Global context im.proves mAP@.5 by about 1 point (Table 9). 
Multi-scale testing. In the above, all results are obtained by single-scale training/testing as in [32], where the imagefis shorter side is s = 600 pixels. Multi-scale training/testing has been developed in [12, 7] by selecting a scale from a feature pyramid, and in [33] by using maxout layers. In our current implementation, we have performed multi-scale testing following [33]; we have not performed multi-scale training because of limited time. In addition, we have per.formed multi-scale testing only for the Fast R-CNN step (but not yet for the RPN step). With a trained model, we compute conv feature maps on an image pyramid, where the imagefis shorter sides are s .{200, 400, 600, 800, 1000}. 

                <<TABLE>>

Table 9. Object detection improvements on MS COCO using Faster R-CNN and ResNet-101. 

    <<TABLE>>

Table 10. Detection results on the PASCAL VOC 2007 test set. The baseline is the Faster R-CNN system. The system fibaseline+++fi include box refinement, context, and multi-scale testing in Table 9. 


system  net  data  mAP  areo bike bird boat bottle bus car cat chair cow table dog horse mbike person plant sheep sofa train tv  
baseline baseline baseline+++  VGG-16 ResNet-101 ResNet-101  07++12 07++12 COCO+07++12  70.4 73.8 83.8  84.9 79.8 74.3 53.9 49.8 77.5 75.9 88.5 45.6 77.1 55.3 86.9 81.7 80.9 79.6 40.1 72.6 60.9 81.2 61.5 86.5 81.6 77.2 58.0 51.0 78.6 76.6 93.2 48.6 80.4 59.0 92.1 85.3 84.8 80.7 48.1 77.3 66.5 84.7 65.6 92.1 88.4 84.8 75.9 71.4 86.3 87.8 94.2 66.8 89.4 69.2 93.9 91.9 90.9 89.6 67.9 88.2 76.8 90.3 80.0  
Table 11. Detection results on the PASCAL VOC 2012 test set (http://host.robots.ox.ac.uk:8080/leaderboard/displaylb.php?challengeid=11&compid=4). The baseline is the Faster R-CNN system. The system baseline+++ include box refinement, context, and multi-scale testing in Table 9. 


We select two adjacent scales from the pyramid following [33]. RoI pooling and subsequent layers are performed on the feature maps of these two scales [33], which are merged by maxout as in [33]. Multi-scale testing improves the mAP by over 2 points (Table 9). 
Using validation data. Next we use the 80k+40k trainval set for training and the 20k test-dev set for evaluation. The test.dev set has no publicly available ground truth and the result is reported by the evaluation server. Under this setting, the results are an mAP@.5 of 55.7% and an mAP@[.5, .95] of 34.9% (Table 9). This is our single-model result. 
Ensemble. In Faster R-CNN, the system is designed to learn region proposals and also object classifiers, so an ensemble can be used to boost both tasks. We use an ensemble for proposing regions, and the union set of proposals are pro.cessed by an ensemble of per-region classifiers. Table 9 shows our result based on an ensemble of 3 networks. The mAP is 59.0% and 37.4% on the test-dev set. This result won the 1st place in the detection task in COCO 2015. 

We revisit the PASCAL VOC dataset based on the above model. With the single model on the COCO dataset (55.7% mAP@.5 in Table 9), we fine-tune this model on the PAS.CAL VOC sets. The improvements of box refinement, con.text, and multi-scale testing are also adopted. By doing so we achieve 85.6% mAP on PASCAL VOC 2007 (Table 10) and 83.8% on PASCAL VOC 2012 (Table 11)6. The result on PASCAL VOC 2012 is 10 points higher than the previ.ous state-of-the-art result [6]. 

<<TABLE>>

Table 12. Our results (mAP, %) on the ImageNet detection dataset. Our detection system is Faster R-CNN [32] with the improvements in Table 9, using ResNet-101. 


ImageNet Detection 
The ImageNet Detection (DET) task involves 200 object categories. The accuracy is evaluated by mAP@.5. Our object detection algorithm for ImageNet DET is the same as that for MS COCO in Table 9. The networks are pre.trained on the 1000-class ImageNet classification set, and are fine-tuned on the DET data. We split the validation set into two parts (val1/val2) following [8]. We fine-tune the detection models using the DET training set and the val1 set. The val2 set is used for validation. We do not use other ILSVRC 2015 data. Our single model with ResNet-101 has 

<<TABLE>>

Table 13. Localization error (%) on the ImageNet validation. In the column of fiLOC error on GT classfi ([41]), the ground truth class is used. In the fitestingfi column, fi1-cropfi denotes testing on a center crop of 224.224 pixels, fidensefi denotes dense (fully convolutional) and multi-scale testing. 

<<TABLE>> 

Table 14. Comparisons of localization error (%) on the ImageNet dataset with state-of-the-art methods. 


58.8% mAP and our ensemble of 3 models has 62.1% mAP on the DET test set (Table 12). This result won the 1st place in the ImageNet detection task in ILSVRC 2015, surpassing the second place by 8.5 points (absolute). 

C. ImageNet Localization 

The ImageNet Localization (LOC) task [36] requires to classify and localize the objects. Following [40, 41], we assume that the image-level classifiers are first adopted for predicting the class labels of an image, and the localiza.tion algorithm only accounts for predicting bounding boxes based on the predicted classes. We adopt the fiper-class re.gressionfi (PCR) strategy [40, 41], learning a bounding box regressor for each class. We pre-train the networks for Im.ageNet classification and then fine-tune them for localiza.tion. We train networks on the provided 1000-class Ima.geNet training set. 
Our localization algorithm is based on the RPN frame.work of [32] with a few modifications. Unlike the way in 
[32] that is category-agnostic, our RPN for localization is designed in a per-class form. This RPN ends with two sib.ling 1.1 convolutional layers for binary classification (cls) and box regression (reg), as in [32]. The cls and reg layers are both in a per-class from, in contrast to [32]. Specifi.cally, the cls layer has a 1000-d output, and each dimension is binary logistic regression for predicting being or not be.ing an object class; the reg layer has a 1000.4-d output consisting of box regressors for 1000 classes. As in [32], our bounding box regression is with reference to multiple translation-invariant fianchorfi boxes at each position. 
As in our ImageNet classification training (Sec. 3.4), we randomly sample 224.224 crops for data augmentation. We use a mini-batch size of 256 images for fine-tuning. To avoid negative samples being dominate, 8 anchors are ran.domly sampled for each image, where the sampled positive and negative anchors have a ratio of 1:1 [32]. For testing, the network is applied on the image fully-convolutionally. 
Table 13 compares the localization results. Following [41], we first perform fioraclefi testing using the ground truth class as the classification prediction. VGGfis paper [41] re-ports a center-crop error of 33.1% (Table 13) using ground truth classes. Under the same setting, our RPN method us.ing ResNet-101 net significantly reduces the center-crop er.ror to 13.3%. This comparison demonstrates the excellent performance of our framework. With dense (fully convolu.tional) and multi-scale testing, our ResNet-101 has an error of 11.7% using ground truth classes. Using ResNet-101 for predicting classes (4.6% top-5 classification error, Table 4), the top-5 localization error is 14.4%. 
The above results are only based on the proposal network (RPN) in Faster R-CNN [32]. One may use the detection network (Fast R-CNN [7]) in Faster R-CNN to improve the results. But we notice that on this dataset, one image usually contains a single dominate object, and the proposal regions highly overlap with each other and thus have very similar RoI-pooled features. As a result, the image-centric training of Fast R-CNN [7] generates samples of small variations, which may not be desired for stochastic training. Motivated by this, in our current experiment we use the original R-CNN [8] that is RoI-centric, in place of Fast R-CNN. 
Our R-CNN implementation is as follows. We apply the per-class RPN trained as above on the training images to predict bounding boxes for the ground truth class. These predicted boxes play a role of class-dependent proposals. For each training image, the highest scored 200 proposals are extracted as training samples to train an R-CNN classi.fier. The image region is cropped from a proposal, warped to 224.224 pixels, and fed into the classification network as in R-CNN [8]. The outputs of this network consist of two sibling fc layers for cls and reg, also in a per-class form. This R-CNN network is fine-tuned on the training set us.ing a mini-batch size of 256 in the RoI-centric fashion. For testing, the RPN generates the highest scored 200 proposals for each predicted class, and the R-CNN network is used to update these proposalsfi scores and box positions. 
This method reduces the top-5 localization error to 10.6% (Table 13). This is our single-model result on the validation set. Using an ensemble of networks for both clas.sification and localization, we achieve a top-5 localization error of 9.0% on the test set. This number significantly out.performs the ILSVRC 14 results (Table 14), showing a 64% relative reduction of error. This result won the 1st place in the ImageNet localization task in ILSVRC 2015. 
<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>
                    Direct Feedback Alignment Scales to Modern Deep Learning Tasks and Architectures

                      Julien Launay 1;2  Iacopo Poli 1  François Boniface 1  Florent Krzakala 1;2

                                     1 LightOn   2 École Normale Supérieure

                                               Abstract

                       Despite being the workhorse of deep learning, the backpropagation algorithm is
                       no panacea. It enforces sequential layer updates, thus preventing efficient paral-
                       lelization of the training process. Furthermore, its biological plausibility is being
                       challenged. Alternative schemes have been devised; yet, under the constraint of
                       synaptic asymmetry, none have scaled to modern deep learning tasks and architec-
                       tures. Here, we challenge this perspective, and study the applicability of Direct
                       Feedback Alignment to neural view synthesis, recommender systems, geometric
                       learning, and natural language processing. In contrast with previous studies lim-
                       ited to computer vision tasks, our findings show that it successfully trains a large
                       range of state-of-the-art deep learning architectures, with performance close to
                       fine-tuned backpropagation. At variance with common beliefs, our work supports
                       that challenging tasks can be tackled in the absence of weight transport.


                 1 Introduction

                 While the backpropagation algorithm (BP) [1,2] is at the heart of modern deep learning achievements,
                 it is not without pitfalls. For one, its weight updates are non-local and rely on upstream layers. Thus,
                 they cannot be easily parallelized [3], incurring important memory and compute costs. Moreover,
                 its biological implementation is problematic [4,5]. For instance, BP relies on the transpose of the
                 weights to evaluate updates. Hence, synaptic symmetry is required between the forward and backward
                 path: this is implausible in biological brains, and known as the weight transport problem [6].
                 Consequently, alternative training algorithms have been developed. Some of these algorithms are
                 explicitly biologically inspired [7–13], while others focus on making better use of available compute
                 resources [3,14–19]. Despite these enticing characteristics, none has been widely adopted, as they
                 are often demonstrated on a limited set of tasks. Moreover, as assessed in [20], their performance on
                 challenging datasets under the constraint of synaptic asymmetry is disappointing.
                 We seek to broaden this perspective, and demonstrate the applicability of Direct Feedback Alignment
                 (DFA) [19] in state-of-the-art settings: from applications of fully connected networks such as neural
                 view synthesis and recommender systems, to geometric learning with graph convolutions, and natural
                 language processing with Transformers. Our results define new standards for learning without weight
                 transport and show that challenging tasks can indeed be tackled under synaptic asymmetry.
                 All code needed to reproduce our experiments is available at https://github.com/lightonai/dfa-scales-to-modern-deep-learning.

                                           1.1 Related work

                 Training a neural network is a credit assignment problem: an update is derived for each parameter
                 from its contribution to a cost function. To solve this problem, a spectrum of algorithms exists [21].

                 Biologically motivated methods Finding a training method applicable under the constraints of
                 biological brains remains an open problem. End-to-end propagation of gradients is unlikely to occur
                 [22], implying local learning is required. Furthermore, the weight transport problem enforces synaptic
                 asymmetry [6]. Inspired by auto-encoders, target propagation methods (TP) [10–12] train distinct
                 feedback connections to invert the feedforward ones. Feedback alignment (FA) [13] replaces the
                 transpose of the forward weights used in the backward pass by a random matrix. Throughout training,
                 the forward weights learn to align with the arbitrary backward weights, eventually approximating BP.

                 Beyond biological considerations As deep learning models grow bigger, large-scale distributed
                 training is increasingly desirable. Greedy layer-wise training [14] allows networks to be built layer
                 by layer, limiting the depth of backpropagation. To enable parallelization of the backward pass,
                 updates must only depend on local quantities. Unsupervised learning is naturally suited for this,
                 as it relies on local losses such as Deep InfoMax [17] and Greedy InfoMax [18]. More broadly,
                 synthetic gradient methods, like decoupled neural interfaces [3,15] and local error signals (LES)
                 [16], approximate gradients using layer-wise trainable feedback networks. DFA [19] expands on FA
                 and directly projects a global error to each layer. A shared feedback path is still needed, but it only
                 depends on a simple random projection operation.

                 Performance of alternative methods Local training methods are successful in unsupervised learn-
                 ing [18]. Even in a supervised setting, they scale to challenging datasets like CIFAR-100 or ImageNet
                 [14,16]. Thus, locality is not too penalizing. However, TP, FA, and DFA are unable to scale to these
                 tasks [20]. In fact, DFA is unable to train convolutional layers [23]. To enable feedback alignment
                 techniques to perform well on challenging datasets, some form of weight transport is necessary:
                 either by explicitly sharing sign information [24–26], or by introducing dedicated phases of alignment
                 for the forward and backward weights where some information is shared [27]. To the best of our
                 knowledge, no method compatible with the weight transport problem has ever been demonstrated on
                 challenging tasks.

                 1.2 Motivations and contributions

                 We focus on DFA, a compromise between biological and computational considerations. Notably,
                 DFA is compatible with synaptic asymmetry: this asymmetry raises important challenges, seemingly
                 preventing learning in demanding settings. Moreover, it allows for asynchronous weight updates,
                 and puts a single operation at the center of the training stage. This enables new classes of training
                 co-processors [28, 29], leveraging dedicated hardware to perform the random projection.

                 Extensive survey We apply DFA in a large variety of settings matching current trends in machine
                 learning. Previous works have found that DFA is unsuitable for computer vision tasks [20,23]; but
                 computer vision alone cannot be the litmus test of a training method. Instead, we consider four vastly
                 different domains, across eight tasks, and with eleven different architectures. This constitutes a survey
                 of unprecedented scale for an alternative training method, and makes a strong case for the possibility
                 of learning without weight transport in demanding scenarios.

                 Challenging settings We demonstrate the ability of DFA to tackle challenging tasks. We success-
                 fully learn and render real-world 3D scenes (section 3.1.1); we perform recommendation at scale
                 (section 3.1.2); we explore graph-based citation networks (section 3.2); and we consider language
                 modelling with a Transformer (section 3.3). We study tasks at the state-of-the-art level, that have
                 only been recently successfully tackled with deep learning.

                 Modern architectures We prove that the previously established failure of DFA to train convolutions
                 does not generalize. By evaluating performance metrics, comparing against a shallow baseline,
                 measuring alignment, and visualizing t-SNE embeddings, we show that learning indeed occurs in
                 layers involving graph convolutions and attention. This significantly broadens the applicability of
                 DFA–previously thought to be limited to simple problems like MNIST and CIFAR-10.

                                                  2 Methods

                 Forward pass In a fully connected network, at layer i out of N, neglecting its biases, with W_i its
                 weight matrix, f_i its non-linearity, and hi its activations, the forward pass is:

                                   <<FORMULA>>               (1)

                 <<FORMULA>> is the input data, and <<FORMULA>> are the predictions. A task-specific cost function
                 <<FORMULA>> is computed to quantify the quality of the predictions with respect to the targets y.

                  Backward pass with BP The weight updates are computed by backpropagation of the error vector.
                 Using the chain-rule of derivatives, each neuron is updated based on its contribution to the cost
                 function. Leaving aside the specifics of the optimizer used, the equation for the weight updates is:

                                     <<FORMULA>>                              (2)

                  Backward pass with DFA The gradient signal <<FORMULA>> of the (i+1)-th layer violates synaptic
                 asymmetry. DFA replaces it with a random projection of the topmost derivative of the loss, <<FORMULA>>.
                 For common classification and regression losses such as the mean squared error or the negative log
                 likelihood, this corresponds to a random projection of the global error <<FORMULA>>. With B_i, a fixed
                 random matrix of appropriate shape drawn at initialization for each layers:

                                                                <<FORMULA>>                  (3)

                 3 Experiments

                 We study the applicability of DFA to a diverse set of applications requiring state-of-the-art architec-
                 tures. We start with fully connected networks, where DFA has already been demonstrated, and address
                 new challenging settings. We then investigate geometric learning: we apply DFA to graph neural net-
                 works in classification tasks on citation networks, as well as graph autoencoders. These architectures
                 feature graph convolutions and attention layers. Finally, we use DFA to train a transformer-based
                 Natural Language Processing (NLP) model on a dataset of more than 100 million tokens.

                 3.1 Fully connected architectures

                 DFA has been successful at training fully connected architectures, with performance on-par with
                 backpropagation [19,20]. However, only computer vision tasks have been considered, where fully
                 connected networks considerably underperform their convolutional counterpart. Here, we focus on
                 tasks where fully connected architectures are state-of-the-art. Moreover, the architectures considered
                 are deeper and more complex than those necessary to solve a simple task like MNIST.

                 3.1.1 Neural view synthesis with Neural Radiance Fields
                 The most recent state-of-the-art neural view synthesis methods are based on large fully connected
                 networks: this is an ideal setting for a first evaluation of DFA on a challenging task.

                 Background There has been growing interest in methods capable of synthesizing novel renders of
                 a 3D scene using a dataset of past renders. The network is trained to learn an inner representation of
                 the scene, and a classical rendering system can then query the model to generate novel views. With
                 robust enough methods, real-world scenes can also be learned from a set of pictures.
                 Until recently, most successful neural view synthesis methods were based on sampled volumetric
                 representations [30–32]. In this context, Convolutional Neural Networks (CNNs) can be used to
                 smooth out the discrete sampling of 3D space [33,34]. However, these methods scale poorly to
                 higher resolutions, as they still require finer and finer sampling. Conversely, alternative schemes
                 based on a continuous volume representation have succeeded in generating high-quality renders [35],
                 even featuring complex phenomenons such as view-dependant scattering [36]. These schemes make
                 point-wise predictions, and use fully connected neural networks to encode the scene.

                                            <<FIGURE>>

                 Figure 1: Comparisons of NeRF-DFA with state-of-the-art methods trained with BP on the most
                 challenging synthetic and real-world scenes. While NeRF-DFA generates render of lower quality,
                 they maintain multi-view consistency and exhibit no geometric artifacts. BP results from [36].


                 Setting We employ Neural Radiance Fields (NeRF) [36], the state-of-the-art for neural view
                 synthesis. NeRF represents scenes as a continuous 5D function of space–three spatial coordinates,
                 two viewing angles–and outputs a point-wise RGB radiance and opacity. A ray-casting renderer can
                 then query the network to generate arbitrary views of the scene. The network modeling the continuous
                 function is 10 layers deep. Two identical networks are trained: the coarse network predictions inform
                 the renderer about the spatial coordinates that the fine network should preferentially evaluate to avoid
                 empty space and occluded regions.

                 Results We report quantitative results of training NeRF with DFA in Table 1. Neural view synthesis
                 methods are often better evaluated qualitatively: we showcase some renders in Figure 1.
                 On a dataset of renders featuring complex scenes with non-Lambertian materials (NeRF-Synthetic
                 [36]), NeRF-DFA outperforms two previous fine-tuned state-of-the-art methods–Scene Representation
                 Networks (SRN) [35] and Local Light Field Fusion (LLFF) [32]–and nearly matches the performance
                 of Neural Volumes (NV) [34]. While DFA underperforms alternative methods trained with BP on
                 the real world view dataset (LLFF-Real [32]), its performance remains significant: real world view
                 synthesis is a challenging tasks, and this level of PSNR indicates that learning is indeed happening.
                 In particular, we find that NeRF-DFA retains the key characteristics of NeRF-BP: it can render view-
                 dependant effects, and is multi-view consistent. The last point is an especially important achievement,
                 and most visible in videos, as it is a challenge for most algorithms [30–32,35]. The main drawback
                 of NeRF-DFA appears to be a seemingly lower render definition. The NeRF architecture has not


                 Table 1: Peak Signal to Noise Ratio (PSNR, higher is better) of neural view synthesis methods
                 trained with backpropagation against NeRF trained with DFA. Even when trained with DFA, NeRF
                 outperforms two state-of-the-art methods on a synthetic dataset (NeRF-Synthetic), and achieves fair
                 performance on a challenging real world views datasets (LLFF-Real). BP results from [36].

                                            <<TABLE>>

                 been fine-tuned to achieve these results: DFA works out-of-the-box on this advanced method. Future
                 research focusing on architectural changes to NeRF could improve performance with DFA; some
                 preliminary results are included in the supplementary material.

                 3.1.2 Click-through rate prediction with recommender systems
                 We have demonstrated that DFA can train large fully connected networks on the difficult task of neural
                 view synthesis. We now seek to use DFA in more complex heterogeneous architectures, combining
                 the use of fully connected networks with other machine learning methods.Recommender systems are
                 an ideal application for such considerations.

                 Background Recommender systems are used to model the behavior of users and predict future
                 interactions. In particular, in the context of click-through rate (CTR) prediction, these systems model
                 the probability of a user clicking on a given item. Building recommender systems is hard [37]: their
                 input is high-dimensional and sparse, and the model must learn to extract high-order combinatorial
                 features from the data. Moreover, they need to do so efficiently, as they are used to make millions of
                 predictions and the training data may contain billions of examples.
                 Factorization Machines (FM) [38] use inner-products of latent vectors between features to extract
                 pairwise feature interactions. They constitute an excellent baseline for shallow recommender systems,
                 but fail to efficiently transcribe higher-level features. To avoid extensive feature engineering, it has
                 been suggested that deep learning can be used in conjunction with wide shallow models to extract
                 these higher-level features [39]. In production, these systems are regularly retrained on massive
                 datasets: the speedup allowed by backward unlocking in DFA is thus of particular interest.

                 Setting Deep Factorization Machines (DeepFM) [40] combine FM and a deep fully connected
                 neural network, which we train with DFA. The input embedding is still trained directly via gradient
                 descent, as weight transport is not necessary to backpropagate through the FM. Deep & Cross
                 Networks (DCN) [41] replace the FM with a Cross Network, a deep architecture without non-
                 linearities capable of extracting high-degree interactions across features. We train the fully connected
                 network, the deep cross network, and the embeddings with DFA. Finally, Adaptative Factorization
                 Network (AFN) [42] uses Logarithmic Neural Networks [43] to enhance the representational power
                 of its deep component. We evaluate these methods on the Criteo dataset [44], which features nearly
                 46 million samples of one million sparse features. This is a difficult task, where performance
                 improvements of the AUC on the 0.001-level can enhance CTR significantly [39].

                 Results Performance metrics are reported in Table 2. To obtain these results, a simple hyperpa-
                 rameter grid search over optimization and regularization parameters was performed for BP and DFA
                 independently. DFA successfully trains all methods above the FM baseline, and in fact matches BP
                 performance in both DeepFM and AFN. Because of their complexity, recommender systems require
                 intensive tuning and feature engineering to perform at the state-of-the-art level–and reproducing
                 existing results can be challenging [45]. Hence, it is not surprising that a performance gap exists with
                 Deep&Cross–further fine-tuning may be necessary for DFA to reach BP performance.
                 Alignment measurements corroborate that learning is indeed occurring in the special layers of
                 Deep&Cross and AFN–see supplementary for details. Our results on recommender systems support
                 that DFA can learn in a large variety of settings, and that weight transport is not necessary to solve a
                 difficult recommendation task.


                 Table 2: AUC (higher is better) and log loss (lower is better) of recommender systems trained on the
                 Criteo dataset [44]. Even in complex heterogeneous architectures, DFA performance is in line with
                 BP. Values in bold indicate DFA AUC within 0.001 from the BP AUC or better.

                                <<TABLE>>


                                         3.2 Geometric Learning with Graph Convolutional Networks

                 The use of sophisticated architectures beyond fully connected layers is necessary for certain tasks,
                 such as geometric learning[46], where information lies in a complex structured domain. To address
                 geometric learning tasks, methods capable of handling graph-based data are commonly needed.
                 Graph convolutional neural networks (GCNNs) [47–50] have demonstrated the ability to process
                 large-scale graph data efficiently. We study the applicability of DFA to these methods, including
                 recent architectures based on an attention mechanism. Overall, this is an especially interesting setting,
                 as DFA fails to train more classic 2D image convolutional layers [23].

                 Background Complex data like social networks or brain connections lie on irregular or non-
                 Euclidean domains. They can be represented as graphs, and efficient processing in the spectral
                 domain is possible. Non-spectral techniques to apply neural networks to graphs have also been
                 developed [51–53], but they exhibit unfavorable scaling properties. The success of CNNs in deep
                 learning can be attributed to their ability to efficiently process structured high-dimensional data
                 by sharing local filters. Thus, a generalization of the convolution operator to the graph domain is
                 desirable: [47] first proposed a spectral convolution operation for graphs, and [48] introduced a form
                 of regularization to enforce spatial locality of the filters. We use DFA to train different such GCNNs
                 implementations. We study both spectral and non-spectral convolutions, as well as methods inspired
                 by the attention mechanism. We consider the task of semi-supervised node classification: nodes from
                 a graph are classified using their relationship to other nodes as well as node-wise features.

                 Setting Fast Localized Convolutions (ChebConv) [49] approximate the graph convolution kernel
                 with Chebyshev polynomials, and are one of the first scalable convolution methods on graph. Graph
                 Convolutions (GraphConv) [50] remove the need for an explicit parametrization of the kernel by
                 enforcing linearity of the convolution operation on the graph Laplacian spectrum. It is often considered
                 as the canonical graph convolution. More recent methods do not operate in the spectral domain. Spline
                 Convolutions (SplineConv) [54] use a spline-based kernel, enabling the inclusion of information
                 about the relative positioning of nodes, enhancing their representational power–for instance in the
                 context of 3D meshes. Graph Attention Networks (GATConv) [55] use self-attention [56] layers to
                 enable predictions at a given node to attend more specifically to certain parts of its neighborhood.
                 Finally, building upon Jumping Knowledge Network [57], Just Jump (DNAConv) [58] uses multi-
                 head attention [59] to enhance the aggregation process in graph convolutions and enable deeper
                 architectures. We use PyTorch Geometric [60] for reference implementation of all of these methods.
                 We evaluate performance on three citation network datasets: Cora, CiteSeer, and PubMed [61].

                 Results We report classification accuracy in Table 3. BP and DFA regularization and optimiza-
                 tion hyperparameters are fine-tuned separately on the Cora dataset. In general, we find that less
                 regularization and lower learning rates are needed with DFA. DFA successfully trains all graph
                 methods, independent of whether they use the spectral domain or not, and even if they use attention.
                 Furthermore, for GraphConv, SplineConv, and GATConv DFA performance nearly matches BP.
                 As GCNNs struggle with learning meaningful representations when stacking many layers [62], all
                 architectures but DNAConv are quite shallow (two layers). However, DFA performance is still
                 significantly higher than that of a shallow training method–see supplementary for details. The lower
                 performance on DNAConv is not a failure to learn: alignment measurements show that learning is
                 indeed occurring. It may be explained instead by a need for more in-depth fine-tuning, as this is a
                 deep architecture with 5 successive attention layers.

                 Table 3: Classification accuracy (%, higher is better) of graph convolution methods trained with BP
                 and DFA, on citation networks [61]. But for ChebConv and DNAConv, DFA performance nearly
                 matches BP performance. Values in bold when DFA is within 2.5% of BP.

                                            <<TABLE>>

                      Table 4: AUC and Average Precision Figure 2: t-SNE visualization of the hidden layer
                      (AP, higher is better) for a Graph- activations of a two-layer GraphConv trained on
                      Conv GAE trained with BP or DFA Cora with DFA. Classes forms clear clusters, indicating 
                      that a useful intermediary representation is learned. Colors represent different classes.
                      on citation networks. DFA reproduces BP performance.         


                 We further demonstrate that DFA helps graph convolutions learn meaningful representations by
                 applying t-SNE [63,64] to the hidden layer activations in GraphConv (Figure 2). Cluster of classes
                 are well-separated, indicating that a useful intermediary representation is derived by the first layer.

                 Graph autoencoders We consider one last application of graph convolutions, in the context of
                 graph autoencoders (GAE). We train a non-probabilistic GAE [65] based on GraphConv with DFA,
                 and report results in Table 4. DFA performance is always in line with BP.

                 3.3 Natural Language Processing with Transformers

                 We complete our study by training a Transformer [59] on a language modelling task. Transformers
                 have proved successful in text, image, music generation, machine translation, and many supervised
                 NLP tasks [59,66–69]. Here, we demonstrate that DFA can train them, and we show the influence of
                 tuning the optimizer hyperparameters in narrowing the gap with BP.

                 Background NLP has largely benefited from advances in deep learning. Recurrent Neural Net-
                 works were responsible for early breakthroughs, but their sequential nature prevented efficient
                 parallelization of data processing. Transformers are attention-based models that do not rely on
                 recurrence or convolution. Their ability to scale massively has allowed the training of models with
                 several billion parameters [70,71], obtaining state-of-the-art results on all NLP tasks: Transformers
                 now top the prominent SQuAD 2.0 [72,73] and SuperGLUE [74] benchmarks. In parallel, transfer
                 learning in NLP has leaped forward thanks to language modelling, the unsupervised task of predicting
                 the next word. It can leverage virtually unlimited data from web scraping [75]. This enabled the
                 training of universal language models[76] on extremely large and diversified text corpora. These
                 models are useful across a wide range of domains, and can solve most NLP tasks after fine-tuning.

                 Setting The prominence of both language modelling and Transformers gives us the ideal candidate
                 for our NLP experiments: we train a Transformer to predict the next word on the WikiText-103
                 dataset [77], a large collection of good and featured Wikipedia articles. We use byte-pair-encoding
                 [78] with 32,000 tokens. Our setup is similar to GPT [66]: we adapt the Transformer, originally an
                 encoder-decoder model designed for machine translation, to language modelling. We keep only the
                 encoder and mask the tokens to predict. Our architecture consists in 6 layers, 8 attention heads, a
                 model dimension of 512, and a hidden size of 2048 in the feed-forward blocks. The text is sliced
                 in chunks of 128 tokens and batches of 64 such chunks, resulting in 8192 tokens per batch. Our
                 baseline is trained with BP using the optimization setup of [59]. We found perplexity after 20 epochs
                 to be an excellent indicator of perplexity at convergence; to maximize the number of experiments
                 we could perform, we report the best validation perplexity after 20 epochs. We study two ways of
                 implementing DFA: applying the feedback after every encoder block (macro) or after every layer in
                 those blocks (micro). The input embedding layer receives gradients from the next feedback point
                 through BP. This leaves some amount of weight transport even in the micro-case.

                 Table 5: Best validation perplexity after 20 epochs of a Transformer trained on WikiText-103 (lower
                 is better). The BP and DFA baselines share all hyper-parameters. In Macro the feedback is applied
                 after every transformer layer, while in Micro the feedback is applied after every sub-layer. The
                 learning rate of Adam without the learning rate scheduler is <<FORMULA>>. With the scheduler, the initial
                 learning rate is <<FORMULA>> and it is multiplied by 0.2 when performance plateaus, with a patience of 1.
                 * score after 22 epochs to let the learning rate scheduler take effect

                                          <<TABLE>>

                 Results Our results are summarized in Table 5. Hyper-parameters fine-tuned for BP did not fare
                 well with DFA, but changes in the optimizer narrowed the gap between BP and DFA considerably.
                 The learning rate schedule used on top of Adam [79] in [59] proved detrimental. Using Adam alone
                 required reducing the learning rate between BP and DFA. Increasing 2 from 0.98 [59] to 0.999
                 improved performance significantly. Finally, a simple scheduler that reduces the learning rate when
                 the validation perplexity plateaus helped reducing it further. Considering that the perplexity of the
                 shallow baseline is over 400, DFA is clearly able to train Transformers. However, our results are not
                 on par with BP, especially in the micro setting. A substantial amount of work remains to make DFA
                 competitive with BP, even more so in a minimal weight transport scenario. The large performance
                 improvements brought by small changes in the optimizer indicate that intensive fine-tuning, common
                 in publications introducing state-of-the-art results, could close the gap between BP and DFA.

                 4 Conclusion and outlooks

                 We conducted an extensive study demonstrating the ability of DFA to train modern architectures. We
                 considered a broad selection of domains and tasks, with complex models featuring graph convolutions
                 and attention. Our results on large networks like NeRF and Transformers are encouraging, suggesting
                 that with further tuning, such leading architectures can be effectively trained with DFA. Future work
                 on principled training with DFA–in particular regarding the influence of common practices and
                 whether new procedures are required–will help close the gap with BP.
                 More broadly, we verified for the first time that learning under synaptic asymmetry is possible beyond
                 fully-connected layers, and in tasks significantly more difficult than previously considered. This
                 addresses a notable concern in biologically-plausible architectures. DFA still requires an implausible
                 global feedback pathway; however, local training has already been demonstrated at scale. The next
                 step towards biologically-compatible learning is a local method without weight transport.
                 While the tasks and architectures we have considered are not biologically inspired, they constitute
                 a good benchmark for behavioral realism[20]. Any learning algorithm claiming to approximate
                 the brain should reproduce its ability to solve complex and unseen task. Furthermore, even though
                 the current implementation of mechanisms like attention is devoid of biological considerations, they
                 represent broader concepts applicable to human brains [80]. Understanding how our brain learns is a
                 gradual process, and future research could incorporate further realistic elements, like spiking neurons.
                 Finally, unlocking the backward pass in large architectures like Transformers is promising. More opti-
                 mized implementation of DFA–built at a lower-level of existing ML libraries–could unlock significant
                 speed-up. Leveraging the use of a single random projection as the cornerstone of training, dedicated
                 accelerators may employ more exotic hardware architectures. This will open new possibilities in the
                 asynchronous training of massive models.

                                                  Broader Impact

                 Of our survey This study is the first experimental validation of DFA as an effective training method
                 in a wide range of challenging tasks and neural networks architectures. This significantly broadens the
                 applications of DFA, and more generally brings new insight on training techniques alternative to back-
                 propagation. From neural rendering and recommender systems, to natural language processing or
                 geometric learning, each of these applications has its own potential impact. Our task selection process
                 was motivated by current trends in deep learning, as well as by technically appealing mechanisms
                 (graph convolutions, attention). A limit of our survey is that our–arguably biased–selection of tasks
                 cannot be exhaustive. Our experiments required substantial cloud compute resources, with state-of-
                 the-art GPU hardware. Nevertheless, as this study provides new perspectives for hardware accelerator
                 technologies, it may favor the application of neural networks in fields previously inaccessible because
                 of computational limits. Future research on DFA should continue to demonstrate its use in novel
                 contexts of interest as they are discovered.

                 Of the considered applications Each of the applications considered in our study has a wide
                 potential impact, consider for example the impact of textual bias in pretrained word embeddings [81].
                 We refer to [82] and references therein for a discussion of ethical concerns of AI applications.

                 Of DFA as a training method DFA enables parallelization of the backward pass and places a
                 single operation at the center of the training process, opening the prospect of reducing the power
                 consumption of training chips by an order of magnitude [28]. Not only is more efficient training a
                 path to more environmentally responsible machine learning [83], but it may lower the barrier of entry,
                 supporting equality and sustainable development goals. A significant downside of moving from BP to
                 DFA is a far more limited understanding of how to train models and how the trained models behave.
                 There is a clear empirical understanding of the impact of techniques such as batch normalization
                 or skip connections on the performance of BP; new insights need to be obtained for DFA. BP also
                 enjoys decades of works on topics like adversarial attacks, interpretability, and fairness. Much of
                 this work has to be cross-checked for alternative training methods, something we encourage further
                 research to consider as the next step towards safely and responsively scaling up DFA.

                 Of biologically motivated method Finally, a key motivation for this study was to demonstrate that
                 learning challenging tasks was possible without weight transport. Biologically motivated methods
                 are a more foundational research direction, and as such the possible long-term impact of our findings
                 is harder to estimate under this light. However, fundamental research of this kind is important to open
                 new pathways for ML and neuroscience.

                 Acknowledgments and Disclosure of Funding

                 We thank Igor Carron and Laurent Daudet for the general guidance on the subject of this investigation
                 and the insightful comments, as well as the larger LightOn team for their support.

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                                                  Appendix


                 We first provide additional elements to corroborate our findings: alignment measurement (Section
                 A), and shallow baselines (Section B). We then discuss the process of adapting the considered
                 architectures for DFA (Section C), and the issue of weight transport in attention layers (Section D).
                 We provide some supplementary results for NeRF (Section E), including details of performance on
                 each scene of each datatset, and a discussion on possible mitigation of DFA shortcomings. Finally,
                 we outline steps necessary for reproduction of this work (Section F).

                 A Alignment

                 Alignment measurement In feedback alignment methods, the forward weights learn toalignwith
                 the random backward weights, making the delivered updates useful. This alignment can be quantified
                 by measuring the cosine similarity between the gradient signal delivered by DFABi ay and the
                 gradient signal BP would have deliveredWT <<FORMULA>>. For learning to occur and DFA to work as
                 a training method, there must be alignment. This can be measured numerically [23]. Measuring
                 alignments allows to check whether or not the layers are effectively being trained by DFA, regardless
                 of performance metrics. We note that any alignment value superior to 0 signifies that learning is
                 occuring. Values closer to 1 indicate a better match with BP, but small alignment values are sufficient
                 to enable learning. We report values measured at the deepest DFA layer.

                 Recommender systems We measure alignment on the Criteo dataset, in the two architectures
                 featuring non-conventional fully-connected layers: Deep & Cross and AFN. Alignment is measured
                 after 15 epochs of training, and averaged over a random batch of 512 samples. Results are reported in
                 table A.1. These alignment measurements indicate that learning is indeed occurring in the cross and
                 logarithmic layers. High-variance of alignment in the cross layers is unique: it may be explained by
                 the absence of non-linearity, and account for the difference in performance between BP and DFA on
                 this architecture–which is higher than on the others.

                 Table A.1: Alignment cosine similarity (higher is better, standard deviation in parenthesis) of
                 recommender systems as measured on the Criteo dataset. Learning occurs in both architectures, and
                 high variance may explain the larger performance gap on Deep & Cross compared to other methods.

                                              <<TABLE>>

                 Graph convolutions We measure alignment on the Cora dataset, after 250 epochs of training,
                 averaging values over every sample available–train, validation, and test split included. Results are
                 reported in Table A.2. We observe high alignment values in all architectures, indicative that learning
                 is indeed occuring. Slightly lower values in SplineConv and GATConv may be explained by the use
                 of the Exponential Linear Unit (ELU) instead of the Rectified Linear Unit (ReLU) used as activation
                 in other architectures.
                 Table A.2: Alignment cosine similarity (standard deviation in parenthesis) of various graph convolu-
                 tions architectures as measured on the Cora dataset. These values corroborate that DFA successfully
                 trains all architectures considered.

                                <<TABLE>>

                 B Shallow baselines

                 Shallow learning We compare DFA to BP, but also to shallow learning–where only the topmost
                 layer is trained. While DFA may not reach the performance level of BP, it should still vastly

                 Figure A.1: Comparisons of Tiny-NeRF trained with BP, DFA, and a shallow approach. Shallow
                 training is insufficient to learn scene geometry. Lego scene from the NeRF synthetic dataset.

                                            <<FIGURE>>

                 outperform shallow learning: failure to do so would mean that the weight updates delivered by DFA
                 are useless. On a simple task like MNIST, a shallow baseline may be as high as 90%. However, given
                 the difficulty of the tasks we consider, the shallow baseline is here usually much lower.

                 NeRF Because NeRF models are expensive to train–up to 15 hours on a V100–we consider a
                 simplified setup for the shallow baseline, NeRF-Tiny. This setup operates at half the full resolution
                 of the training images available, runs for 5000 iterations only, and does away with view-dependant
                 characteristics. Furthermore, the network is cut down to 3 layers of half the width of NeRF, and
                 no coarse network is used to inform the sampling. We train this network on the Lego scene of the
                 NeRF-Synthetic dataset, and compare results.
                 Figure A.1 presents renders generated by NeRF-Tiny trained with BP, DFA, and a shallow approach.
                 While BP and DFA delivers similar renders, shallow training fails to reproduce even basic scene
                 geometry, instead outputting a diffuse cloud of colors. This highlights that while DFA may not reach
                 a level of performance on-par with BP on NeRF, it nonetheless delivers meaningful updates enabling
                 the learning of complex features.

                 Recommender systems Because recommender systems require fine-tuning, we perform the same
                 hyperparameter search for shallow learning than for DFA and BP. Results are detailed in Table A.3.
                 Performance of shallow training is always well under BP and DFA–remember that0.001-levelmatter
                 in recommender systems. In particular, in Deep & Cross, where there was the biggest gap between
                 BP and DFA, the performance of the shallow method is extremely poor, well below the FM baseline.
                 Finally, it is expected to see that DeepFM recovers more or less the performance of FM even with a
                 shallow baseline.

                 Table A.3: Shallow baseline for recommender system models on the Criteo dataset. Performance is
                 always well below BP and DFA, as expected.

                                         <<TABLE>>

                 Graph convolutions We use the same hyperparameters as for DFA to produce the shallow baseline
                 on graph datasets. Results are reported in Table A.4. Performance is always much worse than BP
                 and DFA. GATConv recovers the best performance: random attention layers may still deliver useful
                 features [84], as do random convolutions.

                 Transformers In the baseline setting (optimizer and hyper-parameters of [59]), a Transformer
                 trained in the shallow regime yields a perplexity of 428 on WikiText-103. We do not consider

                 Table A.4: Shallow baseline for GCNNs on Cora, CiteSeer, and PubMed [61]. Performance is always
                 well below BP and DFA.

                               <<TABLE>>

                 other settings, as the cost of training a Transformer is high and we do not expect any meaningful
                 improvements–as with NeRF above.

                 C Adapting architectures to DFA

                 NeRF We use an architecture identical to the one used in [36], but based on the effective code
                 implementation rather than the description in the paper 1 . During our tests, we have found that
                 lowering the learning rate to <<FORMULA>> rather than <<FORMULA>> works best with DFA.


                 Recommender systems For all training methods (BP, DFA, and shallow), we have conducted
                 independent hyperparameter searches. We performed a grid search over the learning rate, from
                 <<FORMULA>> to <<FORMULA>> in <<FORMULA>> steps, as well as over the dropout probability, from <<FORMULA>> to <FORMULA> in <<FORMULA>> steps
                 (where applicable). On DeepFM, this search leads to reduce the learning rate from <<FORMULA>> with BP
                 to <<FORMULA>> with DFA, but to keep the 0.5 dropout rate. On Deep & Cross, we reduce learning rate
                 from <<FORMULA>> to <<FORMULA>>, with no dropout in both cases. In AFN, we reduce dropout from <<FORMULA>> to
                 <<FORMULA>> and dropout from 0.3 to 0.

                 Graph convolutions We manually test for a few hyperparameters configuration on the Cora dataset,
                 focusing on learning rate, weight decay, and dropout. We do not consider architectural changes, such
                 as changing the number of filters or of attention heads. For ChebConv and GraphConv, we reduce
                 weight decay to <<FORMULA>> instead of <<FORMULA>>, and set the dropout rate to 0 and 0.1 respectively, instead
                 of 0.5 with BP. For SplineConv, we find that no change in the hyperparameters are necessary. For
                 GATConv, we reduce weight decay to <<FORMULA>> instead of <<FORMULA>> and reduce dedicated dropout layer
                 to 0.1 instead of 0.6 but keep the 0.6 dropout rate within the GAT layer. Finally, on DNAConv we
                 disable weight decay entirely, instead of an original value of <<FORMULA>>, double the learning rate from
                 <<FORMULA>> to <<FORMULA>>, and disable dropout entirely. In all cases, we share the backward random matrix
                 across all nodes in a graph.

                 Transformers The model hyper-parameters were fixed across all of our experiments, except for
                 the number of attention heads in one case, that we will precise below, and dropout. We tested several
                 values of dropout probability between 0 and 0.5, but found the original value of 0.1 to perform
                 best. We manually tested a number of optimizers, optimizer parameters and attention mechanisms.
                 We tested four combinations of optimizers and schedulers : Adam with the scheduler used in [59],
                 Adam alone, RAdam [85] alone, and Adam with a scheduler that reduces the learning rate when
                 the validation perplexity plateaus. We found it necessary to reduce the initial learning rate of Adam
                 from <<FORMULA>> to <<FORMULA>>, although it could be set back to <<FORMULA>> with a scheduler. We tried two values
                 of 0.98 and 0.999. We also tried to change <<FORMULA>> and observed some small differences that were
                 not significant enough for the main text. Finally, we tried three attention mechanisms in addition to
                 the standard multihead scaled dot-product attention: the dense and random (learnable) Synthesizers
                 of [84], as well as the fixed attention patterns of [86]. The latter needed to be adapted to language
                 modelling to prevent attending to future tokens, which led us to reduced the number of attention
                 heads to 4. The backward random matrix is always shared across all tokens and batches.

                                                D Weight transport and attention

                 We consider an attention layer operating on inputx. The queries, keys, and values are respectively
                 <<FORMULA>>, and <<FORMULA>> is the dimension of the queries and keys. The layer
                 performs:                                
                                     <<FORMULA>>                (4)

                 When using DFA on attention, we deliver the random feedback to the top of the layer. Accordingly,
                 to obtain updates toWQ ;WK ;andWV we still to have to backpropagate through the attention
                 mechanism itself. This involves weight transport onWV , sacrificing some biological realism for
                 simplicity. Overall weight transport between layers still does not occur, and updating the layers in
                 parallel remains possible.
                 Beside using FA or DFA within the attention layer, alternative mechanisms like the synthesizer
                 [84]–which uses random attention in place of the query and key system–or fixed attention [86] can
                 remove the need for weight transport. Implementing these mechanisms in DFA-trained Transformers,
                 or other attention-powered architectures, will require further research.


                 E Supplementary NeRF results

                 Quantitative results We report per-scene scores for each dataset in Table A.5. BP values are taken
                 from [36]. On three scenes of the synthetic datasets, NeRF-DFA even outperforms past state-of-the-art
                 methods trained with BP. Note that Neural Volumes (NV) is not applicable to forward-facing view
                 synthesis–as is required in LLFF-Real–and thus no results are reported.

                 Qualitative results We report sample renders from the NeRF-Synthetic dataset (Figure A.2) and
                 the LLFF-Real dataset (Figure A.2), for every scene available. However, we recommend readers to
                 consult the supplementary video to make better sense of characteristics like multi-view consistency
                 and view-dependent effects (most visible on the LLFF-Real Room scene).


                 Table A.5: Per-scene PSNR for NeRF DFA and BP against other state-of-the-art methods on the
                 Nerf-Synthetic and LLFF-Real. DFA performance is fairly homogeneous across each dataset and in
                 line with the differences in other methods.

                                            <<TABLE>>

                 Possible future directions Despite retranscribing scene geometry in a multi-view consistent way,
                 NeRF produces renders of a lower quality when trained with DFA instead of BP. In particular, it
                 struggles to transcribe small-scale details, resulting in "blurry" renders. Moreover, it displays high-
                 frequency artefacts: not in the scene geometry, but in individual pixels taking values very distant from
                 their neighborhood. Interestingly, this noise phenomenon is unique to NeRF-DFA: it is not observed
                 on NeRF-BP with similar PSNR values (achieved during training) or on other methods with similar
                 or lower PSNR. This leads us to hypothesize this is an aspect unique to DFA, possibly due to the
                 alignment process. Indeed, DFA creates a bias on the weights, by encouraging them to be "aligned"
                 with an arbitrary values dependant on the random matrix used. It is possible this could introduce
                 random noise in the final renders–though we leave a more principled experiment to future research.
                 To attempt to alleviate this issue, we first consider NeRF-Dual. In NeRF-Dual, we average the
                 pixel-wise prediction between the fine and coarse network, to attempt to remove some of the noise.
                 To do so, we first still use the coarse network to create a probability distribution for the hierarchical
                 sampling. Then, we evaluate again both the coarse and fine networks at the locations informed by
                 this probability distribution. Compared to vanilla NeRF, this requires an extra batch of evaluation of
                 the coarse network for all rays–rougly speaking, this increases inference time by 30-50% depending
                 on the coarse network architecture considered. We note that this is not applied during training, so that
                 training times remain identical.
                 Figure A.2 and Figure A.3 showcase comparisons between NeRF and NeRF-Dual trained with DFA
                 on all scenes. When viewed at high resolution–such as in our supplementary video–the NeRF-Dual
                 renders are more pleasing, especially for the full scenes. They remove most of the high-frequency
                 noise, leading to smoother renders. However, this averaging process further blurs small-scale details in
                 the render. This is especially visible in the NeRF-Synthetic dataset, on scenes like Ficus. Furthermore,
                 NeRF-Dual introduces novel artefacts in the Mic and Ship scenes, with areas improperly colored
                 with a violet tint. The cause for these artefacts is unknown, but they show that NeRF-Dual is far from
                 a silver bullet. The PSNR is also minimally increased, by less than 0.5 per scene. Nevertheless, this
                 shows some promise in possibilities to allievate the shortcomings of NeRF-DFA. It is possible that
                 changes to the overall rendering process, or the use of classic image processing techniques, may help
                 enhance the NeRF-DFA images.
                 Finally, we also experimented with increasing the capacity of the fine network, by widening its layers
                 to 512 neurons. We call this architecture NeRF-XL. However, we have not succeeded in getting
                 PSNR values higher than with vanilla NeRF on DFA. In particular, the training process becomes
                 much more cumbersome, as multi-GPU parallelism is needed to fit the model. It is possible that
                 higher network capacity may help learning both the task at hand and to align simultaneously, but
                 further work is required.


                 F Reproducibility

                 Hardware used All main experiments require at most a single NVIDIA V100 GPU with 16GB
                 of memory to reproduce. Alignment measurement on large architectures (NeRF and Transformers)
                 require a second identical GPU to keep a copy of the network to evaluate BP gradients.
                 We estimate that a total of around 10,000 GPU-hours on V100s were necessary for this paper.
                 Accordingly, we estimate the cloud-computing carbon impact of this paper to be of 1700 kgCO 2 eq 2 .
                 However, without hyperparameter searches, our results can be reproduced with less than 500 GPU-
                 hours on V100s, with most of that budget going to NeRF and Transformers.

                 Implementation We use the shared random matrix trick from [23] to reduce memory use in DFA
                 and enable its scaling to large networks. We use PyTorch [87] for all experiments. For reference
                 implementation of the methods considered, we relied on various sources. Our NeRF implementation
                 is based on the PyTorch implementation by Krishna Murthy 3 , with modifications to allow for proper
                 test and validation, as well as DFA and multi-GPU support. For recommender systems, we use
                 PyTorch Geometric [60] for all graph operations. Our Transformer implementation is our own. 
                 Our code is available as supplementary material.

                 NeRF We provide training, testing, and rendering code along with the configurations used to obtain
                 our results. An example to reproduce our results is given in the supplementary code repository. Given
                 the computing cost associated with training a NeRF, we also provide our trained models.

                 Recommender systems We provide bash scripts to reproduce the results in Table 2 and A.3, with
                 the results of our hyperparameter search. We provide code to reproduce the results in Table A.1.

                 Graph convolutions We provide the code to reproduce all of our results. Note that the t-SNE
                 results are not exactly reproducible, as the CUDA implementation used is non-deterministic.

                 Transformers We provide bash scripts to reproduce Table 5 and the shallow results.

                                    <<FIGURE>>

                 Figure A.2: Sample renders for every scene of the NeRF-Synthetic dataset, for NeRF and NeRF-Dual
                 trained with DFA.

                                            <<FIGURE>>

                Figure A.3: Sample renders for every scene of the LLFF-Real dataset, for NeRF and NeRF-Dual
                 trained with DFA.
<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>
Efficient Behavior of Small-World Networks 
 
We introduce the concept of efficiency of a network, measuring how efficiently it exchanges information. By using this simple measure small-world networks are seen as systems that are both globally and locally efficient. This allows to give a clear physical meaning to the concept of small-world, and also to perform a precise quantitative analysis of both weighted and unweighted networks. We study neural networks and man-made communication and transportation systems and we show that the underlying general principle of their construction is in fact a small-world principle of high efficiency. PACS numbers 89.70.+c, 05.90.+m, 87.18.Sn, 89.40.+k 
We live in a world of networks. In fact any complex system in nature can be modeled as a network, where vertices are the elements of the system and edges represent the interactions between them. Coupled biological and chemical systems, neural networks, social interacting species, computer networks or the Internet are only few of such examples [1]. Characterizing the structural properties of the networks is then of fundamental importance to understand the complex dynamics of these systems. A recent paper [2] has shown that the connection topology of some biological and social networks is neither completely regular nor completely random. These networks, there named small-worlds, in analogy with the concept of small-world phenomenon developed 30 years ago in social psychology [3], are in fact highly clustered like regular lattices, yet having small characteristics path lengths like random graphs. The original paper has triggered a large interest in the study of the properties of small-worlds (see ref. [4] for a recent review). Researchers have focused their attention on different aspects: study of the inset mechanism [5,7], dynamics [8] and spreading of diseases on small-worlds [9], applications to social net.works [10,11] and to the Internet [12,13]. In this letter we introduce the concept of efficiency of a network, measuring how efficiently information is exchanged over the net.work. By using efficiency small-world networks results as systems that are both globally and locally efficient. This formalization gives a clear physical meaning to the concept of small-world, and also allows a precise quantitative analysis of unweighted and weighted networks. We study several systems, like brains, communication and transportation networks, and show that the underlying general principle of their construction is in fact a small-world principle, provided attention is taken not to ignore an important observational property (closure). We start by reexamining the original formulation pro.posed in ref. [2]. There, a generic graph G with N vertices and K edges is considered. G is assumed to be unweighted, i.e. edges are all equal, sparse <<(K . N (N . 1)/2)>>, and connected. i.e. there exists at least one path connecting any couple of vertices with a infinite number of steps. G is therefore represented by simply giving the adjacency (or connection) matrix, i.e. the NxN matrix whose entry a_ij is 1 if there is an edge joining vertex i to vertex j, and 0 otherwise. An important quantity of G is the degree of vertex i, i.e. the number ki of edges incident with vertex i (the number of neighbors of i). The average value of ki is <<k =2K/N>>. Once {<<FORMULA>>} is given it can be used to calculate the matrix of the short.est path lengths d_ij between two generic vertices i and j. The fact that G is assumed to be connected implies that dij is positive and infinite .i = j. In order to quantify the structural properties of G, [2] proposes to evaluate two different quantities: the characteristic path length L and the clustering coefficient C. L is the average distance be-
tween two generic vertices <<FORMULA>>, and C is a local property defined as <<FORMULA>>. Here C_i is the number of edges existing in Gi, the subgraph of the neighbors of i, divided by the maximum possible number ki(ki . 1)/2. In [2] a simple method is considered to pro.duce a class of graphs with increasing randomness. The initial graph G is taken to be a one-dimensional lattice with each vertex connected to its k neighbors and with periodic boundary conditions. Rewiring each edge at ran.dom with probability p, G can be tuned in a continuous way from a regular lattice (p = 0) into a random graph (p = 1). For the regular lattice we expect <<FORMULA>> and a high clustering coefficient <<FORMULA>>, while for a random graph <<FORMULA>> and <<FORMULA>> [14,5]. Although in the two limit cases a large C is associated to a large L and vice versa a small C to a small L, the numerical experiment reveals an intermediate regime at small p where the system is highly clustered like regular lattices, yet having small characteristics path lengths like random graphs. This behavior is there called small-world and it is found to be a property of some social and 
biological networks analyzed [2]. 
Now we propose a more general set-up to investigate real networks. We will show that: the definition of small-world behavior can be given in terms of a single variable with a physical meaning, the efficiency E of the network. -1/L and C can be seen as first approximations of E evaluated resp. on a global and on a local scale. -we can drop all the restrictions on the system, like unweightedness, connectedness and sparseness. We represent a real network as a generic weighted (and possibly even non sparse and non connected) graph G. Such a graph needs two matrices to be described: the adjacency matrix {a_ij} defined as for the unweighted graph, and the matrix {<<FORMULA>>} of physical distances. The number <<FORMULA>> can be the space distance between the two vertices or the strength of their possible interaction: we suppose <<FORMULA to be known even if in the graph there is no edge between i and j. To make some examples, <<FORMULA>> can be the geographical distance between stations in transportation systems (in such a case <<FORMULA>> respects the triangle equality, though this is not a necessary assumption), the time taken to ex.change a packet of information between routers in the Internet, or the inverse v<<E_loc>>ity of chemical reactions along a direct connection in a biological system. Of course, in the particular case of an unweighted graph <<FORMULA>>. The shortest path length dij between two generic points i and j is the smallest sum of the physical distances throughout all the possible paths in the graph from i to j. The matrix {<<FORMULA>>} is therefore calculated by using the information contained both in matrix {a_ij} and in matrix {<<FORMULA>>}. We have <<FORMULA>>, the equality being valid when there is an edge between i and j. Let us now suppose that the system is parallel, i.e. every vertex sends information concurrently along the network, through its edges. The efficiency <<FORMULA>> in the communication between vertex i and j can be then defined to be inversely proportional to the shortest distance: <<FORMULA>>. When there is no path in the graph between i and j, <<FORMULA>> and consistently <<FORMULA>>. The average efficiency of G can be defined as: 

<<FORMULA>>

To normalize E we consider the ideal case G_id in which the graph G has all the <<N (N . 1)/2>> possible edges. In such a case the information is propagated in the most efficient way since dij = .ij .i, j, and E assumes its maxi-
<<FORMULA>>. The efficiency <<FORMULA>>
<<E(G)>> considered in the following of the paper is always divided by <<FORMULA>> and therefore <<FORMULA>>. Though the equality E = 1 is valid when there is an edge between each couple of vertices, real networks can reach a high value of E. 
In our formalism, we can define the small-world be.haviour by using the single measure E to analyze both the local and global behavior, rather than two different variables L and C. The quantity in eq. (1) is the global efficiency of G and we therefore name it E_glob. Since E is defined also for a disconnected graph we can characterize the local properties of G by evaluating for each vertex i the efficiency of G_i, the subgraph of the neighbors of i. We define the local efficiency as the average efficiency of the local subgraphs, E loc =1/N E(Gi). 

This quantity plays a role similar to the clustering co.efficient C. Since <<FORMULA>>, the local efficiency <<FORMULA>> tells how much the system is fault tolerant, thus how efficient is the communication between the first neighbors of i when i is removed [15]. The definition of small-world can now be rephrased and generalized in terms of the information <<FORMULA>>: small-world networks have high <<FORMULA>> and <<FORMULA>>, i.e. are very efficient in global and local communication. This definition is valid both for unweighted and weighted graphs, and can also be applied to disconnected and/or non sparse graphs. 
It is interesting to see the correspondence between our measure and the quantities L and C of [2] (or, correspondingly, <<1/L>> and C). The fundamental difference is that E_glob is the efficiency of a parallel systems, where all the nodes in the network concurrently exchange pack.ets of information (such are all the systems in [2], for example), while 1/L measures the efficiency of a sequential system (i.e. only one packet of information goes along the network). <<FORMULA>> is a reasonable approximation of <<E_glob>>when there are not huge differences among the distances in the graph, and this can explain why L works reasonably well in the unweighted examples of [2]. But, in general 1/L can significantly depart from E_glob. For instance, in the Internet, having few computers with an extremely slow connection does not mean that the whole Internet diminishes by far its efficiency: in practice, the presence of such very slow computers goes unnoticed, be.cause the other thousands of computers are exchanging packets among them in a very efficient way. Here 1/L would give a number very close to zero (strictly 0 in the particular case when a computer is disconnected from the others and <<FORMULA>>, while E_glob gives the correct efficiency measure of the Internet. We turn now our attention to the local properties of a network. C is only one among the many possible intuitive measures [10] of how well connected a cluster is. It can be shown that when in a graph most of its local subgraphs Gi are not sparse, then C is a good approximation of E_loc. Summing up there are not two different kinds of analysis to be done for the global and local scales, but just one with a very precise physical meaning: the efficiency in transporting information. We now illustrate the onset of the small-world in an un.weighted graph by means of the same example used in [2]. A regular lattice with <<N = 1000>> and <<k = 20>> is rewired 
2 
with probability p and <<E_glob>> and <<E_loc>> are reported in <<FORMULA>> as functions of p [16]. For <<p = 0>> we expect the system to be inefficient on a global scale (E_glob . k/N log(N/K)) but locally efficient. The situation is inverted for the ran.dom graph. In fact at p =1 E_glob assumes a maximum value of 0.4, meaning 40% the efficiency of the ideal graph with an edge between each couple of vertices. This at the expenses of the fault tolerance (<<FORMULA>>). 

<<FIGURE>>

FIG. 1. FIG.1 Global and local efficiency for the graph example considered in [2]. A regular lattice with <<N = 1000>> and <<k = 20>> is rewired with probability p. The small-world behavior results from the increase of E_glob caused by the introduction of only a few rewired edges (short cuts), which on the other side do not affect <<E_loc>>. At p <20> 0.1, E_glob has almost reached the value of the random graph, though <<E_loc>> has only diminished by very little from the value of 0.82 of the regular lattice. Small worlds have high E_glob and <<E_loc>>. 
The small-world behavior appears for intermediate values of p. It results from the fast increase of E_glob (for small p we find a linear increase of E_glob in the logarithmic horizontal scale) caused by the introduction of only a few rewired edges (short cuts), which on the other side do not affect <<E_loc>>. At p . 0.1, E_glob has almost reached the maximum value of 0.4, though <<E_loc>> has only diminished by very little from the maximum value of 0.82. For an unweighted case the description in terms of network efficiency resembles the approximation given in [2]. In particular we have checked that a good agreement with curves L(p) and C(p) [2] can be obtained by reporting <<FORMULA>> and <<FORMULA>>. Of course in such an example the short cuts connect at almost no cost vertices that would otherwise be much farther apart (because <<FORMULA>>). On the other hand this is not true when we consider a weighted network. As real networks we consider first different examples of natural systems (neural networks), and then we turn our attention to man-made communication and transportation systems. 
1) Neural Networks. Thanks to recent experiments 
neural structures can be studied at several levels of scale. Here we focus first on the analysis of the neuro-anatomical structure of cerebral cortex, and then on a simple nervous system at the level of wiring between neurons. The anatomical connections between cortical areas are of particular importance for their intricate relationship with the functional connectivity of the cerebral cortex [18]. We analyze two databases of cortico-cortical connections in the macaque and in the cat [19]. Tab.1 indicates the two networks are small-worlds [16]: they have high E_glob, respectively 52% and 69% the efficiency of the ideal graph with an edge between each couple of vertices (just slightly smaller than the best possible values of 57% and 70% obtained in random graphs) and high <<E_loc>>, respectively 70% and 83%, i.e. high fault tolerance [22]. These results indicate that in neural cortex each region is intermingled with the others and has grown following a perfect balance between local necessities (fault tolerance) and wide-scope interactions. Next we consider the neural network of C. elegans, the only case of a nervous system completely mapped at the level of neurons and chemical synapses [23]. Tab.1 shows that this is also a small-world network: C. elegans achieves both a 50% of global and local efficiency. Moreover the value of E_glob is similar to <<E_loc>>. This is a difference from cortex databases where fault tolerance is slightly privileged with respect to global communication. 
2) Communication Networks. We have considered two of the most important large-scale communication net.works present nowadays: the World Wide Web and the Internet. Tab.2 shows that they have relatively high val.ues of E_glob (slightly smaller than the best possible val.ues obtained for random graphs) and <<E_loc>>. Despite the WWW is a virtual network and the Internet is a physical network, at a global scale they transport information essentially in the same way (as their E_glob<6F>s are almost equal). At a local scale, the bigger <<E_loc>> in the WWW case can be explained both by the tendency in the WWW to create Web communities (where pages talking about the same subject tend to link to each other), and by the fact that many pages within the same site are often quickly connected to each other by some root or menu page. 
3) Transport Networks. differently from previous databases the Boston subway transportation system (MBTA) can be better described by a weighted graph, the matrix {.ij } being given by the geographical distances between stations. If we consider the MBTA as an unweighted graph we obtain that it is apparently neither locally nor globally efficient (see Tab.3). On the other hand, when we take into account the geographical distances, we have E_glob =0.63: this shows the MBTA is a very efficient transportation system on a global scale, only 37% less efficient than the ideal subway with a di.rect tunnel from each station to the others. Even in the weighted case <<E_loc>> stays low (0.03), indicating a poor local behavior: differently from a neural network the 


MBTA is not fault tolerant and a damage in a station will dramatically affect the connection between the previous and the next station. The difference with respect to neural networks comes from different needs and priorities in the construction and evolution mechanism: when we build a subway system, the priority is given to the achievement of global efficiency, and not to fault tolerance. In fact a temporary problem in a station can be solved by other means: for example, walking, or taking a bus from the previous to the next station. That is to say, the MBTA is not a closed system: it can be considered, after all, as a subgraph of a wider transportation network. This property is of fundamental importance when we analyze a system: while global efficiency is without doubt the major characteristic, it is closure that somehow leads a system to have high local efficiency (without alternatives, there should be high fault-tolerance). The MBTA is not a closed system, and thus this explains why, unlike in the case of the brain, fault tolerance is not a critical issue. Indeed, if we increase the precision of the analysis and change the MBTA subway network by taking into account, for example, the Boston Bus System, this ex.tended transportation system comes back to be a small-world network (<<FORMULA>>). Qualitatively similar results, confirming the similarity of construction principles, have been obtained for other undergrounds and for a wider transportation system consisting of all the main airplane and highway connections throughout the world [25]. Considering all the transportation alter.natives available at that scale makes again the system closed (there are no other reasonable routing alternatives), and so fault-tolerance comes back as a leading construction principle. 
Summing up, the introduction of the efficiency mea.sure allows to give a definition of small-world with a clear physical meaning, and provides important hints on why the original formulas of [2] work reasonably well in some cases, and where they fail. The efficiency measure al.lows a precise quantitative analysis of the information flow, and works both in the unweighted abstraction, and in the more realistic assumption of weighted networks. Finally, analysis of real data indicates that various existing (neural, communication and transport) networks exhibit the small-world behavior (even, in some cases, when their unweighted abstractions do not), substantiating the idea that the diffusion of small-world networks can be interpreted as the need to create networks that are both globally and locally efficient. 
[1] Y. Bar-Yam, Dynamics of Complex Systems (Addison-Wesley, Reading Mass, 1997). 
[2] D.J. Watts and S.H. Strogatz, Nature 393, 440 (1998). 
[3] S. Milgram, Physicol. Today, 2, 60 (1967). 
[4] M.E.J. Newman, cond-mat/0001118. 
[5] A. Barrat, M. Weigt, Europ. Phys. J. B 13, 547 (2000) 
[6] M. Marchiori and V. Latora, Physica A285, 539 (2000). 
[7] M. Barthelemy, L. Amaral, Phys. Rev. Lett. 82, 3180 (1999). 
[8] L. F. Lago-Fernandez et al, Phys. Rev. Lett. 84, 2758 (2000). 
[9] C. Moore and M.E.J. Newman, Phys. Rev. E61, 5678 (2000). 
[10] M.E.J. Newman, cond-mat/0011144. 
[11] L. A. N. Amaral, A. Scala, M. Barth<74>el<65>emy, and H. E. Stanley, Proc. Natl. Acad. Sci. 97, 11149 (2000). 
[12] R. Albert, H. Jeong, and A.-L. Barab<61>asi, Nature 401, 130 (1999). 
[13] A.-L. Barab<61>asi and R. Albert, Science 286, 509 (1999). 
[14] B. Bollob<6F>as, Random Graphs (Academic, London, 1985). 
[15] Our concept of fault tolerance is different from the one adopted in R. Albert, H. Jeong, and A.-L. Barab<61>asi, Na.ture 406, 378 (2000); R. Cohen et al. Phys. Rev. Lett. 85, 2758 (2000), where the authors consider the response of the entire network to the removal of i. 
[16] Here and in the following the matrix {dij }i,j2G has been computed by using two different methods: the Floyd-Warshall (O(N 3 )) [17] and the Dijkstra algorithm (O(N 2 logN )) [10]. 
[17] G. Gallo and S. Pallottino, Ann. Oper. Res. 13, 3 (1988). 
[18] O. Sporns, G. Tononi, G.M. Edelman, Celebral Cortex 10, 127 (2000). [19] J.W.Scannell, Nature 386, 452 (1997). [20] M.P. Young, Phil.Trans.R.Soc B252, 13 (1993). 
[21] J.W. Scannell, M.P. Young and C. Blakemore, J. Neu.rosci. 15, 1463 (1995). 
[22] E. Sivan, H. Parnas and D. Dolev, Biol. Cybern. 81, 11.23 (1999). 
[23] J.G. White et. al., Phil. Trans. R. Soc. London B314,1 (1986). 
[24] T.B. Achacoso and W.S. Yamamoto, AY<41>s Neuroanatomy of C. elegans for Computation (CRC Press, FL, 1992). 
[25] M. Marchiori and V. Latora, in preparation. 

TABLE I. Macaque and cat cortico-cortical connections [19]. The macaque database contains N = 69 cortical areas and K = 413 connections [20]. The cat database has N = 55 cortical areas (including hippocampus, amygdala, entorhinal cortex and subiculum) and K = 564 (revised database and cortical parcellation from [21]). The nervous system of C. elegans consists of N = 282 neurons and K = 2462 links which can be either synaptic connections or gap junctions [24]. 

<<TABLE>>

TABLE II. Communication networks. Data on the World Wide Web from http://www.nd.edu/<2F>networks contains N = 325729 documents and K = 1090108 links [12], while the Internet database is taken from http://moat.nlanr.net and has N = 6474 nodes and K = 12572 links. 

<<TABLE>> 

TABLE III. The Boston underground transportation system (MBTA) consists of N = 124 stations and K = 124 tunnels. The matrix {.ij } of the spatial distances between stations, used for the weighted case, has been calculated us.ing databases from http://www.mbta.com/ and the U.S. Na.tional Mapping Division. 
 
<<TABLE>>

<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>
            Efficient Processing of Deep Neural Networks: A Tutorial and Survey

             Vivienne Sze,Senior Member, IEEE,Yu-Hsin Chen,Student Member, IEEE,Tien-Ju Yang,Student
                                   Member, IEEE,Joel Emer,Fellow, IEEE

                                                     Abstract
         
        Deep neural networks (DNNs) are currently widely representation of an input space. This is different from earlier
        used for many artificial intelligence (AI) applications including approaches that use hand-crafted features or rules designed by
        While DNNs experts. deliver state-of-the-art accuracy on many AI tasks, it comes at the   The superior accuracy of DNNs, however, comes at the cost of high computational complexity. Accordingly, techniques
        that enable efficient processing of DNNs to improve energy cost of high computational complexity. While general-purpose
        efficiency and throughput without sacrificing application accuracy compute engines, especially graphics processing units (GPUs),
        or increasing hardware cost are critical to the wide deployment have been the mainstay for much DNN processing, increasingly of DNNs in AI systems.                            there is interest in providing more specialized acceleration of This article aims to provide a comprehensive tutorial and the DNN computation. This article aims to provide an overview survey about the recent advances towards the goal of enabling
        efficient processing of DNNs. Specifically, it will provide an of DNNs, the various tools for understanding their behavior,
        overview of DNNs, discuss various hardware platforms and and the techniques being explored to efficiently accelerate their
        architectures that support DNNs, and highlight key trends in computation. reducing the computation cost of DNNs either solely via hardware   This paper is organized as follows: design changes or via joint hardware design and DNN algorithm
        changes. It will also summarize various development resources    Section II provides background on the context of why
        that enable researchers and practitioners to quickly get started     DNNs are important, their history and applications.
        in this field, and highlight important benchmarking metrics and    Section III gives an overview of the basic components of design considerations that should be used for evaluating the     DNNs and popular DNN models currently in use. rapidly growing number of DNN hardware designs, optionally
        including algorithmic co-designs, being proposed in academia    Section IV describes the various resources used for DNN
        and industry.                                      research and development.
         The reader will take away the following concepts from this    Section V describes the various hardware platforms used
        article: understand the key design considerations for DNNs; be     to process DNNs and the various optimizations used able to evaluate different DNN hardware implementations with     to improve throughput and energy efficiency without benchmarks and comparison metrics; understand the trade-offs     impacting application accuracy (i.e., produce bit-wise between various hardware architectures and platforms; be able to
        evaluate the utility of various DNN design techniques for efficient     identical results).
        processing; and understand recent implementation trends and    Section VI discusses how mixed-signal circuits and new
        opportunities.                                      memory technologies can be used for near-data processing
                                                       to address the expensive data movement that dominates
                                                       throughput and energy consumption of DNNs.
                                                       
                                                       I. INTRODUCTION                   Section VII describes various joint algorithm and hardware
         Deep neural networks (DNNs) are currently the foundation     optimizations that can be performed on DNNs to improve
        for many modern artificial intelligence (AI) applications [1].     both throughput and energy efficiency while trying to
        Since the breakthrough application of DNNs to speech recogni-     minimize impact on accuracy.
        tion [2] and image recognition [3], the number of applications    Section VIII describes the key metrics that should be
        that use DNNs has exploded. These DNNs are employed in a     considered when comparing various DNN designs.
        myriad of applications from self-driving cars [4], to detecting
        cancer [5] to playing complex games [6]. In many of these  II. B ACKGROUND ON DEEP NEURAL NETWORKS (DNN)
        domains, DNNs are now able to exceed human accuracy. The   In this section, we describe the position of DNNs in thesuperior performance of DNNs comes from its ability to extract context of AI in general and some of the concepts that motivatedhigh-level features from raw sensory data after using statistical its development. We will also present a brief chronology oflearning over a large amount of data to obtain an effective the major steps in its history, and some current domains to
                                                   which it is being applied. V. Sze, Y.-H. Chen and T.-J. Yang are with the Department of Electrical
        Engineering and Computer Science, Massachusetts Institute of Technol-
        ogy, Cambridge, MA 02139 USA. (e-mail: sze@mit.edu; yhchen@mit.edu, A. Artificial Intelligence and DNNs tjy@mit.edu)
         J. S. Emer is with the Department of Electrical Engineering and Computer   DNNs, also referred to as deep learning, are a part of Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA, the broad field of AI, which is the science and engineering and also with Nvidia Corporation, Westford, MA 01886 USA. (e-mail:
        jsemer@mit.edu)                                  of creating intelligent machines that have the ability to                                                                                             2

                                                                <<FIGURE>>

                                                   Fig. 2. Connections to a neuron in the brain. <<FORMULA>>,<<FORMULA>>,<<FORMULA>>, and b are the
                                                   activations, weights, non-linear function and bias, respectively. (Figure adopted
                                                   from [7].)Fig. 1. Deep Learning in the context of Artificial Intelligence.


                                                   to be10 14 to10 15 synapses in the average human brain.
        achieve goals like humans do, according to John McCarthy,   A key characteristic of the synapse is that it can scale the
        the computer scientist who coined the term in the 1950s. signal (x_i) crossing it as shown in Fig. 2. That scaling factor
        The relationship of deep learning to the whole of artificial can be referred to as a weight (<<FORMULA>>), and the way the brain is
        intelligence is illustrated in Fig. 1. believed to learn is through changes to the weights associated
         Within artificial intelligence is a large sub-field called with the synapses. Thus, different weights result in different
        machine learning, which was defined in 1959 by Arthur Samuel responses to an input. Note that learning is the adjustment
        as the field of study that gives computers the ability to learn of the weights in response to a learning stimulus, while the
        without being explicitly programmed. That means a single organization (what might be thought of as the program) of the
        program, once created, will be able to learn how to do some brain does not change. This characteristic makes the brain an
        intelligent activities outside the notion of programming. This is excellent inspiration for a machine-learning-style algorithm.
        in contrast to purpose-built programs whose behavior is defined   Within the brain-inspired computing paradigm there is a
        by hand-crafted heuristics that explicitly and statically define subarea called spiking computing. In this subarea, inspiration
        their behavior.                                  is taken from the fact that the communication on the dendrites
         The advantage of an effective machine learning algorithm and axons are spike-like pulses and that the information being
        is clear. Instead of the laborious and hit-or-miss approach of conveyed is not just based on a spike’s amplitude. Instead,
        creating a distinct, custom program to solve each individual it also depends on the time the pulse arrives and that the
        problem in a domain, the single machine learning algorithm computation that happens in the neuron is a function of not just
        simply needs to learn, via a processes called training, to handle a single value but the width of pulse and the timing relationship
        each new problem.                              between different pulses. An example of a project that was
         Within the machine learning field, there is an area that is inspired by the spiking of the brain is the IBM TrueNorth [8].
        often referred to as brain-inspired computation. Since the brain In contrast to spiking computing, another subarea of brain-
        is currently the best ‘machine’ we know for learning and inspired computing is called neural networks, which is the
        solving problems, it is a natural place to look for a machine focus of this article. 1
        learning approach. Therefore, a brain-inspired computation is
        a program or algorithm that takes some aspects of its basic B. Neural Networks and Deep Neural Networks (DNNs)
        form or functionality from the way the brain works. This is in   Neural networks take their inspiration from the notion that
        contrast to attempts to create a brain, but rather the program a neuron’s computation involves a weighted sum of the input
        aims to emulate some aspects of how we understand the brain values. These weighted sums correspond to the value scaling
        to operate.                                    performed by the synapses and the combining of those values
         Although scientists are still exploring the details of how the in the neuron. Furthermore, the neuron doesn’t just output that
        brain works, it is generally believed that the main computational weighted sum, since the computation associated with a cascade
        element of the brain is the neuron. There are approximately of neurons would then be a simple linear algebra operation.
        86 billion neurons in the average human brain. The neurons Instead there is a functional operation within the neuron that
        themselves are connected together with a number of elements is performed on the combined inputs. This operation appears
        entering them called dendrites and an element leaving them to be a non-linear function that causes a neuron to generate
        called an axon as shown in Fig. 2. The neuron accepts the an output only if the inputs cross some threshold. Thus by
        signals entering it via the dendrites, performs a computation on analogy, neural networks apply a non-linear function to the
        those signals, and generates a signal on the axon. These input weighted sum of the input values. We look at what some of
        and output signals are referred to as activations. The axon of those non-linear functions are in Section III-A1.
        one neuron branches out and is connected to the dendrites of
        many other neurons. The connections between a branch of the   1 Note: Recent work using TrueNorth in a stylized fashion allows it to be
                                                   used to compute reduced precision neural networks [9]. These types of neural axon and a dendrite is called asynapse. There are estimated networks are discussed in Section VII-A.                                                                                             3

                                                <<FIGURE>>
               
        Fig. 3. Simple neural network example and terminology (Figure adopted (a) Compute the gradient of the loss (b) Compute the gradient of the lossfrom [7]).                                      relative to the filter inputs      relative to the weights

                            <<FIGURE>>

           Fig. 4. An example of backpropagation through a neural network.

                                                    <<FIGURE>>

         Fig. 3(a) shows a diagrammatic picture of a computational neural network. The neurons in the input layer receive some
        values and propagate them to the neurons in the middle layer and is referred to as training the network. 
        
        Once trained, the
        of the network, which is also frequently called a ‘hidden program can perform its task by computing the output of
        layer’. The weighted sums from one or more hidden layers are the network using the weights determined during the training
        ultimately propagated to the output layer, which presents the process. Running the program with these weights is referred
        final outputs of the network to the user. To align brain-inspired to as inference.
        terminology with neural networks, the outputs of the neurons   In this section, we will use image classification, as shown
        are often referred to as activations, and the synapses are often in Fig. 6, as a driving example for training and using a DNN.
        referred to as weights as shown in Fig. 3(a). We will use the When we perform inference using a DNN, we give an input
        activation/weight nomenclature in this article.            image and the output of the DNN is a vector of scores, one for
         Fig. 3(b) shows an example of the computation at each each object class; the class with the highest score indicates the
         most likely class of object in the image. The overarching goal layer: <<For>>, where W_ij ,x_i and y_j are the for training a DNN is to determine the weights that maximize
        weights, input activations and output activations, respectively, i=1                                the score of the correct class and minimize the scores of the
        and <<FORMULA>> is a non-linear function described in SectionIII-A1. incorrect classes. When training the network the correct class
        The bias term b is omitted from Fig. 3(b) for simplicity.     is often known because it is given for the images used for
         Within the domain of neural networks, there is an area called training (i.e., the training set of the network). The gap between
        deep learning, in which the neural networks have more than the ideal correct scores and the scores computed by the DNN
        three layers, i.e., more than one hidden layer. Today, the typical based on its current weights is referred to as theloss(L).
        numbers of network layers used in deep learning range from Thus the goal of training DNNs is to find a set of weights to
        five to more than a thousand. In this article, we will generally minimize the average loss over a large training set.
        use the terminologydeep neural networks (DNNs)to refer to   When training a network, the weights (wij ) are usually
        the neural networks used in deep learning.              updated using a hill-climbing optimization process called
         DNNs are capable of learning high-level features with more gradient descent. A multiple of the gradient of the loss relative
        complexity and abstraction than shallower neural networks. An to each weight, which is the partial derivative of the loss with
        example that demonstrates this point is using DNNs to process respect to the weight, is used to update the weight (i.e., updated
        visual data. In these applications, pixels of an image are fed into <<FORMULA>>, where <<FORMULA>> is called the learning rate). 
        Note <<FORMULA>> the first layer of a DNN, and the outputs of that layer can be that this gradient indicates how the weights should change in ij

        interpreted as representing the presence of different low-level order to reduce the loss. The process is repeated iteratively to
        features in the image, such as lines and edges. At subsequent reduce the overall loss.
        layers, these features are then combined into a measure of the   An efficient way to compute the partial derivatives of
        likely presence of higher level features, e.g., lines are combined the gradient is through a process called backpropagation.
        into shapes, which are further combined into sets of shapes. Backpropagation, which is a computation derived from the
        And finally, given all this information, the network provides a chain rule of calculus, operates by passing values backwards
        probability that these high-level features comprise a particular through the network to compute how the loss is affected by
        object or scene. This deep feature hierarchy enables DNNs to each weight.
        achieve superior performance in many tasks.               This backpropagation computation is, in fact, very similar
        in form to the computation used for inference as shown in Fig. 4 [10]. 2 Thus, techniques for efficiently performing

        C. Inference versus Training                        
        
         Since DNNs are an instance of a machine learning algorithm,   2 To backpropagate through each filter: (1) compute the gradient of the loss
        the basic program does not change as it learns to perform its relative to the weights from the filter inputs (i.e., the forward activations) and
        given tasks. In the specific case of DNNs, this learning involves the gradients of the loss relative to the filter outputs; (2) compute the gradient
                                                   of the loss relative to the filter inputs from the filter weights and the gradients determining the value of the weights (and bias) in the network, of the loss relative to the filter outputs.                                                                                             4


        inference can sometimes be useful for performing training.                 DNN Timeline
        It is, however, important to note a couple of points. First,
        backpropagation requires intermediate outputs of the network    1940s - Neural networks were proposed
        to be preserved for the backwards computation, thus training    1960s - Deep neural networks were proposed
        has increased storage requirements. Second, due to the gradients    1989 - Neural networks for recognizing digits (LeNet)
        use for hill-climbing, the precision requirement for training    1990s - Hardware for shallow neural nets (Intel ETANN)
        is generally higher than inference. Thus many of the reduced    2011 - Breakthrough DNN-based speech recognition
                                                       (Microsoft)precision techniques discussed in Section VII are limited to
        inference only.                                    2012 - DNNs for vision start supplanting hand-crafted
                                                       approaches (AlexNet)A variety of techniques are used to improve the efficiency
        and robustness of training. For example, often the loss from    2014+ - Rise of DNN accelerator research (Neuflow,
                                                       DianNao...)multiple sets of input data, i.e., abatch, are collected before a
        single pass of weight update is performed; this helps to speed Fig. 5. A concise history of neural networks. ’Deep’ refers to the number of
        up and stabilize the training process.                  layers in the network.
         There are multiple ways to train the weights. The most
        common approach, as described above, is called supervised
        learning, where all the training samples are labeled (e.g., with amount of available information to train the networks. To learn
        the correct class).Unsupervised learning is another approach a powerful representation (rather than using a hand-crafted
        where all the training samples are not labeled and essentially approach) requires a large amount of training data. For example,
        the goal is to find the structure or clusters in the data.Semi- Facebook receives over 350 millions images per day, Walmart
        supervised learning falls in between the two approaches where creates 2.5 Petabytes of customer data hourly and YouTube
        only a small subset of the training data is labeled (e.g., use has 300 hours of video uploaded every minute. As a result,
        unlabeled data to define the cluster boundaries, and use the the cloud providers and many businesses have a huge amount
        small amount of labeled data to label the clusters). Finally, of data to train their algorithms.
        reinforcement learning can be used to the train weights such   The second factor is the amount of compute capacity
        that given the state of the current environment, the DNN can available. Semiconductor device and computer architecture
        output what action the agent should take next to maximize advances have continued to provide increased computing
        expected rewards; however, the rewards might not be available capability, and we appear to have crossed a threshold where the
        immediately after an action, but instead only after a series of large amount of weighted sum computation in DNNs, which
        actions.                                      is required for both inference and training, can be performed
         Another commonly used approach to determine weights is in a reasonable amount of time.
        fine-tuning, where previously-trained weights are available and   The successes of these early DNN applications opened the
        are used as a starting point and then those weights are adjusted floodgates of algorithmic development. It has also inspired the
        for a new dataset (e.g., transfer learning) or for a new constraint development of several (largely open source) frameworks that
        (e.g., reduced precision). This results in faster training than make it even easier for researchers and practitioners to explore
        starting from a random starting point, and can sometimes result and use DNNs. Combining these efforts contributes to the third
        in better accuracy.                               factor, which is the evolution of the algorithmic techniques that
         This article will focus on the efficient processing of DNN have improved application accuracy significantly and broadened
        inference rather than training, since DNN inference is often the domains to which DNNs are being applied.
        performed on embedded devices (rather than the cloud) where   An excellent example of the successes in deep learning can
        resources are limited as discussed in more details later.      be illustrated with the ImageNet Challenge [14]. This challenge
                                                   is a contest involving several different components. One of the
                                                   components is an image classification task where algorithmsD. Development History                           are given an image and they must identify what is in the image,Although neural nets were proposed in the 1940s, the first as shown in Fig. 6. The training set consists of 1.2 millionpractical application employing multiple digital neurons didn’t images, each of which is labeled with one of 1000 objectappear until the late 1980s with the LeNet network for hand- categories that the image contains. For the evaluation phase,written digit recognition [11]3 . Such systems are widely used the algorithm must accurately identify objects in a test set ofby ATMs for digit recognition on checks. However, the early images, which it hasn’t previously seen.2010s have seen a blossoming of DNN-based applications with   Fig. 7 shows the performance of the best entrants in thehighlights such as Microsoft’s speech recognition system in ImageNet contest over a number of years. One sees that 2011 [2] and the AlexNet system for image recognition in the accuracy of the algorithms initially had an error rate2012 [3]. A brief chronology of deep learning is shown in of 25% or more. In 2012, a group from the University ofFig. 5.                                       Toronto used graphics processing units (GPUs) for their highThe deep learning successes of the early 2010s are believed compute capability and a deep neural network approach, namedto be a confluence of three factors. The first factor is the AlexNet, and dropped the error rate by approximately 10% [3].
                                                   Their accomplishment inspired an outpouring of deep learning In the early 1960s, single analog neuron systems were used for adaptive
                                                   style algorithms that have resulted in a steady stream of filtering [12, 13].                                                                                   5

                                                       Speech and LanguageDNNs have significantly improved
                                                       the accuracy of speech recognition [21] as well as many
                                                       related tasks such as machine translation [2], natural
                                                        language processing [22], and audio generation [23]. Machines Learning                      
                                                         MedicalDNNs have played an important role in genomic                        
                                                          to gain insight into the genetics of diseases such as autism,
                                                        cancers, and spinal muscular atrophy [24–27]. 
                   <<FIGURE>>                           They have also been used in medical imaging to detect skin cancer [5],
                                                       brain cancer [28] and breast cancer [29]. 
        Fig. 6. Example of an image classification task. 
        
        The machine learning platform takes in an image and outputs the confidence scores for a predefined set of classes.        
                                                        Game PlayRecently, many of the grand AI challenges
                                                       involving game play have been overcome using DNNs.
                                                       These successes also required innovations in training
                                                       techniques and many rely on reinforcement learning [30].
                                                             DNNs have surpassed human level accuracy in playing
                                                               Atari [31] as well as Go [6], where an exhaustive search
                                	                         of all possibilities is not feasible due to the unimaginably
                                                             huge number of possible moves.
                                             	             RoboticsDNNs have been successful in the domain of
                            <<FIGURE>>                        robotic tasks such as grasping with a robotic arm [32],
                                                              motion planning for ground robots [33], visual navigation [4,34], control to stabilize a quadcopter [35] and
        Fig. 7. Results from the ImageNet Challenge [14]. driving strategies for autonomous vehicles [36].

                                                     DNNs are already widely used in multimedia applications
                                                   today (e.g., computer vision, speech recognition). Looking
        improvements.                                 forward, we expect that DNNs will likely play an increasingly
         In conjunction with the trend to deep learning approaches important role in the medical and robotics fields, as discussed
        for the ImageNet Challenge, there has been a corresponding above, as well as finance (e.g., for trading, energy forecasting,
        increase in the number of entrants using GPUs. From 2012 and risk assessment), infrastructure (e.g., structural safety, and
        when only 4 entrants used GPUs to 2014 when almost all traffic control), weather forecasting and event detection [37].
        the entrants (110) were using them. This reflects the almost The myriad application domains pose new challenges to the
        complete switch from traditional computer vision approaches efficient processing of DNNs; the solutions then have to be
        to deep learning-based approaches for the competition.      adaptive and scalable in order to handle the new and varied
         In 2015, the ImageNet winning entry, ResNet [15], exceeded forms of DNNs that these applications may employ.
        human-level accuracy with a top-5 error rate 4 below 5%. Since
        then, the error rate has dropped below 3% and more focus F. Embedded versus Cloud
        is now being placed on more challenging components of the   The various applications and aspects of DNN processing competition, such as object detection and localization. These (i.e., training versus inference) have different computational successes are clearly a contributing factor to the wide range needs. Specifically, training often requires a large dataset 5 and of applications to which DNNs are being applied.         
        significant computational resources for multiple weight-update
         iterations. In many cases, training a DNN model still takes several hours to multiple days and thus is typically performed

         E. Applications of DNN                   

         Many applications can benefit from DNNs ranging from in the cloud. Inference, on the other hand, can happen either
        multimedia to medical space. In this section, we will provide in the cloud or at the edge (e.g., IoT or mobile).
        examples of areas where DNNs are currently making an impact   In many applications, it is desirable to have the DNN
        and highlight emerging areas where DNNs hope to make an inference processing near the sensor. For instance, in computer
        impact in the future.                             vision applications, such as measuring wait times in stores
          Image and VideoVideo is arguably the biggest of the or predicting traffic patterns, it would be desirable to extract
           big data. It accounts for over 70% of today’s Internet meaningful information from the video right at the image
           traffic [16]. For instance, over 800 million hours of video sensor rather than in the cloud to reduce the communication
           is collected daily worldwide for video surveillance [17]. cost. For other applications such as autonomous vehicles,
           Computer vision is necessary to extract meaningful infor- drone navigation and robotics, local processing is desired since
           mation from video. DNNs have significantly improved the the latency and security risks of relying on the cloud are
           accuracy of many computer vision tasks such as image too high. However, video involves a large amount of data,
           classification [14], object localization and detection [18], which is computationally complex to process; thus, low cost
           image segmentation [19], and action recognition [20].   hardware to analyze video is challenging yet critical to enabling

         4 The top-5 error rate is measured based on whether the correct answer   5 One of the major drawbacks of DNNs is their need for large datasets to
        appears in one of the top 5 categories selected by the algorithm.        prevent over-fitting during training.                                                                                             6


                                                   attention has been given to hardware acceleration specifically Feed Forward     Recurrent     Fully-Connected   Sparsely-Connected    for RNNs.
                                                     DNNs can be composed solely offully-connected(FC)
                                                   layers (also referred to as multi-layer perceptrons, or MLP)
                                                   as shown in the leftmost layer of Fig. 8(b). In a FC layer,
                                                   all output activations are composed of a weighted sum of
                                                   all input activations (i.e., all outputs are connected to all
                                                   inputs). This requires a significant amount of storage and
        Thankfully, in many applications, we can remove current) networks some connections between the activations by setting the weights
        to zero without affecting accuracy. This results in a sparsely connected layer. A sparsely connected layer is illustrated in
                                                   the rightmost layer of Fig. 8(b).these applications. Speech recognition enables us to seamlessly   We can also make the computation more efficient by limitinginteract with electronic devices, such as smartphones. While the number of weights that contribute to an output. This sort ofcurrently most of the processing for applications such as Apple structured sparsity can arise if each output is only a functionSiri and Amazon Alexa voice services is in the cloud, it is of a fixed-size window of inputs. Even further efficiency canstill desirable to perform the recognition on the device itself to be gained if the same set of weights are used in the calculationreduce latency and dependency on connectivity, and to improve of every output. This repeated use of the same weight values is privacy and security.                             calledweight sharingand can significantly reduce the storageMany of the embedded platforms that perform DNN infer- requirements for weights.ence have stringent energy consumption, compute and memory   An extremely popular windowed and weight-shared DNNcost limitations; efficient processing of DNNs have thus become layer arises by structuring the computation as a convolution,of prime importance under these constraints. Therefore, in this as shown in Fig. 9(a), where the weighted sum for each outputarticle, we will focus on the compute requirements for inference activation is computed using only a small neighborhood of inputrather than training.                              activations (i.e., all weights beyond beyond the neighborhood
                                                   are set to zero), and where the same set of weights are shared for
                                                    every output (i.e., the filter is space invariant). Such convolution-

                                            III. OVERVIEW OF DNN'S  

         DNNs come in a wide variety of shapes and sizes depending based layers are referred to as convolutional (CONV) layers. 
        on the application. The popular shapes and sizes are also
        evolving rapidly to improve accuracy and efficiency. In all A. Convolutional Neural Networks (CNNs)cases, the input to a DNN is a set of values representing the   A common form of DNNs isConvolutional Neural Netsinformation to be analyzed by the network. For instance, these (CNNs), which are composed of multiple CONV layers asvalues can be pixels of an image, sampled amplitudes of an shown in Fig. 10. In such networks, each layer generates aaudio wave or the numerical representation of the state of some successively higher-level abstraction of the input data, calledsystem or game.                                afeature map(fmap), which preserves essential yet uniqueThe networks that process the input come in two major information. Modern CNNs are able to achieve superior per-forms: feed forward and recurrent as shown in Fig. 8(a). In formance by employing a very deep hierarchy of layers. CNNfeed-forward networks all of the computation is performed as a are widely used in a variety of applications including imagesequence of operations on the outputs of a previous layer. The understanding [3], speech recognition [39], game play [6],final set of operations generates the output of the network, for robotics [32], etc. This paper will focus on its use in imageexample a probability that an image contains a particular object, processing, specifically for the task of image classification [3].the probability that an audio sequence contains a particular   Each of the CONV layers in CNN is primarily composed ofword, a bounding box in an image around an object or the high-dimensional convolutions as shown in Fig. 9(b). In thisproposed action that should be taken. In such DNNs, the computation, the input activations of a layer are structured asnetwork has no memory and the output for an input is always a set of 2-Dinput feature maps(ifmaps), each of which isthe same irrespective of the sequence of inputs previously given called achannel. Each channel is convolved with a distinctto the network.                                 2-D filter from the stack of filters, one for each channel; thisIn contrast, recurrent neural networks (RNNs), of which stack of 2-D filters is often referred to as a single 3-D filter.Long Short-Term Memory networks (LSTMs) [38] are a The results of the convolution at each point are summed acrosspopular variant, have internal memory to allow long-term all the channels. In addition, a 1-D bias can be added to thedependencies to affect the output. In these networks, some filtering results, but some recent networks [15] remove itsintermediate operations generate values that are stored internally usage from parts of the layers. The result of this computationin the network and used as inputs to other operations in is the output activations that comprise one channel ofoutputconjunction with the processing of a later input. In this article, feature map(ofmap). Additional 3-D filters can be used onwe will focus on feed-forward networks since (1) the major
        computation in RNNs is still the weighted sum, which is   6 Note: the structured sparsity in CONV layers is orthogonal to the sparsity covered by the feed-forward networks, and (2) to-date little that occurs from network pruning as described in Section VII-B2.                                                                                             7

                 after the CONV layers for classification purposes. A FC layer Fully	
                                     Connected	          also applies filters on the ifmaps as in the CONV layers, but
                  ×	                           ×	     the filters are of the same size as the ifmaps. Therefore, it
                                                   does not have the weight sharing property of CONV layers. Optional                      
                                                   Eq. (1) still holds for the computation of FC layers with a
        Fig. 10. Convolutional Neural Networks.       few additional constraints on the shape parameters: <<FORMULA>>,
                                                   <<FORMULA>>,<<FORMULA>>, and <<FORMULA>>.
                                                     In addition to CONV and FC layers, various optional layers
        the same input to create additional output channels. Finally, can be found in a DNN such as the non-linearity, pooling,
        multiple input feature maps may be processed together as a and normalization. The function and computations for each of
        batchto potentially improve reuse of the filter weights.      these layers are discussed next.
         Given the shape parameters in Table I, the computation of   1) Non-Linearity:A non-linear activation function is typically 
         applied after each CONV or FC layer. Various non-linear
                                                   functions are used to introduce non-linearity into the DNN as
                                                   shown in Fig. 11. These include historically conventional non- <<FORMULA>>
        <<FORMULA>>                             linear functions such as sigmoid or hyperbolic tangent as well
         <<FORMULA>>                             as rectified linear unit (ReLU) [40], which has become popular
        <<FORMULA>>;                            in recent years due to its simplicity and its ability to enable
        <<FORMULA>>;                           fast training. Variations of ReLU, such as leaky ReLU [41], (1) parametric ReLU [42], 
        and exponential LU [43] have also been O,I,W and B are the matrices of the of_maps, if_maps, filters explored for improved accuracy. 
        Finally, a non-linearity called and biases, respectively.Uis a given stride size. Fig. 9(b) maxout, which takes the max value of two intersecting linear shows a visualization of this computation (ignoring biases).   
        functions, has shown to be effective in speech recognition To align the terminology of CNNs with the generic DNN,  tasks [44, 45].
          filters are composed of weights (i.e., synapses)          2) Pooling: A variety of computations that reduce the
          input and output feature maps (if_maps, of_maps) are dimensionality of a feature map are referred to as pooling.
           composed of activations (i.e., input and output neurons)  Pooling, which is applied to each channel separately, enables                                                                                             
                                                 DNN is run only once), which is more consistent with what
                                                   would likely be deployed in real-time and/or energy-constrained 
                                                   LeNet[11] was one of the first CNN approaches introduced
                                                    in 1989. It was designed for the task of digit classification in
           <<FIGURE>>                            grayscale images of size 28x28. The most well known version,
                                                   LeNet-5, contains two CONV layers and two FC layers [48].
        Fig. 12. Various forms of pooling (Figure adopted from Caffe Tutorial [46]). Each CONV layer uses filters of size 5x5 (1 channel per filter)
                                                   with 6 filters in the first layer and 16 filters in the second layer.
        the network to be robust and invariant to small shifts and Average pooling of 2x2 is used after each convolution and a
        distortions. Pooling combines, or pools, a set of values in sigmoid is used for the non-linearity. In total, LeNet requires
        its receptive field into a smaller number of values. It can be 60k weights and 341k multiply-and-accumulates (MACs) per
        configured based on the size of its receptive field (e.g., 2x2) image. LeNet led to CNNs’ first commercial success, as it was
        and pooling operation (e.g., max or average), as shown in deployed in ATMs to recognize digits for check deposits.
        Fig. 12. Typically pooling occurs on non-overlapping blocks   AlexNet[3] was the first CNN to win the ImageNet Challenge
        (i.e., the stride is equal to the size of the pooling). Usually a in 2012. It consists of five CONV layers and three FC layers.
        stride of greater than one is used such that there is a reduction Within each CONV layer, there are 96 to 384 filters and the
        in the dimension of the representation (i.e., feature map).    filter size ranges from 3x3 to 11x11, with 3 to 256 channels
         3) Normalization:Controlling the input distribution across each. In the first layer, the 3 channels of the filter correspond
        layers can help to significantly speed up training and improve to the red, green and blue components of the input image.
        accuracy. Accordingly, the distribution of the layer input A ReLU non-linearity is used in each layer. Max pooling of
        activations <<FORMULA>> are normalized such that it has a zero mean 3x3 is applied to the outputs of layers 1, 2 and 5. To reduce
        and a unit standard deviation. In batch normalization (BN), computation, a stride of 4 is used at the first layer of the
        the normalized value is further scaled and shifted, as shown network. AlexNet introduced the use of LRN in layers 1 and
        in Eq. (2), where the parameters <<FORMULA>> are learned from 2 before the max pooling, though LRN is no longer popular
        training [47].X is a small constant to avoid numerical problems. in later CNN models. One important factor that differentiates
        Prior to this, local response normalization (LRN) [3] was AlexNet from LeNet is that the number of weights is much
        used, which was inspired by lateral inhibition in neurobiology larger and the shapes vary from layer to layer. To reduce the
        where excited neurons (i.e., high value activations) should amount of weights and computation in the second CONV layer,
        subdue its neighbors (i.e., cause low value activations); however, the 96 output channels of the first layer are split into two groups
        BN is now considered standard practice in the design of of 48 input channels for the second layer, such that the filters in
        CNNs while LRN is mostly deprecated. Note that while LRN the second layer only have 48 channels. Similarly, the weights
        usually is performed after the non-linear function, BN is mostly in fourth and fifth layer are also split into two groups. In total,
        performed between the CONV or FC layer and the non-linear AlexNet requires 61M weights and 724M MACs to process
        one 227x227 input image.
                                                       Overfeat[49] has a very similar architecture to AlexNet with
                      <<FORMULA>>           (2)      five CONV layers and three FC layers. The main differences <<FORMULA>>                    
                                                    are that the number of filters is increased for layers 3 (384
                                                   to 512), 4 (384 to 1024), and 5 (256 to 1024), layer 2 is notB. Popular DNN Models                          
                                                   split into two groups, the first fully connected layer only has
         Many DNN models have been developed over the past 3072 channels rather than 4096, and the input size is 231x231
        two decades. Each of these models has a different ‘network rather than 227x227. As a result, the number of weights grows
        architecture’ in terms of number of layers, layer types, layer to 146M and the number of MACs grows to 2.8G per image.
        shapes (i.e., filter size, number of channels and filters), and Overfeat has two different models: fast (described here) and
        connections between layers. Understanding these variations accurate. The accurate model used in the ImageNet Challenge
        and trends is important for incorporating the right flexibility gives a 0.65% lower top-5 error rate than the fast model at the
        in any efficient DNN engine.                        cost of 1.9% more MACs
         In this section, we will give an overview of various popular   VGG-16[50] goes deeper to 16 layers consisting of 13
        DNNs such as LeNet [48] as well as those that competed in CONV layers and 3 FC layers. In order to balance out the
        and/or won the ImageNet Challenge [14] as shown in Fig. 7, cost of going deeper, larger filters (e.g., 5x5) are built from
        most of whose models with pre-trained weights are publicly multiple smaller filters (e.g., 3x3), which have fewer weights,
        available for download; the DNN models are summarized in to achieve the same receptive fields as shown in Fig. 13(a).
        Table II. Two results for top-5 error results are reported. In the As a result, all CONV layers have the same filter size of 3x3.
        first row, the accuracy is boosted by using multiple crops from In total, VGG-16 requires 138M weights and 15.5G MACs
        the image and an ensemble of multiple trained models (i.e., to process one 224224 input image. VGG has two different
        the DNN needs to be run several times); these results were models: VGG-16 (described here) and VGG-19. VGG-19 gives
        used to compete in the ImageNet Challenge. The second row a 0.1% lower top-5 error rate than VGG-16 at the cost of
        reports the accuracy if only a single crop was used (i.e., the 1.27more MACs.                                                                                             9


                     <<FIGURE>>                                                                         <<FIGURE>>

        Fig. 13. Decomposing larger filters into smaller filters.             Fig. 14. Inception module from GoogleNet [51] with example channel lengths.


         GoogLeNet[51] goes even deeper with 22 layers. It in-
        troduced an inception module, shown in Fig. 14, which is                              
        composed of parallel connections, whereas previously there
        was only a single serial connection. Different sized filters (i.e.,                  
        1x1, 3x3, 5x5), along with 3x3 max-pooling, are used for
        each parallel connection and their outputs are concatenated                       
        for the module output. Using multiple filter sizes has the                          
        effect of processing the input at multiple scales. For improved                  	
        training speed, GoogLeNet is designed such that the weights         
        ReLU  and the activations, which are stored for backpropagation during         	<<FORMULA>>
        training, could all fit into the GPU memory. In order to reduce               
        the number of weights, 1x1 filters are applied as a ‘bottleneck’        
        to reduce the number of channels for each filter [52]. The 22        
        layers consist of three CONV layers, followed by 9 inceptions                     	
        layers (each of which are two CONV layers deep), and one FC         
        layer. Since its introduction in 2014, GoogleNet (also referred                 <<FIGURE>>
        to as Inception) has multiple versions: v1 (described here), v3 7     
        smaller 1-D filters as shown in Fig. 13(b) to reduce number           Fig. 15. Shortcut module from ResNet [15]. 
        Note that ReLU following last
        of MACs and weights in order to go deeper to 42 layers. CONV layer in short cut is after the addition.
        In conjunction with batch normalization [47], v3 achieves
        over 3% lower top-5 error than v1 with 2.5% increase in is used. This is similar to the LSTM networks that are used for computation [53]. 
        Inception-v4 uses residual connections [54], sequential data. ResNet also uses the ‘bottleneck’ approach of described in the next section, 
        for a 0.4% reduction in error.   using 1x1 filters to reduce the number of weight parameters.ResNet[15], also known as Residual Net, uses residual 
        As a result, the two layers in the shortcut module are replace d connections to go even deeper (34 layers or more). It was by three layers (1x1, 3x3, 1x1) where the 1x1 reduces and
        the first entry DNN in ImageNet Challenge that exceeded then increases (restores) the number of weights. ResNet-50human-level accuracy with a top-5 error rate below 5%. 
        One consists of one CONV layer, followed by 16 shortcut layers of the challenges with deep networks is the vanishing gradient (each of which are three CONV layers deep), and one FC
        during training: as the error backpropagates through the network layer; it requires 25.5M weights and 3.9G MACs per image.the gradient shrinks, which affects the ability to update the There are various versions of ResNet with multiple depths
        weights in the earlier layers for very deep networks. Residual (e.g.,without bottleneck:18, 34;with bottleneck:50, 101, 152).net introduces a ‘shortcut’ module which contains an identity The ResNet with 152 layers was the winner of the ImageNet
        connection such that the weight layers (i.e., CONV layers) Challenge requiring 11.3G MACs and 60M weights. Compared can be skipped as shown in Fig. 15. Rather than learning the to ResNet-50, it reduces the top-5 error by around 1% at the
        function for the weight layersF(x), the shortcut module learns cost of 2.9% more MACs and 2.5% more weights.the residual mapping <<FORMULA>>. Initially, <<FORMULA>> is
        zero and the identity connection is taken; then gradually during   Several trends can be observed in the popular DNNs shown
        training, the actual forward connection through the weight layer in Table II. Increasing the depth of the network tends to provide
                                                   higher accuracy. Controlling for number of weights, a deeper
         7 v2 is very similar to v3.                            network can support a wider range of non-linear functions                                                                                           


        that are more discriminative and also provides more levels B. Models
        of hierarchy in the learned representation [15,50,51,55].   Pretrained DNN models can be downloaded from variousThe number of filter shapes continues to vary across layers, websites [56–59] for the various different frameworks. It shouldthus flexibility is still important. Furthermore, most of the be noted that even for the same DNN (e.g., AlexNet) thecomputation has been placed on CONV layers rather than FC accuracy of these models can vary by around 1% to 2%layers. In addition, the number of weights in the FC layers is depending on how the model was trained, and thus the resultsreduced and in most recent networks (since GoogLeNet) the do not always exactly match the original publication.CONV layers also dominate in terms of weights. Thus, the
        focus of hardware implementations should be on addressing
        the efficiency of the CONV layers, which in many domains C. Popular Datasets for Classification
        are increasingly important.                           It is important to factor in the difficulty of the task when
                                                   comparing different DNN models. For instance, the task of
               IV. DNN DEVELOPMENT RESOURCES         classifying handwritten digits from the MNIST dataset [62]
                                                   is much simpler than classifying an object into one of 1000
                                                   One of the key factors that has enabled the rapid development classes as is required for the ImageNet dataset [14](Fig. 16).
                                                   of DNNs is the set of development resources that have been It is expected that the size of the DNNs (i.e., number ofmade available by the research community and industry. 
                                                   These weights) and the number of MACs will be larger for the moreresources are also key to the development of DNN accelerators difficult task than the simpler task and thus 
                                                   require moreby providing characterizations of the workloads and facilitating energy and have lower throughput. For instance, LeNet-5[48]the exploration of trade-offs in 
                                                   model complexity and accuracy. is designed for digit classification, while AlexNet[3], VGG-This section will describe these resources such that those who 16[50], GoogLeNet[51], 
                                                   and ResNet[15] are designed for theare interested in this field can quickly get started.         
                                                     There are many AI tasks that come with publicly availableA. Frameworks                                
         For ease of DNN development and to enable sharing of Public datasets are important for comparing the accuracy of
        trained networks, several deep learning frameworks have been different approaches. The simplest and most common task
        developed from various sources. These open source libraries is image classification, which involves being given an entire
        contain software libraries for DNNs. Caffe was made available image, and selecting 1 of N classes that the image most likely
        in 2014 from UC Berkeley [46]. It supports C, C++, Python belongs to. There is no localization or detection.
        and MATLAB. Tensorflow was released by Google in 2015,   MNISTis a widely used dataset for digit classification
        and supports C++ and python; it also supports multiple CPUs that was introduced in 1998 [62]. It consists of 2828 pixel
        and GPUs and has more flexibility than Caffe, with the grayscale images of handwritten digits. There are 10 classes
        computation expressed as dataflow graphs to manage the (for 10 digits) and 60,000 training images and 10,000 test
        tensors (multidimensional arrays). Another popular framework images. LeNet-5 was able to achieve an accuracy of 99.05%
        is Torch, which was developed by Facebook and NYU and when MNIST was first introduced. Since then the accuracy has
        supports C, C++ and Lua. There are several other frameworks increased to 99.79% using regularization of neural networks
        such as Theano, MXNet, CNTK, which are described in [60]. with dropconnect [63]. Thus, MNIST is now considered a fairly
        There are also higher-level libraries that can run on top of easy dataset.
        the aforementioned frameworks to provide a more universal   CIFARis a dataset that consists of 3232 pixel colored
        experience and faster development. One example of such images of of various objects, which was released in 2009 [64].
        libraries is Keras, which is written in Python and supports CIFAR is a subset of the 80 million Tiny Image dataset [65].
        Tensorflow, CNTK and Theano.                      CIFAR-10 is composed of 10 mutually exclusive classes. There
         The existence of such frameworks are not only a convenient are 50,000 training images (5000 per class) and 10,000 test
        aid for DNN researchers and application designers, but they images (1000 per class). A two-layer convolutional deep belief
        are also invaluable for engineering high performance or more network was able to achieve 64.84% accuracy on CIFAR-10
        efficient DNN computation engines. In particular, because the when it was first introduced [66]. Since then the accuracy has
        frameworks make heavy use of a set primitive operations, increased to 96.53% using fractional max pooling [67].
        such processing of a CONV layer, they can incorporate use of   ImageNetis a large scale image dataset that was first
        optimized software or hardware accelerators. This acceleration introduced in 2010; the dataset stabilized in 2012 [14]. It
        is transparent to the user of the framework. Thus, for example, contains images of 256256 pixel in color with 1000 classes.
        most frameworks can use Nvidia’s cuDNN library for rapid The classes are defined using the WordNet as a backbone to
        execution on Nvidia GPUs. Similarly, transparent incorporation handle ambiguous word meanings and to combine together
        of dedicated hardware accelerators can be achieved as was synonyms into the same object category. In otherwords, there
        done with the Eyeriss chip [61].                     is a hierarchy for the ImageNet categories. The 1000 classes
         Finally, these frameworks are a valuable source of workloads were selected such that there is no overlap in the ImageNet
        for hardware researchers. They can be used to drive experi- hierarchy. The ImageNet dataset contains many fine-grained
        mental designs for different workloads, for profiling different categories including 120 different breeds of dogs. There are
        workloads and for exploring hardware-software trade-offs.    1.3M training images (732 to 1300 per class), 100,000 testing                                                                                            11

               <<TABLE>>
                                                TABLE II
        SUMMARY OF POPULAR DNN S [3,15,48,50,51]. y ACCURACY IS MEASURED BASED ON TOP -5 ERROR ON IMAGE NET [14]. z THIS VERSION OF LE NET -5
                HAS 431 K WEIGHTS FOR THE FILTERS AND REQUIRES 2.3M MAC S PER IMAGE ,AND USES RE LU RATHER THAN SIGMOID .



                                                   be localized and classified (out of 1000 classes). The DNN
                                                   outputs the top five categories and top five bounding box
                                                   locations. There is no penalty for identifying an object that
                                                   is in the image but not included in the ground truth. For
                                                   object detection, all objects in the image must be localized
                                                   and classified (out of 200 classes). The bounding box for all
                                                   objects in these categories must be labeled. Objects that are
                                                   not labeled are penalized as are duplicated detections. Fig. 16. 
                                                   MNIST (10 classes, 60k training, 10k testing) [62] vs. ImageNet
        (1000 classes, 1.3M training, 100k testing)[14] dataset.               Beyond ImageNet, there are also other popular image
                                                   datasets for computer vision tasks. For object detection, there
        images (100 per class) and 50,000 validation images (50 per is the PASCAL VOC (2005-2012) dataset that contains 11k
        class).                                       images representing 20 classes (27k object instances, 7k of
         The accuracy of the ImageNet Challenge are reported using which has detailed segmentation) [68]. For object detection,
        two metrics: Top-5 and Top-1 error. Top-5 error means that if segmentation and recognition in context, there is the MS COCO
        any of the top five scoring categories are the correct category, dataset with 2.5M labeled instances in 328k images (91 object
        it is counted as a correct classification. The Top-1 requires categories) [69]; compared to ImageNet, COCO has fewer
        that the top scoring category be correct. In 2012, the winner categories but more instances per category, which is useful for
        of the ImageNet Challenge (AlexNet) was able to achieve an precise 2-D localization. COCO also has more labeled instances
        accuracy of 83.6% for the top-5 (which is substantially better per image to potentially help with contextual information.
        than the 73.8% which was second place that year that did not   Most recently even larger scale datasets have been made
        use DNNs); it achieved 61.9% on the top-1 of the validation available. For instance, Google has an Open Images dataset
        set. In 2017, the highest accuracy was 97.7% for the top-5.   with over 9M images [70], spanning 6000 categories. There is
         In summary of the various image classification datasets, it also a YouTube dataset with 8M videos (0.5M hours of video)
        is clear that MNIST is a fairly easy dataset, while ImageNet covering 4800 classes [71]. Google also released an audio
        is a challenging one with a wider coverage of classes. Thus dataset comprised of 632 audio event classes and a collection
        in terms of evaluating the accuracy of a given DNN, it is of 2M human-labeled 10-second sound clips [72]. These large
        important to consider that dataset upon which the accuracy is datasets will be evermore important as DNNs become deeper
        measured.                                    with more weight parameters to train.
                                                     Undoubtedly, both larger datasets and datasets for new
        D. Datasets for Other Tasks                        domains will serve as important resources for profiling and
                                                   exploring the efficiency of future DNN engines.Since the accuracy of the state-of-the-art DNNs are perform-
        ing better than human-level accuracy on image classification
        tasks, the ImageNet Challenge has started to focus on more         V. H ARDWARE FOR DNN P ROCESSING
        difficult tasks such as single-object localization and object   Due to the popularity of DNNs, many recent hardware
        detection. For single-object localization, the target object must platforms have special features that target DNN processing. For                                                                                            12


        instance, the Intel Knights Landing CPU features special vector     
        instructions for deep learning; the Nvidia PASCAL GP100        
        GPU features 16-bit floating point (FP16) arithmetic support     
        to perform two FP16 operations on a single precision core for   
        faster deep learning computation. Systems have also been built                       
        specifically for DNN processing such as Nvidia DGX-1 and    
        Facebook’s Big Basin custom DNN server [73]. DNN inference
        has also been demonstrated on various embedded System-on-   
        Chips (SoC) such as Nvidia Tegra and Samsung Exynos as    
        well as FPGAs. Accordingly, it’s important to have a good                          
        understanding of how the processing is being performed on     
        these platforms, and how application-specific accelerators can                       <<FIGURE>>                   
        be designed for DNNs for further improvement in throughput
        and energy efficiency.                                                   Fig. 17. Highly-parallel compute paradigms.
         The fundamental component of both the CONV and FC lay-
        ers are the multiply-and-accumulate (MAC) operations, which
        can be easily parallelized. In order to achieve high performance,       
        highly-parallel compute paradigms are very commonly used,                                   
        including both temporal and spatial architectures as shown in                       <<FORMULA>> 
        Fig. 17. The temporal architectures appear mostly in CPUs                            
        parallelism such as vectors (SIMD) or parallel threads (SIMT).
        Such temporal architecture use a centralized control for a large
        number of ALUs. These ALUs can only fetch data from the
        memory hierarchy and cannot communicate directly with each   
        other. In contrast, spatial architectures use dataflow processing,   
        i.e., the ALUs form a processing chain so that they can pass data
        from one to another directly. Sometimes each ALU can have       
        its own control logic and local memory, called a scratchpad or       
        register file. We refer to the ALU with its own local memory as
        a processing engine (PE). Spatial architectures are commonly              
        used for DNNs in ASIC and FPGA-based designs. In this   
        section, we will discuss the different design strategies for
        efficient processing on these different platforms, without any
        impact on accuracy (i.e., all approaches in this section produce
        bit-wise identical results); specifically,                                           <<FIGURE>>
         * For temporal architectures such as CPUs and GPUs, we
           will discuss howcomputational transformson the kernel            Fig. 18. Mapping to matrix multiplication for fully connected layers
           can reduce the number of multiplications to increase
           throughput.                                
         * For spatial architectures used in accelerators, we will
           discuss howdataflowscan increase data reuse from low andNin Fig. 18(b)); finally, the height of the output feature
           cost memories in the memory hierarchy toreduce energy map matrix is the number of channels in the output feature
           consumption.                               maps (M), and the width is the number of output feature maps
                                                   (N), where each output feature map of the FC layer has the
                                                   dimension of 1x1 number of output channels (M).
        A. Accelerate Kernel Computation on CPU and GPU Platforms   The CONV layer in a DNN can also be mapped to a matrix
         CPUs and GPUs use parallelizaton techniques such as SIMD multiplication using a relaxed form of the Toeplitz matrix as
        or SIMT to perform the MACs in parallel. All the ALUs share shown in Fig. 19. The downside for using matrix multiplication
        the same control and memory (register file). On these platforms, for the CONV layers is that there is redundant data in the input
        both the FC and CONV layers are often mapped to a matrix feature map matrix as highlighted in Fig. 19(a). This can lead
        multiplication (i.e., the kernel computation). Fig. 18 shows how to either inefficiency in storage, or a complex memory access
        a matrix multiplication is used for the FC layer. The height of pattern.
        the filter matrix is the number of filters and the width is the   There are software libraries designed for CPUs (e.g., Open-
        number of weights per filter (input channels (C) width (W) BLAS, Intel MKL, etc.) and GPUs (e.g., cuBLAS, cuDNN,
        height (H), sinceR=WandS=Hin the FC layer); etc.) that optimize for matrix multiplications. The matrix
        the height of the input feature maps matrix is the number of multiplication is tiled to the storage hierarchy of these platforms,
        activations per input feature map <<FORMULA>>, and the which are on the order of a few megabytes at the higher levels.                                                                                         
 
                    <<FIGURE>>

            Fig. 21. Read and write access per MAC.
 
                    <<FIGURE>>

        Fig. 19. Mapping to matrix multiplication for convolutional layers.     
        
         for a 3x3 filter, respectively, at the cost of reduced numerical stability, increased storage requirements, and specialized
         The matrix multiplications on these platforms can be further processing depending on the size of the filter.
        sped up by applying computational transforms to the data to   In practice, different algorithms might be used for different
        reduce the number of multiplications, while still giving the layer shapes and sizes (e.g., FFT for filters greater than 5x5,
        same bit-wise result. Often this can come at a cost of increased and Winograd for filters 3x3 and below). Existing platform
        number of additions and a more irregular data access pattern. libraries, such as MKL and cuDNN, dynamically chose the
                                                   appropriate algorithm for a given shape and size [77, 78].Fast Fourier Transform (FFT) [10,74] is a well known
        approach, shown in Fig. 20 that reduces the number of
        multiplications from <<O(N^2 N^2)>> to <<O(N^2 log N)>>            B. Energy-Efficient Dataflow for Accelerators <<FORMULA>>, where the
        output size is <<FORMULA>> and the filter size is <<FORMULA>>. To   For DNNs, the bottleneck for processing is in the memory perform 
        the convolution, we take the FFT of the filter and access. Each MAC requires three memory reads (for filterinput feature map, and then 
        perform the multiplication in weight, fmap activation, and partial sum) and one memorythe frequency domain; we then apply an inverse 
        FFT to the write (for the updated partial sum) as shown in Fig. 21. In theresulting product to recover the output feature map in the 
        worst case, all of the memory accesses have to go through the spatial domain. However, there are several drawbacks to using off-chip 
        DRAM, which will severely impact both throughput FFT: (1) the benefits of FFTs decrease with filter size; (2) the and energy efficiency. 
        For example, in AlexNet, to support itssize of the FFT is dictated by the output feature map size which 724M MACs, nearly 3000M DRAM 
        accesses will be required. is often much larger than the filter; (3) the coefficients in the Furthermore, DRAM accesses require up to 
        several orders offrequency domain are complex. As a result, while FFT reduces magnitude higher energy than computation [79].computation, 
        it requires larger storage capacity and bandwidth.   Accelerators, such as spatial architectures as shown inFinally, a popular 
        approach for reducing complexity is to make Fig. 17, provide an opportunity to reduce the energy cost ofthe weights sparse, which will 
        be discussed in SectionVII-B2; data movement by introducing several levels of local memoryusing FFTs makes it difficult for this sparsity 
        to be exploited. hierarchy with different energy cost as shown in Fig. 22. This
         Several optimizations can be performed on FFT to make it includes a large global buffer with a size of several hundred
        more effective for DNNs. To reduce the number of operations, kilobytes that connects to DRAM, an inter-PE network that
        the FFT of the filter can be precomputed and stored. In addition, can pass data directly between the ALUs, and a register file
        the FFT of the input feature map can be computed once and (RF) within each processing element (PE) with a size of a
        used to generate multiple channels in the output feature map. few kilobytes or less. The multiple levels of memory hierarchy
        Finally, since an image contains only real values, its Fourier help to improve energy efficiency by providing low-cost data
        Transform is symmetric and this can be exploited to reduce accesses. For example, fetching the data from the RF or
        storage and computation cost.                       neighbor PEs is going to cost 1 or 2 orders of magnitude
         Other approaches include Strassen [75] and Winograd [76], lower energy than from DRAM.
        which rearrange the computation such that the number of   Accelerators can be designed to support specialized process-
        multiplications reduce from <<O(N^3)>> to <<FORMULA>> and by 2.25% ing dataflows that leverage this memory hierarchy. The dataflow                                                                                            14

                                                        <<FORMULA>>

                                                   Fig. 23. Data reuse opportunities in DNNs [80].

                        <<FORMULA>>

        Fig. 22. Memory hierarchy and data movement energy [80].

                                  
        decides what data gets read into which level of the memory
        hierarchy and when are they getting processed. Since there is   
        no randomness in the processing of DNNs, it is possible to
        design a fixed dataflow that can adapt to the DNN shapes and                 
        sizes and optimize for the best energy efficiency. The optimized      
        dataflow minimizes access from the more energy consuming                     <<FORMULA>>
        levels of the memory hierarchy. Large memories that can store                 
         a significant amount of data consume more energy than smaller                  
        memories. For instance, DRAM can store gigabytes of data, but                            
        consumes two orders of magnitude higher energy per access                            
        than a small on-chip memory of a few kilobytes. Thus, every
        time a piece of data is moved from an expensive level to a Fig. 24. An analogy between the operation of DNN accelerators (texts in
        lower cost level in terms of energy, we want to reuse that piece black) and that of general-purpose processors (texts in red). Figure adopted
        from [81]. of data as much as possible to minimize subsequent accesses
        to the expensive levels. The challenge, however, is that the
        storage capacity of these low cost memories are limited. Thus program into machine-readable binary codes for executionwe need to explore different dataflows that maximize reuse given the hardware architecture (e.g., x86 or ARM); in theunder these constraints.                           processing of DNNs, the mapper translates the DNN shapeFor DNNs, we investigate dataflows that exploit three forms and size into a hardware-compatible computation mappingof input data reuse (convolutional, feature map and filter) as for execution given the dataflow. While the compiler usuallyshown in Fig. 23. For convolutional reuse, the same input optimizes for performance, the mapper optimizes for energyfeature map activations and filter weights are used within efficiency.a given channel, just in different combinations for different   The following taxonomy (Fig. 25) can be used to classifyweighted sums. For feature map reuse, multiple filters are the DNN dataflows in recent works [82–93] based on their applied to the same feature map, so the input feature map data handling characteristics [80]: activations are used multiple times across filters. Finally, for   1) Weight stationary (WS):The weight stationary dataflow
        filter reuse, when multiple input feature maps are processed at is designed to minimize the energy consumption of reading
        once (referred to as a batch), the same filter weights are used weights by maximizing the accesses of weights from the register
        multiple times across input features maps.               
        file (RF) at the PE (Fig. 25(a)). Each weight is read from
         If we can harness the three types of data reuse by storing DRAM into the RF of each PE and stays stationary for further
        the data in the local memory hierarchy and accessing them accesses. The processing runs as many MACs that use the
        multiple times without going back to the DRAM, it can save same weight as possible while the weight is present in the RF;
        a significant amount of DRAM accesses. For example, in it maximizes convolutional and filter reuse of weights. The
        AlexNet, the number of DRAM reads can be reduced by up to inputs and partial sums must move through the spatial array
        500in the CONV layers. The local memory can also be used and global buffer. The input fmap activations are broadcast to
        for partial sum accumulation, so they do not have to reach all PEs and then the partial sums are spatially accumulated
        DRAM. In the best case, if all data reuse and accumulation across the PE array.
        can be achieved by the local memory hierarchy, the 3000M   One example of previous work that implement weight
        DRAM accesses in AlexNet can be reduced to only 61M.    stationary dataflow is nn-X, or neuFlow [85], which uses
         The operation of DNN accelerators is analogous to that of eight 2-D convolution engines for processing a 1010 filter.
        general-purpose processors as illustrated in Fig. 24 [81]. In There are total 100 MAC units, i.e. PEs, per engine with each
        conventional computer systems, the compiler translates the PE having a weight that stays stationary for processing. The    
        
                                               <<FIGURE>>

                                Fig. 26. Variations of output stationary [80].(b) Output Stationary

                                                            are [89], [88], and [90], respectively.
                                                         No local reuse (NLR): While small register files are
                                                        efficient in terms of energy (pJ/bit), they are inefficient in terms Psum                             
               <<FORMULA>>                              of area (<<FORMULA>>). In order to maximize the storage capacity,                
                                                        and minimize the off-chip memory bandwidth, no local storage
        Fig. 25. Dataflows for DNNs [80].                is allocated to the PE and instead all that area is allocated
                                                   to the global buffer to increase its capacity (Fig. 25(c)). The
                                                   no local reuse dataflow differs from the previous dataflows in
        input fmap activations are broadcast to all MAC units and the that nothing stays stationary inside the PE array. As a result,
        partial sums are accumulated across the MAC units. In order to there will be increased traffic on the spatial array and to the
        accumulate the partial sums correctly, additional delay storage global buffer for all data types. Specifically, it has to multicast
        elements are required, which are counted into the required size the activations, single-cast the filter weights, and then spatially
        of local storage. Other weight stationary examples are found accumulate the partial sums across the PE array.
        in [82–84, 86, 87].                                In an example of the no local reuse dataflow from
         2) Output stationary (OS):The output stationary dataflow is UCLA [91], the filter weights and input activations are read
        designed to minimize the energy consumption of reading and from the global buffer, processed by the MAC units with custom
        writing the partial sums (Fig. 25(b)). It keeps the accumulation adder trees that can complete the accumulation in a single cycle,
        of partial sums for the same output activation value local in the and the resulting partial sums or output activations are then put
        RF. In order to keep the accumulation of partial sums stationary back to the global buffer. Another example is DianNao [92],
        in the RF, one common implementation is to stream the input which also reads input activations and filter weights from
        activations across the PE array and broadcast the weight to all the buffer, and processes them through the MAC units with
        PEs in the array.                                custom adder trees. However, DianNao implements specialized
         One example that implements the output stationary dataflow registers to keep the partial sums in the PE array, which helps
        is ShiDianNao [89], where each PE handles the processing for to further reduce the energy consumption of accessing partial
        each output activation value by fetching the corresponding input sums. Another example of no local reuse dataflow is found
        activations from neighboring PEs. The PE array implements in [93].
        dedicated networks to pass data horizontally and vertically.   4) Row stationary (RS): A row stationary dataflow is
        Each PE also has data delay registers to keep data around for proposed in [80], which aims to maximize the reuse and
        the required amount of cycles. At the system level, the global accumulation at the RF level foralltypes of data (weights,
        buffer streams the input activations and broadcasts the weights pixels, partial sums) for the overall energy efficiency. This
        into the PE array. The partial sums are accumulated inside differs from WS or OS dataflows, which optimize for only
        each PE and then get streamed out back to the global buffer. weights and partial sums, respectively.
        Other examples of output stationary are found in [88, 90].     The row stationary dataflow assigns the processing of a
         There are multiple possible variants of output stationary as 1-D row convolution into each PE for processing as shown
        shown in Fig. 26 since the output activations that get processed in Fig. 27. It keeps the row of filter weights stationary inside
        at the same time can come from different dimensions. For the RF of the PE and then streams the input activations into
        example, the variantOS A targets the processing of CONV the PE. The PE does the MACs for each sliding window at a
        layers, and therefore focuses on the processing of output time, which uses just one memory space for the accumulation
        activations from the same channel at a time in order to of partial sums. Since there are overlaps of input activations
        maximize data reuse opportunities. The variantOS C targets between different sliding windows, the input activations can
        the processing of FC layers, and focuses on generating output then be kept in the RF and get reused. By going through all the
        activations from all different channels, since each channel only sliding windows in the row, it completes the 1-D convolution
        has one output activation. The variantOS B is something in and maximize the data reuse and local accumulation of data
        betweenOS A andOS C . Example of variantsOS A ,OS B , and in this row.                                                                                            16

                                        <<FIGURE>>

        Fig. 27. 1-D Convolutional reuse within PE for Row Stationary Dataflow [80].      
        
                          <<FIGURE>>    
          
          Fig. 29. Multiple rows of different input feature maps, filters and channels are
          mapped to same PE within array for additional reuse in the Row Stationary

                   <<FIGURE>>
        Fig. 28. 2-D convolutional reuse within spatial array for Row Stationary   

        shown in Fig. 28. For example, to generate the first row of
        output activations with a filter having three rows, three 1-D Fig. 30. Mapping optimization takes in hardware and DNNs shape constraints
        convolutions are required. Therefore, we can use three PEs in to determine optimal energy dataflow [80].
        a column, each running one of the three 1-D convolutions. The
        partial sums are further accumulated vertically across the three
        PEs to generate the first output row. To generate the second different channels are interleaved, and run through the same PE
        row of output, we use another column of PEs, where three as a 1-D convolution. The partial sums from different channels
        rows of input activations are shifted down by one row, and use then naturally get accumulated inside the PE.
        the same rows of filters to perform the three 1-D convolutions.   The number of filters, channels, and fmaps that can be
        Additional columns of PEs are added until all rows of the processed at the same time is programmable, and there exists an
        output are completed (i.e., the number of PE columns equals optimal mapping for the best energy efficiency, which depends
        the number of output rows).                        on the shape configuration of the DNN as well as the hardware
         This 2-D array of PEs enables other forms of reuse to reduce resources provided, e.g., the number of PEs and the size of the
        accesses to the more expensive global buffer. For example, each memory in the hierarchy. Since all of the variables are known
        filter row is reused across multiple PEs horizontally. Each row before runtime, it is possible to build a compiler (i.e., mapper)
        of input activations is reused across multiple PEs diagonally. to perform this optimization off-line to configure the hardware
        And each row of partial sums are further accumulated across for different mappings of the RS dataflow for different DNNs
        the PEs vertically. Therefore, 2-D convolutional data reuse and as shown in Fig. 30.
        accumulation are maximized inside the 2-D PE array.        One example that implements the row stationary dataflow
         To address the high-dimensional convolution of the CONV is Eyeriss [94]. It consists of a 14x12 PE array, a 108KB
        layer (i.e., multiple fmaps, filters, and channels), multiple rows global buffer, ReLU and fmap compression units as shown
        can be mapped onto the same PE as shown in Fig. 29. The in Fig. 31. The chip communicates with the off-chip DRAM
        2-D convolution is mapped to a set of PEs, and the additional using a 64-bit bidirectional data bus to fetch data into the
        dimensions are handled by interleaving or concatenating the global buffer. The global buffer then streams the data into the
        additional data. For filter reuse within the PE, different rows PE array for processing.
        of fmaps are concatenated and run through the same PE   In order to support the RS dataflow, two problems need to be
        as a 1-D convolution. For input fmap reuse within the PE, solved in the hardware design. First, how can the fixed-size PE
        different filter rows are interleaved and run through the same array accommodate different layer shapes? Second, although
        PE as a 1-D convolution. Finally, to increase local partial sum the data will be passed in a very specific pattern, it still changes
        accumulation within the PE, filter rows and fmap rows from with different shape configurations. How can the fixed design                                                                                           


                                                        needs of each dataflow under the same area constraint. For
                                                      example, since the no local reuse dataflow does not require any Processing 
                                                            RF in PE, it is allocated with a much larger global buffer. The If map 
                                                       simulation uses the layer configurations from AlexNet with a Buffer             
                                                      batch size of 16. The simulation also takes into account the bits  
                                                    fact that accessing different levels of the memory hierarchy Enc.                          
                                                       requires different energy cost.                    
                                                    of each dataflow for the CONV layers of AlexNet with a
                                                   batch size of 16. The WS and OS dataflows have the lowest
                                                 energy consumption for accessing weights and partial sums,
                                                   respectively. However, the RS dataflow has the lowest total 13                    
                                                     energy consumption since it optimizes for the overall energy
                                                   efficiency instead of only for a certain data type.

                                                     Fig. 33(a) shows the same results with breakdown in terms of
                                                   memory hierarchy. The RS dataflow consumes the most energy           
                                                   in the RF, since by design most of the accesses have been 
                                                    moved to the lowest level of the memory hierarchy. This helps
                                                    to achieve the lowest total energy consumption since RF has
                                                   the lowest energy per access. The NLR dataflow has the lowest Clock Gated                                      
                                                        energy consumption at the DRAM level, since it has a much
                                                 larger global buffer and thus higher on-chip storage capacity
                                                   compared to others. However, most of the data accesses in
                                                        relatively large energy consumption per access compared to
                                                   accessing data from RF or inside the PE array. As a result, the
                                                   overall energy consumption of the NLR dataflow is still fairly
                                                  high. Overall, RS dataflow uses 1.4% to 2.5% lower energy
                                                  pass data in different patterns?

                                                    <<FIGURE>>

        Fig. 32. Mapping uses replication and folding to maximized utilization of the NLR dataflow is from the global buffer, which still has a
        PE array [94]. 

         Two mapping strategies can be used to solve the first problem than other dataflows.
        as shown in Fig. 32. First, replication can be used to map shapes   Fig. 34 shows the energy efficiency between different 
        that do not use up the entire PE array. For example, in the dataflows in the FC layers of AlexNet with a batch size of 16.
        third to fifth layers of AlexNet, each 2-D convolution only uses Since there is not as much data reuse in the FC layers as in a 
        13x3 PE array. This structure is then replicated four times, the CONV layers, all dataflows spend a significant amount of
        and runs different channels and filters in each replication. The energy on reading weights. However, RS dataflow still has the
        second strategy is called folding. For example, in the second lowest energy consumption because it optimizes for the energy
        layer of AlexNet, it requires a 27x5 PE array to complete the of accessing input activations and partial sums. For the OS2-D 
        convolution. In order to fit it into the 14x12 physical PE dataflows,OSarray, it is folded into two parts, 14x5 and 13x5, and each           
        C now consumes lower energy thanOS A since it is designed for the FC layers. Overall, RS still consumesare vertically mapped into 
        the physical PE array. Since not all 1.3% lower energy compared to other dataflows at the batchPEs are used by the mapping, the 
        unused PEs can be clock size of 16.gated to save energy consumption.
         A custom multicast network is used to solve the second   Fig. 35 shows the RS dataflow design with energy breakdown
        problem about flexible data delivery. The simplest way to pass in terms of different layers of AlexNet. In the CONV layers, the
        data to multiple destinations is to broadcast the data to all PEs energy is mostly consumed by the RF, while in the FC layers,
        and let each PE decide if it has to process the data or not. the energy is mostly consumed by DRAM. However, most
        However, it is not very energy efficient especially when the of the energy is consumed by the CONV layers, which takes
        size of PE array is large. Instead, a multicast network is used around 80% of the energy. As recent DNN models go deeper
        to send data to only the places where it is needed.         with more CONV layers, the ratio between number of CONV
         5) Energy comparison of different dataflows:To evaluate and FC layers only gets larger. Therefore, moving forward,
        and compare different dataflows, the same total hardware area significant effort should be placed on energy optimizations for
        and number of PEs (256) are used in the simulation of a spatial CONV layers.
        architecture for all dataflows. The local memory (register file) at   Finally, up until now, we have been looking at architec-
        each processing element (PE) is on the order of 0.5 – 1.0kB and tures with relatively limited storage on the order of a few
        a shared memory (global buffer) is on the order of 100 – 500kB. hundred kilobytes. With much larger storage on the order of
        The sizes of these memories are selected to be comparable to a few megabytes, additional dataflows can be considered. For
        a typical accelerator for multimedia processing, such as video example, Fused-Layer looks at dataflow optimizations across
        coding [95]. The memory sizes are further adjusted for the layers [96].                                                                                            18

                                                             <<FORMULA>>

                                                   Fig. 35. Energy breakdown across layers of the AlexNet [80]. RF energy
                                                   dominates in convolutional layers. DRAM energy dominates in the fully
                                                  connected layer. Convolutional layer dominate energy consumption.
                                                    In this section, we will discuss how moving compute and data Normalized 
                                                    closer to reduce data movement (i.e., near-data processing) can pixels   
                                                    be achieved using mixed-signal circuit design and advanced                                 
                                                    memory technologies.
                                                     Many of these works use analog processing which has the 
                                                     drawback of increased sensitivity to circuit and device non- 
                                                    idealities. Consequentially, the computation is often performed
                                                   at reduced precision, which can be accounted for during (b) Energy breakdown across data type            
                                                   the training of the DNNs using the techniques discussed in
                                                   Section VII. Another factor to take into consideration is that Fig. 33. 
                                                   Comparison of energy efficiency between different dataflows in the DNNs are 
                                                   often trained in the digital domain; thus for analog CONV layers of AlexNet with a batch size of 16 [3]: 
                                                   (a) breakdown in terms of storage levels and ALU, (b) breakdown in terms of data types. OS      
                                                   processing, there is an additional overhead cost for analog- A , OS B and OS C are three variants of the 
                                                   OS dataflow that are commonly seen in to-digital conversion (ADC) and digital-to-analog conversion different implementations [80].                          (DAC).

                                                    A. DRAM 

                                                     Advanced memory technology can reduce the access energy
                                                     for high density memories such as DRAMs. For instance, psums
                                                    embedded DRAM (eDRAM)brings high density memory on-                                  
                                                    chip to avoid the high energy cost of switching off-chip pixels    
                                                    capacitance [97]; eDRAM is 2.85higher density than SRAM 0.5                                 
                                                    and 32% more energy efficient than DRAM (DDR3) [93].
                                                    eDRAM also offers higher bandwidth and lower latency                  
                                                    compared to DRAM. In DNN processing, eDRAM can be used DNN Dataflows                
                                                    to store tens of megabytes of weights and activations on-chip
                                                   to avoid off-chip access, as demonstrated in DaDianNao [93]. 
                                                   off-chip DRAM and can increase the cost of the chip.
                                                     Rather than integrating DRAM into the chip itself, the
                                                   DRAM can also be stacked on top of the chip using throughVI. N EAR -D ATA PROCESSING            silicon vias (TSV). This technology is often referred to as3-D
         The previous section highlighted that data movement domi- memory, and has been commercialized in the form of Hybrid
        nates energy consumption. While spatial architectures distribute Memory Cube (HMC) [98] and High Bandwidth Memory
        the on-chip memory such that it is closer to the computation (HBM) [99]. 3-D memory delivers an order of magnitude higher
        (e.g., into the PE), there have also been efforts to bring the bandwidth and reduces access energy by up to 5relative to
        off-chip high density memory closer to the computation or to existing 2-D DRAMs, as TSV have lower capacitance than
        integrate the computation into the memory itself; the latter is typical off-chip interconnects. Recent works have explored the
        often referred to asprocessing-in-memoryorlogic-in-memory. use of HMC for efficient DNN processing in a variety of ways.
        In embedded systems, there have also been efforts to bring the For instance, Neurocube [100] integrates SIMD processors into
        computation into the sensor where the data is first collected. the logic die of the HMC to bring the memory and computation                                                                                            19
        voltage as the input, and the current as the output as shown in resistive memory. 

                        <<FIGURE>>

        Fig. 36. Analog computation by (a) SRAM bit-cell and (b) non-volatile   
        
        
                                                    Processing with non-volatile resistive memories has several drawbacks as described in [108]. 
                                                   First, it suffers from the
                                                   reduced precision and ADC/DAC overhead of analog process-
                                                   ing described earlier. Second, the array size is limited by thecloser together. Tetris [101] explores the use of HMC with wires that connect the resistive devices; specifically, wire energythe Eyeriss spatial architecture and row stationary dataflow. dominates for large arrays (e.g., 1k1k), and the IR drop alongIt proposes allocating more area to computation than on-chip wire can degrade the read accuracy. Third, the write energymemory (i.e., larger PE array and smaller global buffer) in to program the resistive devices can be costly, in some casesorder to exploit the low energy and high throughput properties requiring multiple pulses. Finally, the resistive devices can alsoof the HMC. It also adapts the dataflow to account for the suffer from device-to-device and cycle-to-cycle variations withHMC memory and smaller on-chip memory. Tetris achieves non-linear conductance across the conductance range.a 1.5reduction in energy consumption and 4.1increase   There have been several recent works that explore the use ofin throughput over a baseline system with conventional 2-D memristors for DNNs. ISAAC [104] replaces the eDRAM inDRAM.                                      DaDianNao with memristors. To address the limited precision
                                                   support, ISAAC computes a 16-bit dot product operation with
        B. SRAM                                     8 memristors each storing 2-bits; a 1-bit2-bit multiplication
         Rather than bringing the memory near the compute, recent is performed at each memristor, where a 16-bit input requires
        work has also investigated bringing the compute into the 16 cycles to complete. In other words, the ISAAC architecture
        memory. For instance, the multiply and accumulate operation trades off area and time for increased precision. Finally, ISAAC
        can be directly integrated into the bit-cells of an SRAM arranges its 25.1M memristors in a hierarchical structure to
        array [102], as shown in Fig. 36(a). In this work, a 5-bit avoid issues with large arrays. PRIME [109] also replaces the
        DAC is used to drive the word line (WL) to an analog voltage DRAM main memory with memristors; specifically, it uses
        that represents the feature vector, while the bit-cells store the 256256 memristor arrays that can be configured for 4-bit
        binary weights1. The bit-cell current (I              multi-level cell computation or 1-bit single level cell storage. BC ) is effectively
        a product of the value of the feature vector and the value of It should be noted that results from ISAAC and PRIME are
        the weight stored in the bit-cell; the currents from the bit- obtained from simulations. The task of actually fabricating
        cells within a column add together to discharge the bitline large memristors arrays is still very much a research challenge;
        (V                                          for instance, [110] uses a fabricated 1212 memristor array BL ). This approach gives 12energy savings compared to
        reading the 1-bit weights from the SRAM and performing the to demonstrate a linear classifier.
        computation separately. To counter circuit non-idealities, the
        DAC accounts for the non-linear bit-line discharge with respect D. Sensors
        to the WL voltage, and boosting is used to combine the weak   In certain applications, such as image processing, the dataclassifiers that are susceptible to device variations to form a movement from the sensor itself can account for a significantstrong classifier [103].                            portion of the system energy consumption. Thus there has
                                                   also been research on performing the computation as close
        C. Non-volatile Resistive Memories                   as possible to the sensor. In particular, much of the work
                                                   focuses on moving the computation into the analog domain toThe multiply and accumulate operation can also be directly avoid using the ADC within the sensor, which accounts for aintegrated into advancednon-volatilehigh density memories significant portion of the sensor power. However, as mentionedby using them as programmable resistive elements, commonly
        referred to asmemristors[105]. Specifically, a multiplication   8 The resistive devices can be inserted between the cross-point of two wires is performed with the resistor’s conductance as the weight, the and in certain cases can avoid the need for an access transistor.                                                                                            20


        earlier, lower precision is required for analog computation due
        to circuit non-idealities.
         In [111], the matrix multiplication is integrated into the
        ADC, where the most significant bits of the multiplications
        are performed using switched capacitors in an 8-bit successive
        approximation format. This is extended in [112] to not only
        perform the multiplications, but also the accumulations in the
        analog domain. In this work, it is assumed that 3-bits and
        6-bits are sufficient to represent the weights and activations,     
        respectively. This reduces the number of ADC conversions in
        the sensor by 21. RedEye [113] takes this approach even
        further by performing the entire convolution layer (including
        convolution, max pooling and quantization) in the analog
        domain at the sensor. It should be noted that [111] and [112]
        report measured results from fabricated test chips, while results
        in [113] are from simulations.                                              <<FIGURE>>
         It is also feasible to embed the computation not just before
        the ADC, but into the sensor itself. For instance, in [114] an      Fig. 37. Various methods of quantization (Figures from [117, 118]).
        Angle Sensitive Pixels sensor is used to compute the gradient
        of the input, which along with compression, reduces the data the number of bits. The benefits of reduced precision includemovement from the sensor by 10. In addition, since the reduced storage cost and/or reduced computation requirements.first layer of the DNN often outputs a gradient-like feature
        map, it maybe possible to skip the computations in the first   There are several ways to map the data to quantization levels.
        layer, which further reduces energy consumption as discussed The simplest method is a linear mapping with uniform distance
        in [115, 116].                                  between each quantization level (Fig. 37(a)). Another approach
                                                   is to use a simple mapping function such as alog function
                                                   (Fig. 37(b)) where the distance between the levels varies; thisVII. C O -DESIGN OF DNN MODELS AND HARDWARE    mapping can often be implemented with simple logic such as aIn earlier work, the DNN models were designed to maximize shift. Alternatively, a more complex mapping function can beaccuracy without much consideration of the implementation used where the quantization levels are determined or learnedcomplexity. However, this can lead to designs that are chal- from the data (Fig. 37(c)), e.g., using k-means clustering; forlenging to implement and deploy. To address this, recent this approach, the mapping is usually implemented with a lookwork has shown that DNN models and hardware can be co- up table.designed to jointly maximize accuracy and throughput, while   Finally, the quantization can be fixed (i.e., the same methodminimizing energy and cost, which increases the likelihood of of quantization is used for all data types and layers, filters, andadoption. In this section, we will highlight various efforts that channels in the network); or it can be variable (i.e., differenthave been made towards the co-design of DNN models and methods of quantization can be used for weights and activations,hardware. Note that unlike Section V, the techniques discussed and different layers, filters, and channels in the network).in this section can affect the accuracy; thus, the goal is to   Reduced precision research initially focused on reducingnot only substantially reduce energy consumption and increase the precision of the weights rather than the activations, sincethroughput, but also to minimize any degradation in accuracy. weights directly increase the storage capacity requirement,The co-design approaches can be loosely grouped into the while the impact of activations on storage capacity depends onfollowing categories:                             the network architecture and dataflow. However, more recent
          Reduce precision of operations and operands.This in- works have also started to look at the impact of quantizationcludes going from floating point to fixed point, reducing on activations. Most reduced precision research also focusesthe bitwidth, non-linear quantization and weight sharing. on reducing the precision for inference rather than training
          Reduce number of operations and model size. This (with some exceptions [88,119,120]) due to the sensitivity ofincludes techniques such as compression, pruning and the gradients to quantization.compact network architectures.                     The key techniques used in recent work to reduce precision
                                                   are summarized in Table III; both linear and non-linear
        A. Reduce Precision                             quantization applied to weights and activations are explored.
         Quantization involves mapping data to a smaller set of The impact on accuracy is reported relative to a baseline
        quantization levels. The ultimate goal is to minimize the error precision of 32-bit floating point, which is the default precision
        between the reconstructed data from the quantization levels and used on platforms such as GPUs and CPUs.
        the original data. The number of quantization levels reflects the   1) Linear quantization:The first step of reducing precision
        precisionand ultimately the number of bits required to represent is usually to convert values and operations from floating point
        the data (usuallylog 2 of the number of levels); thus,reduced to fixed point. A 32-bit floating point number, as shown in
        precisionrefers to reducing the number of levels, and thus Fig. 38(a), is represented by <<FORMULA>>, wheres                                                                                    
        product; that output would need to be accumulated with <<FORMULA>>   
        bit precision, where M is determined based on the largest filter (b) 8-bit dynamic fixed point examples            
        size <<FORMULA>> (<<FORMULA>> from Fig. 9(b)), which is in the range of 0 to 16 bits for the popular DNNs described in SectionIII-B.

        Fig. 38. Various methods of number representations.              1
        
                                                   After accumulation, the precision of the final output activation
                                                   is typically reduced to N-bits [88,121], as shown in Fig. 39.is the sign bit, e is the 
                                                   8-bit exponent, andmis the 23-bit The reduced output precision does not have a significant impact 
                                                   mantisa, and covers the range of <<FORMULA>>.          
                                                   on accuracy if the distribution of the weights and activationsAn N-bit fixed point number is 
                                                   represented by <<FORMULA>> are centered near zero such that the accumulation would not 
                                                   2f , wheresis the sign bit,mis the (N-1)-bit mantissa, and move only in one direction; 
                                                   this is particularly true when batchfdetermines the location of the decimal point and acts as a normalization is used.
                                                   scale factor. For instance, for an 8-bit integer, whenf= 0,   
                                                   The reduced precision is not only explored in research,the dynamic range is -128 to 127, 
                                                   whereas whenf= 10, the but has been used in recent commercial platforms for DNN 
                                                   dynamic range is -0.125 to 0.124023438.Dynamicfixed point processing. For instance, Google’s 
                                                   Tensor Processing Unitrepresentation allowsfto vary based on the desired dynamic (TPU) 
                                                   which was announced in May 2016, was designed forrange as shown in Fig. 38(b). 
                                                   This is useful for DNNs, since 8-bit integer arithmetic [123]. Similarly, Nvidia’s PASCAL
                                                   the dynamic range of the weights and activations can be quite GPU, which was announced in 
                                                   April 2016, also has 8-bitdifferent. In addition, the dynamic range can also vary across 
                                                   \integer instructions for deep learning inference [124]. In generallayers and layer types 
                                                   (e.g., convolutional vs. fully connected). purpose platforms such as CPUs and GPUs, the main benefit
                                                   Using dynamic fixed point, the bitwidth can be reduced to 8 of using 8-bit computation is an increase 
                                                   in throughput, asbits for the weights and 10 bits for the activations without any four 8-bit 
                                                   operations rather than one 32-bit operation can befine-tuning of the weights [121]; with fine-tuning, 
                                                   both weights performed for a given clock cycle.and activations can reach 8-bits [122].                   
                                                   While general purpose platforms usually support 8-bit,Using 8-bit fixed point has the following 
                                                   impact on energy 16-bit and/or 32-bit operations, it has been shown that theand area [79]:                                  
                                                   minimum bit precision for DNNs can actually vary in a more
          An 8-bit fixed point add consumes 3.3% less energy fine grained manner. For instance, the weight and activation
           (3.8less area) than a 32-bit fixed point add, and 30% precision can vary between 4 and 9 bits for AlexNet across
           less energy (116less area) than a 32-bit floating point different layers without significant impact on accuracy (i.e., a
           add. The energy and area of a fixed-point add scales change of less than 1%) [125,126]. This fine-grained variation
           approximately linearly with the number of bits.        can be exploited for increased throughput or reduced energy
          An 8-bit fixed point multiply consumes 15.5% less energy consumption with specialized hardware. For instance, if bit-
           (12.4% less area) than a 32-bit fixed point multiply, serial processing is used, where the number of clock cycles to
           and 18.5% less energy (27.5% less area) than a 32-bit complete an operation is proportional to the bitwidth, adapting
           floating point multiply. The energy and area of a fixed- to fine-grain variations in bit precision can result in a 2.24%
           point multiply scales approximately quadratically with the speed up versus 16-bits [125]. Alternatively, a multiplier can
           number of bits.                              be designed such that its critical path reduces based on the bit
         Reducing the precision also reduces the energy and area cost precision as fewer adders are needed to resolve the product;
        for storage, which is important since memory access and data this can be combined with voltage scaling for a 2.56energy
        movement dominate energy consumption as described earlier. savings versus 16-bits [126]. While these bit scaling results
        The energy and area of the memory scale approximately linearly are reported relative to 16-bit, it would be interesting to see
        with number of bits. It should be noted, however, that changing their impact relative to the maximum precision required across
        from floating point to fixed point, without reducing bit-width, layers (i.e., 9-bits for [125, 126]).
        does not reduce the energy or area cost of the memory.       The precision can be reduced even more aggressively to a
         For completeness, it should be noted that the precision of single bit; this area of research is often referred to asbinary nets.
        the internal values of a fixed-point multiply and accumulate BinaryConnect (BC) [127] introduced the concept of binary
        (MAC) operation are typically higher than the weights and weights (i.e., -1 and 1), where using a binary weight reduced
        activations. To guarantee no precision loss, weights and input the multiplication in the MAC to addition and subtraction
        activations with N-bit fixed-point precision would require an only. This was later extended in Binarized Neural Networks
        N-bitxN-bit multiplication which generates a 2N-bit output (BNN) [128] that uses binary weightsandactivations, which     
        
                <<FIGURE>>

         Fig. 40. Weight sharing hardware.                                                                                       

        w, where w is the average of the absolute values of the
        weights in the filter) 9 , keeping the first and last layers at 32-bit
        floating point precision, and performing normalization before VGG-16 [117]. Furthermore, when weights are quantized to
        convolution to reduce the dynamic range of the activations. powers of two, the multiplication can be replaced with a bit-
        With these changes, BWN reduced the accuracy loss to 0.8%, shift [122,135]. 10 Incremental Network Quantization (INQ)
        while XNOR-Nets reduced the loss to 11%. The loss of XNOR- can be used to further reduce the loss in accuracy by dividing
        Net can be further reduced by increasing the precision of the the large and small weights into different groups, and then
        activations to be slightly larger than one bit. For instance, iteratively quantizing and re-training the weights [136].
        Quantized Neural Networks (QNN) [119], DoReFa-Net [120],   Weight Sharingforces several weights to share a single value.
        and HWGQ-Net [130] allow the activations to have 2-bits, This reduces the number of unique weights in a filter or a
        while the weights remain at 1-bit; in HWGQ-Net, this reduces layer. One example is to group the weights by using a hashing
        the accuracy loss to 5.2%.                         function and use one value for each group [137]. Alternatively,
         All the previously described binary nets limit the weights the weights can be grouped by the k-means algorithm [118].
        to two values (-wandw); however, there may be benefits Both the shared weights and the indexes indicating which
        for allowing weights to be zero (i.e., -w, 0,w). Although weight to use at each position of the filter are stored. This
        this requires an additional bit per weight compared to binary leads to a two step process to fetch the weight: (1) read the
        weights, the sparsity of the weights can be exploited to reduce weight index; (2) using the weight index, read the shared
        computation and storage cost, which can potentially cancel weights. This approach can reduce the cost of reading and
        out the cost of the additional bit. This is explored in Ternary storing the weights if the weight index (log 2 of the number of
        Weight Nets (TWN) [131] and then extended in Trained Ternary unique weights) is less than the bitwidth of the weight itself.
        Quantization (TTQ) where a different scale is trained for each   For instance, in Deep Compression [118], the number of
        weight (i.e., -w                                 unique weights per layer is reduced to 256 for convolutional 1 , 0,w2 ) for an accuracy loss of 0.6% [132],
        assuming 32-bit floating point for the activations.          layers and 16 for fully-connected layers in AlexNet, requiring
         Hardware implementations for binary/ternary nets have 8-bit and 4-bit weight indexes, respectively. Assuming there
        been explored in recent publications. YodaNN [133] uses areUunique weights and the size of the filters in the layer
        binary weights, while BRein [134] uses binary weights and is <<FORMULA>> from Fig. 9(b), there will be energy savings
        activations. Binary weights are also used in the compute if reading from a CRSM <<log(2)>> U-bit memory plus aU16-
        in SRAM work [102] described in Section VI. Finally, the bit memory (as shown in Fig. 40) cost less than reading
        nominally spike-inspired TrueNorth chip can implement a from a CRSM 16-bit memory. Note that unlike the previous
        reduced precision neural network with binary activations and quantization methods, the weight sharing approach does not
        ternary weights using TrueNorth’s quantized weight table [9]. reduce the precision of the MAC computation itself and only
        These works tend not to support state-of-the-art DNN models reduces the weight storage requirement.
        (with the exception of YodaNN).
         2) Non-linear quantization:The previous works described B. Reduce Number of Operations and Model Size
        involve linear quantization where the levels are uniformly   In addition to reducing the size of each operation or operandspaced out. It has been shown that the distributions of the (weight/activation), there is also a significant amount of researchweights and activations are not uniform [118,135], and thus on methods to reduce the number of operations and modela non-linear quantization can potentially improve accuracy. size. These techniques can be loosely classified as exploitingSpecifically, there have been two popular approaches taken activation statistics, network pruning, network architecturein recent works: (1) log domain quantization; (2) learned design and knowledge distillation.quantization or weight sharing.                        1) Exploiting Activation Statistics: As discussed in Sec-Log domain quantizationIf the quantization levels are tionIII-A1, ReLU is a popular form of non-linearity used inassigned based on a logarithmic distribution as shown in DNNs that sets all negative values to zero as shown in Fig. 41(a). Fig 37(b), the weights and activations are more equally As a result, the output activations of the feature maps after the distributed across the different levels and each level is used ReLU are sparse; for instance, the feature maps in AlexNetmore efficiently resulting in less quantization error. For instance, have sparsity between 19% to 63% as shown in Fig. 41(b).using 4 bits in linear quantization results in a 27.8% loss in This sparsity gives ReLU an implementation advantage overaccuracy versus a 5% loss for log base-2 quantization for other non-linearities such as sigmoid, etc.

         9 This can also be thought of as a form of weights sharing, where only two   10 Note however that multiplications do not account for a significant portion
        weights are used per filter.                             of the total energy.                                                                                            

                                                     <<FORMULA>>      

                                                TABLE III
        METHODS TO REDUCE NUMERICAL PRECISION FOR ALEX NET . ACCURACY MEASURED FOR TOP-5 ERROR ON IMAGE NET . 



                                                      a cost of reduced accuracy.
                                                      2) Network Pruning:To make network training easier, the
                                                     networks are usually over-parameterized. Therefore, a large
                                                     amount of the weights in a network are redundant and can
                                                   be removed (i.e., set to zero). This process is called network
                                                   pruning. Aggressive network pruning often requires some fine-
                                                   tuning of the weights to maintain the original accuracy. This
                                                    was first proposed in 1989 through a technique called Optimal
                                                    Brain Damage [140]. The idea was to compute the impact of
                                                    each weight on the training loss (discussed in SectionII-C),
                                                    referred to as the weight saliency. The low-saliency weights (Normalized)                                
                                                    were removed and the remaining weights were fine-tuned; this
                                                    process was repeated until the desired weight reduction and
                                                    accuracy were reached.
                                                    In 2015, a similar idea was applied to modern DNNs in [141].
                    <<FORMULA>>                     Rather than using the saliency as a metric, which is too difficult
                                                   to compute for the large-scaled DNNs, the pruning was simply
        Fig. 41. Sparsity in activations due to ReLU.                  based on the magnitude of the weights. Small weights were
                                                   pruned and the model was fine-tuned to restore the accuracy.
                                                   Without fine-tuning the weights, about 50% of the weightsThe sparsity can be exploited for energy and area savings could be pruned. With fine-tuning, over 80% of the weightsusing compression, particularly for off-chip DRAM access were pruned. Overall this approach can reduce the numberwhich is expensive. For instance, a simple run length coding of weights in AlexNet by 9and the number of MACsthat involves signaling non-zero values of 16-bits and then runs by 3. Most of the weight reduction comes from the fully-of zeros up to 31 can reduce the external memory bandwidth connected layers (9.9for fully-connected layers versus 2.7of the activations by 2.1and the overall external bandwidth for convolutional layers).(including weights) by 1.5[61]. 11 In addition to compression,
        the hardware can also be modified such that it skips reading the   However, the number of weights alone is not a good metric
        weights and performing the MAC for zero-valued activations for energy. For instance, in AlexNet, the number of weights
        to reduce energy cost by 45% [94]. Rather than just gating the in the fully-connected layers is much larger than in the
        read and MAC computation, the hardware could also skip the convolutional layers; however, the energy of the convolutional
        cycle to increase the throughput by 1.37%[138].          layers is much higher than the fully-connected layers as shown
         The activations can be made to be even more sparse by prun- in Fig. 35 [80]. Rather than using the number of weights
        ing the low-valued activations. For instance, if all activations and MAC operations as proxies for energy, the pruning of
        with small values are pruned, this can be translated into an the weights can be directly driven by energy itself [142]. An
        additional 11% speed up [138] or 2power reduction [139] energy evaluation method can be used to estimate the DNN
        with little impact on accuracy. Aggressively pruning more energy that accounts for the data movement from different
        activations can provide additional throughput improvement at levels of the memory hierarchy, the number of MACs, and the
                                                   data sparsity as shown in Fig. 42; this energy estimation tool
                                                   is available at [143]. The resulting energy values for popular This simple run length compression is within 5-10% of the theoretical
        entropy limit.                                    DNN models are shown in Fig. 43(a). Energy-aware pruning                                                                                            24


                                <<FIGURE>> 

        Fig. 42. Energy estimation methodology from [142], which estimates the
        energy based on data movement from different levels of the memory hierarchy,           

            <<FIGURE>>
            
        Fig.  43. Energy values estimated with methodology in [142].          a time [144]. The CSC format will provide an overall lower
                                                   memory bandwidth than CSR if the output is smaller than the
                                                   input, or in the case of DNN, if the number of filters isnot
        can then be used to prune weights based on energy to reduce significantly larger than the number of weights in the filter
        the overall energy across all layers by 3.7% for AlexNet, which (<<FORMULA>> from Fig. 9(b)). Since this is often true, CSC can
        is 1.74more efficient than magnitude-based approaches [141] be an effective format for sparse DNN processing.
        as shown in Fig. 43(b). As mentioned previously, it is well   Custom hardware has been explored to efficiently supportknown that AlexNet is over-parameterized. The energy-aware pruned DNN models. Many works aim to perform the process-pruning can also be applied to GoogleNet, which is already a ing without decompressing the weights or activations. EIE [145]small DNN model, for a 1.6energy reduction.          performs the sparse matrix-vector multiplication specifically for
         Recent works have examine how to efficiently support the fully connected layers. It stores the weights in a CSC format
        processing of sparse weights in hardware. One area of interest along with the start location of each column, which needs to be
        is how to best store the sparse weights after pruning. Similar to stored since the compressed weights have variable length. When
        compressing the sparse activations discussed in SectionVII-B1, the input is not zero, the compressed weight column is read and
        the sparse weights can be compressed to reduce memory access the output is updated. To handle the sparsity, additional logic
        bandwidth by 20 to 30% [118].                      is used to keep track of the location of the output that should
         When DNN processing is performed as a matrix-vector be updated. SCNN [146] supports processing of convolutional                                                                                            25


        layers in a compressed format. It uses an input stationary weights [154]. It proposes afiremodule that first ‘squeezes’
        dataflow to deliver the compressed weights and activations to the network with 1x1 convolution filters and then expands
        a multiplier array followed by a scatter network to add the it with multiple 1x1 and 3x3 convolution filters. It achieves
        scattered partial sums.                            an overall 50% reduction in number of weights compared to
         Recent works have also explored the use of structured AlexNet, while maintaining the same accuracy. It should be
        pruning to avoid the need for custom hardware [147,148]. noted, however, that reducing the number of weights does not
        Rather than pruning individual weights (also referred to as fine- necessarily reduce energy; for instance, SqueezeNet consumes
        grained pruning), structured pruning involves pruning groups more energy than AlexNet, as shown in Fig. 43(a).
        of weights (also referred to as coarse-grained pruning). The     b) After Training:Tensor decomposition can be used to
        benefits of structured pruning are (1) the resulting weights can decompose filters in a trained network without impacting the
        better align with the data-parallel architecture (e.g., SIMD) accuracy. It treats weights in a layer as a 4-D tensor and breaks
        found in existing general purpose hardware, which results in it into a combination of smaller tensors (i.e., several layers).
        more efficient processing [149]; (2) it amortizes the overhead Low-rank approximation can then be applied to further increase
        cost required to signal the location of the non-zero weights the compression rate at the cost of accuracy degradation, which
        across a group of weights, which improves compression and can be restored by fine-tuning the weights.
        thus reduces storage cost. These groups of weights can include   This approach is demonstrated using Canonical Polyadic (CP)
        a pair of neighboring weights, an entire row or column of a decomposition, a high-order extension of singular value decom-
        filter, an entire channel of a filter or the entire filter itself; using position that can be solved by various methods, such as a greedy
        larger groups tends to result in higher loss in accuracy [150]. algorithm [155] or a non-linear least-square method [156].
         3) Compact Network Architectures:The number of weights Combining CP-decomposition with low-rank approximation
        and operations can also be reduced by improving the network achieves a 4.5% speed-up on CPUs [156]. However, CP-
        architecture itself. The trend is to replace a large filter with a decomposition cannot be computed in a numerically stable
        series of smaller filters, which have fewer weights in total; when way when the dimension of the tensor, which represents the
        the filters are applied sequentially, they achieve the same overall weights, is larger than two [156]. To alleviate this problem,
        effective receptive field (i.e., the region the filter uses from input Tucker decomposition is adopted instead in [157].
        image to compute an output). This approach can be applied   4) Knowledge Distillation:Using a deep network or av-
        during the network architecture design (before training) or by eraging the predictions of different models (i.e., ensemble)
        decomposing the filters of a trained network (after training). gives a better accuracy than using a single shallower network.
        The latter one avoids the hassle of training networks from However, the computational complexity is also higher. To get
        scratch. However, it is less flexible than the former one. For the best of both worlds, knowledge distillation transfers the
        example, existing methods can only decompose a filter in a knowledge learned by the complex model (teacher) to the
        trained network into a series of filters without non-linearity simpler model (student). The student network can therefore
        between them.                                 achieve an accuracy that would be unachievable if it was
           a) Before Training:In recent DNN models, filters with directly trained with the same dataset [158,159]. For example,
        a smaller width and height are used more frequently because [160] shows how using knowledge distillation can improve the
        concatenating several of them can emulate a larger filter as speech recognition accuracy of a student net by 2%, which is
        shown in Fig. 13. For example, one 5x5 convolution can be similar to the accuracy of a teacher net that is composed of
        replaced with two 3x3 convolutions. Alternatively, one NxN an ensemble of 10 networks.
        convolution can be decomposed into two 1-D convolutions, one   Fig. 45 shows the simplest knowledge distillation
        1xN and one Nx1 convolution [53]; this basically imposes method [158]. The softmax layer is commonly used as the
        a restriction that the 2-D filter must be separable, which is output layer in the image classification networks to generate
        a common constraint in image processing [151]. Similarly, a the class probabilities from the class scores 12 ; it squashes the
        3-D convolution can be replaced by a set of 2-D convolutions class scores into values between 0 and 1 that sum up to 1.
        (i.e., applied only on one of the input channels) followed by For this knowledge distillation method, soft targets (values
        1x1 3-D convolutions as demonstrated in Xception [152] and between 0 and 1) such as the class scores of the teacher DNN
        MobileNets [153]. The order of the 2-D convolutions and 1x1 (or an ensemble of teacher DNNs) are used instead of the
        3-D convolutions can be switched.                    hard targets (values of either 0 or 1) such as the labels in the
         1x1 convolutional layers can also be used to reduce the dataset. The objective is to minimize the squared difference
        number of channels in the output feature map for a given between the soft targets and the class scores of the student DNN.
        layer, which reduces the number of filter channels and thus Class scores are used as the soft targets instead of the class
        computation cost for the filters in the next layer as demonstrated probabilities because small values in the class scores contain
        in [15,51,52]; this is often referred to as a ‘bottleneck’ as important information that may be eliminated by the softmax.
        discussed in SectionIII-B. For this purpose, the number of 1x1 Alternatively, class probabilities after the softmax layer can be
        filters has to be less than the number of channels in the 1x1 used as soft targets if the softmax is configured to generate
        filter. For example, 32 filters of 1x164 can transform an input softer class probabilities where the smaller values retain more
        with 64 channels to an output of 32 channels and reduce the information [160]. Finally, the intermediate representations of
        number of filter channels in the next layer to 32. SqueezeNet
        uses many 1x1 filters to aggressively reduce the number of   12 Also commonly referred to as logits.                                                                                   


                                                           robotics. For data analytics, high throughput means that more
                                            data can be analyzed in a given amount of time. As the amount

                                                      of visual data is growing exponentially, high-throughput big
                                                     data analytics becomes important, particularly if an action needs 
                                                     to be taken based on the analysis (e.g., security or terrorist
                                                   prevention; medical diagnosis). Try to match                   
                                                   Low latencyis necessary for real-time interactive applications.
                                                   Latency measures the time between when the pixel arrives
                                                      to a system and when the result is generated. Latency is Simple DNN 
                                                      measured in terms of seconds, while throughput is measured
                                                   in operations/second. Often high throughput is obtained by
                                                   batching multiple images/frames together for processing; this Fig. 45. 
                                                   Knowledge distillation matches the class scores of a small DNN to results 
                                                   in multiple frame latency (e.g., at 30 frames per second, an ensemble of large DNNs.                            
                                                   a batch of 100 frames results in a 3 second delay). This delay
                                                   is not acceptable for real-time applications, such as high-speed
                                                   navigation where it would reduce the time available for coursethe teacher DNN can 
                                                   also be incorporated as the extra hints correction. Thus achieving low latency and
                                                    high throughputto train the student DNN [161].                      
                                                     Hardware costis in large part dictated by the amount of 
                                                     on-chip storage and the number of cores. Typical embedded 
                                                    processors have limited on-chip storage on the order of a few
                                                    simultaneously can be a challenge.

                                                   VIII. B ENCHMARKING METRICS FOR DNN EVALUATION  AND COMPARISON                
         As we have seen in this article, there has been a significant hundred kilobytes. Since there is a trade-off between the amount
        amount of research on efficient processing of DNNs. We should of on-chip memory and the external memory bandwidth, both
        consider several key metrics to compare the various strengths metrics should be reported. Similarly, there is a correlation
        and weaknesses of different designs and proposed techniques. between the number of cores and the throughput. In addition,
        These metrics should cover important attributes such as accu- while many cores can be built on a chip, the number of cores
        racy/robustness, power/energy consumption, throughput/latency that can actually be used at a given time should be reported. It is
        and cost. Reporting all these metrics is important in order often unrealistic to assume peak utilization and performance due
        to provide a complete picture of the trade-offs made by a to limitations of mapping and memory bandwidth. Accordingly,
        proposed design or technique. We have prepared a website to the power and throughput should be reported for running actual
        collect these metrics from various publications [162].       DNNs as opposed to only reporting theoretical limits.
         In terms ofaccuracyandrobustness, it is important that the
        accuracy be reported on widely-accepted datasets as discussed
        in Section IV. The difficulty of the dataset and/or task should A. Metrics for DNN Models
        be considered when measuring the accuracy. For instance, the   To evaluate the properties of a given DNN model, we should
        MNIST dataset for digit recognition is significantly easier than consider the following metrics:the ImageNet dataset. 
        As a result, a DNN that performs well
        on MNIST may not necessarily perform well on ImageNet.    Theaccuracy of the model in terms of the top-5 error
        Thus it is important that the same dataset and task is used when     on datasets such as ImageNet. Also, the type of data
        comparing the accuracy of different DNN models; currently     augmentation used (e.g., multiple crops, ensemble models)
        ImageNet is preferred since it presents a challenge for DNNs,     should be reported.
        as opposed to MNIST, which can also be addressed with simple    Thenetwork architectureof the model should be reported,
        non-DNN techniques. To demonstrate primarily hardware     including number of layers, filter sizes, number of filters
        innovations, it would be desirable to report results for widely-     and number of channels.
        used DNN models (e.g., AlexNet, GoogLeNet) whose accuracy    Thenumber of weightsimpact the storage requirement of
        and robustness have been well studied and tested.             the model and should be reported. If possible, the number
         Energyandpowerare important when processing DNNs at     of non-zero weights should be reported since this reflects
        the edge in embedded devices with limited battery capacity     the theoretical minimum storage requirements.
        (e.g., smart phones, smart sensors, UAVs, and wearables), or in    Thenumber of MACsthat needs to be performed should
        the cloud in data centers with stringent power ceilings due to     be reported as it is somewhat indicative of the number
        cooling costs, respectively. Edge processing is preferred over     of operations and potential throughput of the given DNN.
        the cloud for certain applications due to latency, privacy or     If possible, the number of non-zero MACs should also
        communication bandwidth limitations. When evaluating the     be reported since this reflects the theoretical minimum
        power and energy consumption, it is important to account     compute requirements.
        for all aspects of the system including the chip and external   Table IV shows how these metrics are reported for various
        memory accesses.                               well known DNNs. The accuracy is reported for the case where
         High throughputis necessary to deliver real-time perfor- only a single crop for a single model is used for classification,
        mance for interactive applications such as navigation and such that the number of weights and MACs in the table are                                                                                           
                                                           reported in terms of the core area in squared millimeters 
                                                           per multiplier along with process technology.
                                                           In terms of cost, different platforms will have different
                                                            implementation-specific metrics. For instance, for an FPGA, (Number of CONV Layers)
                                                             the specific device should be reported, along with the utilization
                                                          of resources such as DSP, BRAM, LUT and FF; performance
                                                            density such as GOPs/slice can also be reported. Stride                   
                                                            Each processor should report various specifications for each NZ Weights      
                                                            metric as shown in Table V, using the Eyeriss chip as an
                                                         example. It is important that all metrics and specifications are
                                                           accounted for in order fairly evaluate all the design trade-offs. Number of Channels            
                                                         For instance, without the accuracy given for a specific dataset Number of Filters       
                                                            and task, one could run a simple DNN and easily claim low
                                                            power, high throughput, and low cost – however, the processor
                                                         might not be usable for a meaningful task; alternatively, without Total NZ MACs         
                                                          reporting the off-chip bandwidth, one could build a processor 
                                                          with only multipliers and easily claim low cost, high throughput,
                                                        high accuracy, and lowchippower – however, when evaluating
                                                   systempower, the off-chip memory access would be substantial.
                                                   Finally, the test setup should also be reported, including whether
                                                   the results are measured or obtained from simulation and consistent. 
                                                    (NZ) operations significantly reduces the number of MACs   
                                                    In summary, the evaluation process for whether a DNNand weights. 
                                                    Since the number of NZ MACs depends on the system is a viable solution 
                                                    for a given application might go asinput data, we propose using the publicly available 50,000 follows:
                                                   (1) the accuracy determines if it can perform the givenvalidation images from ImageNet for the 
                                                   computation. Finally, task; (2) the latency and throughput determine if it can run fast there are 
                                                   various methods to reduce the weights in a DNN enough and in real-time; (3) the energy and power consumption
                                                   (e.g., network pruning in SectionVII-B2). Table IV shows will primarily dictate the form factor of the device 
                                                   where the another example of these DNN model metrics, by comparing processing can operate; (4) the cost, 
                                                   which is primarily dictatedsparse DNNs pruned using [142] to dense DNNs.         
                                                   by the chip area, determines how much one would pay for this
                                                   solution.
                            <<TABLE>>
                            TABLE IV
        METRICS FOR POPULAR DNN MODELS. SPARSITY IS ACCOUNT FOR BY  
              REPORTING NON-ZERO (NZ) WEIGHTS AND MACS.

        B. Metrics for DNN Hardware
         To measure the efficiency of the DNN hardware, we should                 IX. SUMMARY
        consider the following additional metrics:                 The use of deep neural networks (DNNs) has seen explosive
          Thepower and energyconsumption of the design should growth in the past few years. They are currently widely used
           be reported for various DNN models; the DNN model for many artificial intelligence (AI) applications including
           specifications should be provided including which layers computer vision, speech recognition and robotics and are often
           and bit precision are supported by the hardware during delivering better than human accuracy. However, while DNNs
           measurement. In addition, the amount of off-chip accesses can deliver this outstanding accuracy, it comes at the cost
           (e.g., DRAM accesses) should be included since it of high computational complexity. Consequently, techniques
           accounts for a significant portion of the system power; it that enable efficient processing of deep neural network to
           can be reported in terms of the total amount of data that improveenergy-efficiencyandthroughputwithout sacrificing
           is read and written off-chip per inference.           accuracywith cost-effective hardware are critical to expanding
          Thelatency and throughputshould be reported in terms the deployment of DNNs in both existing and new domains.
           of the batch size and the actual run time for various   Creating a system for efficient DNN processing should
           DNN models, which accounts for mapping and memory begin with understanding the current and future applications
           bandwidth effects. This provides a more useful and and the specific computations required both now and the
           informative metric than peak throughput.            potential evolution of those computations. This article surveys a
          Thecostof the chip depends on the area efficiency, which number of the current applications, focusing on computer vision
           accounts for the size and type of memory (e.g., registers applications, the associated algorithms, and the data being used
           or SRAM) and the amount of control logic. It should be to drive the algorithms. These applications, algorithms and
                                                   input data are experiencing rapid change. So extrapolating
         13 Data augmentation is often used to increase accuracy. This includes using these trends to determine the degree of flexibility desired to
        multiple crops of an image to account for misalignment; in addition, an handle next generation computations, becomes an important ensemble of multiple models can be used where each model has different
        weights due to different training settings, such as using different initializations ingredient of any design project.
        or datasets, or even different network architectures. If multiple crops and
        models are used, then the number of MACs and weights required would   
                                               
                                               <<TABLE>>
                                                TABLE V
                                    EXAMPLE BENCHMARK METRICS FOR EYERISS [94].

         During the design-space exploration process, it is critical to article both reviews a variety of these techniques and discusses
        understand and balance the important system metrics. For DNN the frameworks that are available for describing, running and
        computation these include the accuracy, energy, throughput training networks.
        and hardware cost. Evaluating these metrics is, of course,   Finally, DNNs afford the opportunity to use mixed-signal
        key, so this article surveys the important components of circuit design and advanced technologies to improve efficiency.
        a DNN workload. In specific, a DNN workload has two These include using memristors for analog computation and 3-D
        major components. First, the workload is the form of each stacked memory. Advanced technologies can also can facilitate
        DNN network including the ‘shape’ of each layer and the moving computation closer to the source by embedding compu-
        interconnections between layers. These can vary both within tation near or within the sensor and the memories. Of course, all
        and between applications. Second, the workload consists of of these techniques should also be considered in combination,
        the specific the data input to the DNN. This data will vary while being careful to understand their interactions and looking
        with the input set used for training or the data input during for opportunities for joint hardware/algorithm co-optimization.
        operation for inference.                             In conclusion, although much work has been done, deep
         This article also surveys a number of avenues that prior neural networks remain an important area of research with
        work have taken to optimize DNN processing. Since data many promising applications and opportunities for innovation
        movement dominates energy consumption, a primary focus at various levels of hardware design.
        of some recent research has been to reduce data movement
        while maintaining accuracy, throughput and cost. This means               ACKNOWLEDGMENTS
        selecting architectures with favorable memory hierarchies like   Funding provided by DARPA YFA, MIT CICS, and gifts
        a spatial array, and developing dataflows that increase data from Nvidia and Intel. The authors thank the anonymous
        reuse at the low-cost levels of the memory hierarchy. We reviewers as well as James Noraky, Mehul Tikekar and
        have included a taxonomy of dataflows and an analysis of Zhengdong Zhang for providing valuable feedback on this
        their characteristics. Other work is presented that aims to save paper.
        space and energy by changing the representation of data values
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<<END> <<END>> <<END>>


<<START>> <<START>> <<START>>
EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks 

Abstract 

Convolutional Neural Networks (ConvNets) are commonly developed at a fixed resource budget, and then scaled up for better accuracy if more resources are available. In this paper, we systematically study model scaling and identify that carefully balancing network depth, width, and resolution can lead to better performance. Based on this observation, we propose a new scaling method that uniformly scales all dimensions of depth/width/resolution using a simple yet highly effective compound coefficient. We demonstrate the effectiveness of this method on scaling up MobileNets and ResNet. 
To go even further, we use neural architecture search to design a new baseline network and scale it up to obtain a family of models, called EfficientNets, which achieve much better accuracy and efficiency than previous ConvNets. In particular, our EfficientNet-B7 achieves state-of-the-art 84.4% top-1 / 97.1% top-5 accuracy on ImageNet, while being 8.4x smaller and 6.1x faster on inference than the best existing ConvNet. Our EfficientNets also transfer well and achieve state-of-the-art accuracy on CIFAR-100 (91.7%), Flowers (98.8%), and 3 other transfer learning datasets, with an order of magnitude fewer parameters. Source code is at https: //github.com/tensorflow/tpu/tree/master/models/official/efficientnet. 

1. Introduction

Scaling up ConvNets is widely used to achieve better accuracy. For example, ResNet (He et al., 2016) can be scaled up from ResNet-18 to ResNet-200 by using more layers; Recently, GPipe (Huang et al., 2018) achieved 84.3% Ima.
geNet top-1 accuracy by scaling up a baseline model four 

<<FIGURE>>

Figure 1. Model Size vs. ImageNet Accuracy. All numbers are for single-crop, single-model. Our EfficientNets significantly out.perform other ConvNets. In particular, EfficientNet-B7 achieves new state-of-the-art 84.4% top-1 accuracy but being 8.4x smaller and 6.1x faster than GPipe. EfficientNet-B1 is 7.6x smaller and 5.7x faster than ResNet-152. Details are in Table 2 and 4. 
time larger. However, the process of scaling up ConvNets has never been well understood and there are currently many ways to do it. The most common way is to scale up Con.vNets by their depth (He et al., 2016) or width (Zagoruyko & Komodakis, 2016). Another less common, but increasingly popular, method is to scale up models by image resolution (Huang et al., 2018). In previous work, it is common to scale only one of the three dimensions <20> depth, width, and image size. Though it is possible to scale two or three dimensions arbitrarily, arbitrary scaling requires tedious manual tuning and still often yields sub-optimal accuracy and efficiency. 
In this paper, we want to study and rethink the process of scaling up ConvNets. In particular, we investigate the central question: is there a principled method to scale up ConvNets that can achieve better accuracy and efficiency? Our empirical study shows that it is critical to balance all dimensions of network width/depth/resolution, and surpris.ingly such balance can be achieved by simply scaling each of them with constant ratio. Based on this observation, we propose a simple yet effective compound scaling method. Unlike conventional practice that arbitrary scales these fac.tors, our method uniformly scales network width, depth, 

<<FIGURE>>

Figure 2. Model Scaling. (a) is a baseline network example; (b)-(d) are conventional scaling that only increases one dimension of network width, depth, or resolution. (e) is our proposed compound scaling method that uniformly scales all three dimensions with a fixed ratio. 
and resolution with a set of fixed scaling coefficients. For example, if we want to use 2N times more computational resources, then we can simply increase the network depth by <<FORMULA>>, width by <<FORMULA>> , and image size by <<FORMULA>> are constant coefficients determined by a small grid search on the original small model. Figure 2 illustrates the difference between our scaling method and conventional methods. 
Intuitively, the compound scaling method makes sense be.cause if the input image is bigger, then the network needs more layers to increase the receptive field and more channels to capture more fine-grained patterns on the bigger image. In fact, previous theoretical (Raghu et al., 2017; Lu et al., 2018) and empirical results (Zagoruyko & Komodakis, 2016) both show that there exists certain relationship between network width and depth, but to our best knowledge, we are the first to empirically quantify the relationship among all three dimensions of network width, depth, and resolution. 
We demonstrate that our scaling method work well on exist.ing MobileNets (Howard et al., 2017; Sandler et al., 2018) and ResNet (He et al., 2016). Notably, the effectiveness of model scaling heavily depends on the baseline network; to go even further, we use neural architecture search (Zoph & Le, 2017; Tan et al., 2019) to develop a new baseline network, and scale it up to obtain a family of models, called EfficientNets. Figure 1 summarizes the ImageNet performance, where our EfficientNets significantly outperform other ConvNets. In particular, our EfficientNet-B7 surpasses the best existing GPipe accuracy (Huang et al., 2018), but using 8.4x fewer parameters and running 6.1x faster on inference. Compared to the widely used ResNet-50 (He et al., 2016), our EfficientNet-B4 improves the top-1 accuracy from 76.3% to 83.0% (+6.7%) with similar FLOPS. Besides ImageNet, EfficientNets also transfer well and achieve state-of-the-art accuracy on 5 out of 8 widely used datasets, while reducing parameters by up to 21x than existing ConvNets. 

2. Related Work 

ConvNet Accuracy: Since AlexNet (Krizhevsky et al., 2012) won the 2012 ImageNet competition, ConvNets have become increasingly more accurate by going bigger: while the 2014 ImageNet winner GoogleNet (Szegedy et al., 2015) achieves 74.8% top-1 accuracy with about 6.8M parameters, the 2017 ImageNet winner SENet (Hu et al., 2018) achieves 82.7% top-1 accuracy with 145M parameters. Recently, GPipe (Huang et al., 2018) further pushes the state-of-the-art ImageNet top-1 validation accuracy to 84.3% using 557M parameters: it is so big that it can only be trained with a specialized pipeline parallelism library by partitioning the network and spreading each part to a different accelerator. While these models are mainly designed for ImageNet, recent studies have shown better ImageNet models also per.form better across a variety of transfer learning datasets (Kornblith et al., 2019), and other computer vision tasks such as object detection (He et al., 2016; Tan et al., 2019). Although higher accuracy is critical for many applications, we have already hit the hardware memory limit, and thus further accuracy gain needs better efficiency. 
ConvNet efficiency: Deep ConvNets are often over-parameterized. Model compression (Han et al., 2016; He et al., 2018; Yang et al., 2018) is a common way to re.duce model size by trading accuracy for efficiency. As mo.bile phones become ubiquitous, it is also common to hand.craft efficient mobile-size ConvNets, such as SqueezeNets (Iandola et al., 2016; Gholami et al., 2018), MobileNets (Howard et al., 2017; Sandler et al., 2018), and ShuffleNets (Zhang et al., 2018; Ma et al., 2018). Recently, neural architecture search becomes increasingly popular in designing efficient mobile-size ConvNets (Tan et al., 2019; Cai et al., 2019), and achieves even better efficiency than hand-crafted mobile ConvNets by extensively tuning the network width, depth, convolution kernel types and sizes. However, it is unclear how to apply these techniques for larger models that have much larger design space and much more expensive tuning cost. In this paper, we aim to study model efficiency for super large ConvNets that surpass state-of-the-art accuracy. To achieve this goal, we resort to model scaling. 

Model Scaling: There are many ways to scale a Con.vNet for different resource constraints: ResNet (He et al., 2016) can be scaled down (e.g., ResNet-18) or up (e.g., ResNet-200) by adjusting network depth (#layers), while WideResNet (Zagoruyko & Komodakis, 2016) and Mo.bileNets (Howard et al., 2017) can be scaled by network width (#channels). It is also well-recognized that bigger input image size will help accuracy with the overhead of more FLOPS. Although prior studies (Raghu et al., 2017; Lin & Jegelka, 2018; Sharir & Shashua, 2018; Lu et al., 2018) have shown that network depth and width are both important for ConvNets<74> expressive power, it still remains an open question of how to effectively scale a ConvNet to achieve better efficiency and accuracy. Our work systematically and empirically studies ConvNet scaling for all three dimensions of network width, depth, and resolutions. 

3. Compound Model Scaling

In this section, we will formulate the scaling problem, study different approaches, and propose our new scaling method. 

3.1. Problem Formulation 
A ConvNet Layer i can be defined as a function: <<FORMULA>>, where Fi is the operator, Yi is output tensor, Xi is input tensor, with tensor shape <<FORMULA>>, where H_i and W_i are spatial dimension and C_i is the channel dimension. A ConvNet N can be represented by a list of composed lay-

<<FORMULA>>

practice, ConvNet layers are often partitioned into multiple stages and all layers in each stage share the same architecture: for example, ResNet (He et al., 2016) has <20>ve stages, and all layers in each stage has the same convolutional type except the first layer performs down-sampling. Therefore, we can define a ConvNet as: 

<<FORMULA>>

<<FORMULA>> where <<FORMULA>> denotes layer F_i is repeated L_i times in stage i,

<<FORMULA>> denotes the shape of input tensor X of layer 1For the sake of simplicity, we omit batch dimension. 
i. Figure 2(a) illustrate a representative ConvNet, where the spatial dimension is gradually shrunk but the channel dimension is expanded over layers, for example, from initial input shape h224, 224, 3i to final output shape h7, 7, 512i. 
Unlike regular ConvNet designs that mostly focus on find.ing the best layer architecture Fi, model scaling tries to expand the network length (Li), width (Ci), and/or resolution (Hi,Wi) without changing Fi predefined in the baseline network. By <20>xing Fi, model scaling simplifies the design problem for new resource constraints, but it still remains a large design space to explore different <<FORMULA>> for each layer. In order to further reduce the design space, we restrict that all layers must be scaled uniformly with constant ratio. Our target is to maximize the model accuracy for any given resource constraints, which can be formulated as an optimization problem: 

<<FORMULA>>         (2) 

where <<FORMULA>> are coefficients for scaling network width, depth, and resolution; <<FORMULA>> are predefined parameters in baseline network (see Table 1 as an example). 
3.2. Scaling Dimensions 
The main difficulty of problem 2 is that the optimal d, w, r depend on each other and the values change under different resource constraints. Due to this difficulty, conventional methods mostly scale ConvNets in one of these dimensions: 
Depth (d): Scaling network depth is the most common way used by many ConvNets (He et al., 2016; Huang et al., 2017; Szegedy et al., 2015; 2016). The intuition is that deeper ConvNet can capture richer and more complex features, and generalize well on new tasks. However, deeper networks are also more difficult to train due to the vanishing gradient problem (Zagoruyko & Komodakis, 2016). Although several techniques, such as skip connections (He et al., 2016) and batch normalization (Ioffe & Szegedy, 2015), alleviate the training problem, the accuracy gain of very deep network diminishes: for example, ResNet-1000 has similar accuracy as ResNet-101 even though it has much more layers. Figure 3 (middle) shows our empirical study on scaling a baseline model with different depth coefficient d, further suggesting the diminishing accuracy return for very deep ConvNets. 
Width (w): Scaling network width is commonly used for small size models (Howard et al., 2017; Sandler et al., 2018; 

<<FIGURE>>

Figure 3. Scaling Up a Baseline Model with Different Network Width (w), Depth (d), and Resolution (r) coefficients. Bigger networks with larger width, depth, or resolution tend to achieve higher accuracy, but the accuracy gain quickly saturate after reaching 80%, demonstrating the limitation of single dimension scaling. Baseline network is described in Table 1. 
Tan et al., 2019)2. As discussed in (Zagoruyko & Komodakis, 2016), wider networks tend to be able to capture more fine-grained features and are easier to train. However, extremely wide but shallow networks tend to have dif<69>cul.ties in capturing higher level features. Our empirical results in Figure 3 (left) show that the accuracy quickly saturates when networks become much wider with larger w. 
Resolution (r): With higher resolution input images, Con.vNets can potentially capture more fine-grained patterns. Starting from 224x224 in early ConvNets, modern Con.vNets tend to use 299x299 (Szegedy et al., 2016) or 331x331 (Zoph et al., 2018) for better accuracy. Recently, GPipe (Huang et al., 2018) achieves state-of-the-art ImageNet ac.curacy with 480x480 resolution. Higher resolutions, such as 600x600, are also widely used in object detection ConvNets (He et al., 2017; Lin et al., 2017). Figure 3 (right) shows the results of scaling network resolutions, where indeed higher resolutions improve accuracy, but the accuracy gain dimin.ishes for very high resolutions (r =1.0 denotes resolution 224x224 and r =2.5 denotes resolution 560x560). 
The above analyses lead us to the first observation: 
Observation 1 <20> Scaling up any dimension of network width, depth, or resolution improves accuracy, but the accuracy gain diminishes for bigger models. 
3.3. Compound Scaling 
We empirically observe that different scaling dimensions are not independent. Intuitively, for higher resolution images, we should increase network depth, such that the larger receptive fields can help capture similar features that include more pixels in bigger images. Correspondingly, we should also increase network width when resolution is higher, in 
In some literature, scaling number of channels is called depth multiplier, which means the same as our width coefficient w. 

<<FIGURE>>

Figure 4. Scaling Network Width for Different Baseline Net.works. Each dot in a line denotes a model with different width coefficient (w). All baseline networks are from Table 1. The first baseline network <<FORMULA>> has 18 convolutional layers with resolution 224x224, while the last baseline <<FORMULA>> has 36 layers with resolution 299x299. 
order to capture more fine-grained patterns with more pixels in high resolution images. These intuitions suggest that we need to coordinate and balance different scaling dimensions rather than conventional single-dimension scaling. 
To validate our intuitions, we compare width scaling under different network depths and resolutions, as shown in Figure 4. If we only scale network width w without changing depth <<(d=1.0)>> and resolution <<(r=1.0)>>, the accuracy saturates quickly. With deeper (d=2.0) and higher resolution <<(r=2.0)>>, width scaling achieves much better accuracy under the same FLOPS cost. These results lead us to the second observation: 
Observation 2 In order to pursue better accuracy and efficiency, it is critical to balance all dimensions of network width, depth, and resolution during ConvNet scaling. 

In fact, a few prior work (Zoph et al., 2018; Real et al., 2019) have already tried to arbitrarily balance network width and depth, but they all require tedious manual tuning. 
In this paper, we propose a new compound scaling method, which use a compound coefficient . to uniformly scales network width, depth, and resolution in a principled way: 

<<FORMULA>>      (3) 

where <<FORMULA>> are constants that can be determined by a small grid search. Intuitively, . is a user-specified coefficient that controls how many more resources are available for model scaling, while <<FORMULA>> specify how to assign these extra resources to network width, depth, and resolution respectively. Notably, the FLOPS of a regular convolution op
is proportional to <<FORMULA>> i.e., doubling network depth will double FLOPS, but doubling network width or resolution will increase FLOPS by four times. Since convolution ops usually dominate the computation cost in ConvNets, scaling a ConvNet with equation 3 will approximately in.
crease total FLOPS by <<FORMULA>> In this paper, we constraint <<FORMULA>> such that for any new <<FORMULA>>, the total FLOPS will approximately3 increase by 2.
4. EfficientNet Architecture 
Since model scaling does not change layer operators F_i in baseline network, having a good baseline network is also critical. We will evaluate our scaling method using existing ConvNets, but in order to better demonstrate the effectiveness of our scaling method, we have also developed a new mobile-size baseline, called EfficientNet. 
Inspired by (Tan et al., 2019), we develop our baseline net.work by leveraging a multi-objective neural architecture search that optimizes both accuracy and FLOPS. Specifically, we use the same search space as (Tan et al., 2019), and use <<FORMULA>> as the optimization goal, where <<ACC(m)>> and <<FLOPS(m)>> denote the accuracy and FLOPS of model m, T is the target FLOPS and w=-0.07 is a hyperparameter for controlling the trade-off between accuracy and FLOPS. Unlike (Tan et al., 2019; Cai et al., 2019), here we optimize FLOPS rather than latency since we are not targeting any specific hardware de.vice. Our search produces an efficient network, which we name EfficientNet-B0. Since we use the same search space as (Tan et al., 2019), the architecture is similar to <<FORMULA>>.
FLOPS may differ from theoretical value due to rounding. 

Table 1. EfficientNet-B0 baseline network <<FORMULA>> Each row describes a stage i with L_i layers, with input resolution <<FORMULA>> and output channels C_i. Notations are adopted from equation 2. 

<<FORMULA>>

Net, except our EfficientNet-B0 is slightly bigger due to the larger FLOPS target (our FLOPS target is 400M). Ta.ble 1 shows the architecture of EfficientNet-B0. Its main building block is mobile inverted bottleneck MBConv (Sandler et al., 2018; Tan et al., 2019), to which we also add squeeze-and-excitation optimization (Hu et al., 2018). 
Starting from the baseline EfficientNet-B0, we apply our compound scaling method to scale it up with two steps: 

STEP 1: we first <<FORMULA>> assuming twice more re.sources available, and do a small grid search of <<FORMULA>> based on Equation 2 and 3. In particular, we find the best values for EfficientNet-B0 are <<FORMULA>>, under constraint of <<FORMULA>>. 

STEP 2: we then <<FORMULA>> as constants and scale up baseline network with different . using Equation 3, to obtain EfficientNet-B1 to B7 (Details in Table 2). 

Notably, it is possible to achieve even better performance by searching for <<FORMULA>> directly around a large model, but the search cost becomes prohibitively more expensive on larger models. Our method solves this issue by only doing search once on the small baseline network (step 1), and then use the same scaling coefficients for all other models (step 2). 

5. Experiments 

In this section, we will first evaluate our scaling method on existing ConvNets and the new proposed EfficientNets. 

5.1. Scaling Up MobileNets and ResNets 
As a proof of concept, we first apply our scaling method to the widely-used MobileNets (Howard et al., 2017; Sandler et al., 2018) and ResNet (He et al., 2016). Table 3 shows the ImageNet results of scaling them in different ways. Compared to other single-dimension scaling methods, our compound scaling method improves the accuracy on all these models, suggesting the effectiveness of our proposed scaling method for general existing ConvNets. 

Table 2. EfficientNet Performance Results on ImageNet (Russakovsky et al., 2015). All EfficientNet models are scaled from our baseline EfficientNet-B0 using different compound coefficient . in Equation 3. ConvNets with similar top-1/top-5 accuracy are grouped together for efficiency comparison. Our scaled EfficientNet models consistently reduce parameters and FLOPS by an order of magnitude (up to 8.4x parameter reduction and up to 16x FLOPS reduction) than existing ConvNets. 

<<TABLE>>

We omit ensemble and multi-crop models (Hu et al., 2018), or models pretrained on 3.5B Instagram images (Mahajan et al., 2018). 

Table 3. Scaling Up MobileNets and ResNet. 

<<TABLE>>

Table 5. EfficientNet Performance Results on Transfer Learning Datasets. Our scaled EfficientNet models achieve new state-of-the.art accuracy for 5 out of 8 datasets, with 9.6x fewer parameters on average. 
Comparison to best public-available results Comparison to best reported results Model Accuracy. 

<<TABLE>>

Figure 6. Model Parameters vs. Transfer Learning Accuracy weight decay 1e-5; initial learning rate 0.256 that decays by 0.97 every 2.4 epochs. 

<<FIGURE>>

We also use swish activation (Ramachandran et al., 2018; Elfwing et al., 2018), fixed Au.to Augment policy (Cubuk et al., 2019), and stochastic depth (Huang et al., 2016) with survival probability 0.8. As commonly known that bigger models need more regularization, we linearly increase dropout (Srivastava et al., 2014) ratio from 0.2 for EfficientNet-B0 to 0.5 for EfficientNet-B7. 
Table 2 shows the performance of all EfficientNet models that are scaled from the same baseline EfficientNet-B0. Our EfficientNet models generally use an order of magnitude fewer parameters and FLOPS than other ConvNets with similar accuracy. In particular, our EfficientNet-B7 achieves 84.4% top1 / 97.1% top-5 accuracy with 66M parameters and 37B FLOPS, being more accurate but 8.4x smaller than the previous best GPipe (Huang et al., 2018). 
All models are pretrained on ImageNet and fine tuned on new datasets. Figure 1 and Figure 5 illustrates the parameters-accuracy and FLOPS-accuracy curve for representative ConvNets, where our scaled EfficientNet models achieve better accuracy with much fewer parameters and FLOPS than other ConvNets. Notably, our EfficientNet models are not only small, but also computational cheaper. For example, our EfficientNet-B3 achieves higher accuracy than ResNeXt.101 (Xie et al., 2017) using 18x fewer FLOPS. 
To validate the computational cost, we have also measured the inference latency for a few representative CovNets on a real CPU as shown in Table 4, where we report average latency over 20 runs. Our EfficientNet-B1 runs 5.7x faster than the widely used ResNet-152 (He et al., 2016), while EfficientNet-B7 runs about 6.1x faster than GPipe (Huang et al., 2018), suggesting our EfficientNets are indeed fast on real hardware. 

Figure 7. Class Activation Map (CAM) (Zhou et al., 2016) for Models with different scaling methods-Our compound scaling method allows the scaled model (last column) to focus on more relevant regions with more object details. Model details are in Table 7. 

<<FIGURE>>

Table 6. Transfer Learning Datasets. 
  
<<TABLE>>

5.3. Transfer Learning Results for EfficientNet 
We have also evaluated our EfficientNet on a list of commonly used transfer learning datasets, as shown in Table 6. We borrow the same training settings from (Kornblith et al., 2019) and (Huang et al., 2018), which take ImageNet pretrained checkpoints and fine tune on new datasets. 
Table 5 shows the transfer learning performance: (1) Com.pared to public available models, such as NASNet-A (Zoph et al., 2018) and Inception-v4 (Szegedy et al., 2017), our EfficientNet models achieve better accuracy with 4.7x average (up to 21x) parameter reduction. (2) Compared to state-of-the-art models, including DAT (Ngiam et al., 2018) that dynamically synthesizes training data and GPipe (Huang et al., 2018) that is trained with specialized pipeline parallelism, our EfficientNet models still surpass their accuracy in 5 out of 8 datasets, but using 9.6x fewer parameters 
Figure 6 compares the accuracy-parameters curve for a variety of models. In general, our EfficientNets consistently achieve better accuracy with an order of magnitude fewer parameters than existing models, including ResNet (He et al., 2016), DenseNet (Huang et al., 2017), Inception (Szegedy et al., 2017), and NASNet (Zoph et al., 2018). 

6. Discussion 

Figure 8. Scaling Up EfficientNet-B0 with Different Methods. Table 7. Scaled Models Used in Figure 7. 

<<FIGURE>>

To disentangle the contribution of our proposed scaling method from the EfficientNet architecture, Figure 8 com.pares the ImageNet performance of different scaling methods for the same EfficientNet-B0 baseline network. In general, all scaling methods improve accuracy with the cost of more FLOPS, but our compound scaling method can further improve accuracy, by up to 2.5%, than other single-dimension scaling methods, suggesting the importance of our proposed compound scaling. 
In order to further understand why our compound scaling method is better than others, Figure 7 compares the class activation map (Zhou et al., 2016) for a few representative models with different scaling methods. All these models are scaled from the same baseline, and their statistics are shown in Table 7. Images are randomly picked from ImageNet validation set. As shown in the figure, the model with com.pound scaling tends to focus on more relevant regions with more object details, while other models are either lack of object details or unable to capture all objects in the images. 

7. Conclusion 

In this paper, we systematically study ConvNet scaling and identify that carefully balancing network width, depth, and resolution is an important but missing piece, preventing us from better accuracy and efficiency. To address this issue, we propose a simple and highly effective compound scaling method, which enables us to easily scale up a baseline Con.vNet to any target resource constraints in a more principled way, while maintaining model efficiency. Powered by this compound scaling method, we demonstrate that a mobile-size EfficientNet model can be scaled up very effectively, surpassing state-of-the-art accuracy with an order of magnitude fewer parameters and FLOPS, on both ImageNet and five commonly used transfer learning datasets. 

Acknowledgements 

We thank Ruoming Pang, Vijay Vasudevan, Alok Aggarwal, Barret Zoph, Hongkun Yu, Xiaodan Song, Samy Bengio, Jeff Dean, and Google Brain team for their help. 

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<<END>> <<END>> <<END>>


<<START>> <<START>> <<START>>
Energy and Policy Considerations for Deep Learning in NLP 

Emma Strubell Ananya Ganesh Andrew McCallum College of Information and Computer Sciences University of Massachusetts Amherst 
{strubell, aganesh, 

Abstract 

Recent progress in hardware and methodology for training neural networks has ushered in a new generation of large networks trained on abundant data. These models have obtained notable gains in accuracy across many NLP tasks. However, these accuracy improvements depend on the availability of exception.ally large computational resources that necessitate similarly substantial energy consumption. As a result these models are costly to train and develop, both financially, due to the cost of hardware and electricity or cloud compute time, and environmentally, due to the car.bon footprint required to fuel modern tensor processing hardware. In this paper we bring this issue to the attention of NLP researchers by quantifying the approximate financial and environmental costs of training a variety of recently successful neural network models for NLP. Based on these findings, we propose actionable recommendations to reduce costs and improve equity in NLP research and practice. 

1 Introduction 

Advances in techniques and hardware for train.ing deep neural networks have recently enabled impressive accuracy improvements across many fundamental NLP tasks (Bahdanau et al., 2015; Luong et al., 2015; Dozat and Manning, 2017; Vaswani et al., 2017), with the most computationally-hungry models obtaining the highest scores (Peters et al., 2018; Devlin et al., 2019; Radford et al., 2019; So et al., 2019). As a result, training a state-of-the-art model now re.quires substantial computational resources which demand considerable energy, along with the associated financial and environmental costs. Research and development of new models multiplies these costs by thousands of times by requiring re.training to experiment with model architectures and hyperparameters. Whereas a decade ago most 

<<TABLE>>

Table 1: Estimated CO2 emissions from training com.mon NLP models, compared to familiar consumption.

NLP models could be trained and developed on a commodity laptop or server, many now require multiple instances of specialized hardware such as GPUs or TPUs, therefore limiting access to these highly accurate models on the basis of finances. 
Even when these expensive computational resources are available, model training also incurs a substantial cost to the environment due to the energy required to power this hardware for weeks or months at a time. Though some of this energy may come from renewable or carbon credit-offset resources, the high energy demands of these models are still a concern since (1) energy is not currently derived from carbon-neural sources in many locations, and (2) when renewable energy is available, it is still limited to the equipment we have to pro.duce and store it, and energy spent training a neural network might better be allocated to heating a family's home. It is estimated that we must cut carbon emissions by half over the next decade to deter escalating rates of natural disaster, and based on the estimated CO2 emissions listed in Table 1, 
1Sources: (1) Air travel and per-capita consumption: https://bit.ly/2Hw0xWc; (2) car lifetime: https://bit.ly/2Qbr0w1. 
model training and development likely make up a substantial portion of the greenhouse gas emissions attributed to many NLP researchers. 
To heighten the awareness of the NLP community to this issue and promote mindful practice and policy, we characterize the dollar cost and carbon emissions that result from training the neural net.works at the core of many state-of-the-art NLP models. We do this by estimating the kilowatts of energy required to train a variety of popular off-the-shelf NLP models, which can be converted to approximate carbon emissions and electricity costs. To estimate the even greater resources re.quired to transfer an existing model to a new task or develop new models, we perform a case study of the full computational resources required for the development and tuning of a recent state-of-the-art NLP pipeline (Strubell et al., 2018). We conclude with recommendations to the community based on our findings, namely: (1) Time to retrain and sensitivity to hyperparameters should be reported for NLP machine learning models; (2) academic Researchers need equitable access to computational resources; and (3) researchers should prioritize developing efficient models and hardware. 
2 Methods 
To quantify the computational and environmental cost of training deep neural network models for NLP, we perform an analysis of the energy required to train a variety of popular off-the-shelf NLP models, as well as a case study of the complete sum of resources required to develop LISA (Strubell et al., 2018), a state-of-the-art NLP model from EMNLP 2018, including all tuning and experimentation. 
We measure energy use as follows. We train the models described in 2.1 using the default settings provided, and sample GPU and CPU power con.sumption during training. Each model was trained for a maximum of 1 day. We train all models on a single NVIDIA Titan X GPU, with the exception of ELMo which was trained on 3 NVIDIA GTX 1080 Ti GPUs. While training, we repeatedly query the NVIDIA System Management Interface to sample the GPU power consumption and report the average over all samples. To sample CPU power consumption, we use Intel's Running Average Power Limit interface.

<<TABLE>>

Table 2: Percent energy sourced from: Renewable (e.g. hydro, solar, wind), natural gas, coal and nuclear for the top 3 cloud compute providers (Cook et al., 2017), compared to the United States,4 China5 and Germany (Burger, 2019). 

We estimate the total time expected for models to train to completion using training times and hardware reported in the original papers. We then calculate the power consumption in kilowatt-hours (kWh) as follows. Let pc be the average power draw (in watts) from all CPU sockets during train.ing, let pr be the average power draw from all DRAM (main memory) sockets, let pg be the aver.age power draw of a GPU during training, and let g be the number of GPUs used to train. We esti.mate total power consumption as combined GPU, CPU and DRAM consumption, then multiply this by Power Usage Effectiveness (PUE), which ac.counts for the additional energy required to sup.port the compute infrastructure (mainly cooling). We use a PUE coefficient of 1.58, the 2018 global average for data centers (Ascierto, 2018). Then the total power pt required at a given instance during training is given by: 

<<FORMULA>> (1) 

The U.S. Environmental Protection Agency (EPA) provides average CO2 produced (in pounds per kilowatt-hour) for power consumed in the U.S. (EPA, 2018), which we use to convert power to estimated CO2 emissions: 

<<FORMULA>> (2) 

This conversion takes into account the relative pro.portions of different energy sources (primarily nat.ural gas, coal, nuclear and renewable) consumed to produce energy in the United States. Table 2 lists the relative energy sources for China, Ger.many and the United States compared to the top 
three cloud service providers. The U.S. break.down of energy is comparable to that of the most popular cloud compute service, Amazon Web Ser.vices, so we believe this conversion to provide a reasonable estimate of CO2 emissions per kilowatt hour of compute energy used. 

2.1 Models 

We analyze four models, the computational requirements of which we describe below. All models have code freely available online, which we used out-of-the-box. For more details on the models themselves, please refer to the original papers. 
Transformer. The Transformer model (Vaswani et al., 2017) is an encoder-decoder architecture primarily recognized for efficient and accurate ma.chine translation. The encoder and decoder each consist of 6 stacked layers of multi-head self-attention. Vaswani et al. (2017) report that the Transformer base model (65M parameters) was trained on 8 NVIDIA P100 GPUs for 12 hours, and the Transformer big model (213M parameters) was trained for 3.5 days (84 hours; 300k steps). This model is also the basis for recent work on neural architecture search (NAS) for ma.chine translation and language modeling (So et al., 2019), and the NLP pipeline that we study in more detail in 4.2 (Strubell et al., 2018). So et al. (2019) report that their full architecture search ran for a total of 979M training steps, and that their base model requires 10 hours to train for 300k steps on one TPUv2 core. This equates to 32,623 hours of TPU or 274,120 hours on 8 P100 GPUs. 
ELMo. The ELMo model (Peters et al., 2018) is based on stacked LSTMs and provides rich word representations in context by pre-training on a large amount of data using a language model.ing objective. Replacing context-independent pre.trained word embeddings with ELMo has been shown to increase performance on downstream tasks such as named entity recognition, semantic role labeling, and coreference. Peters et al. (2018) report that ELMo was trained on 3 NVIDIA GTX 1080 GPUs for 2 weeks (336 hours). 
BERT. The BERT model (Devlin et al., 2019) provides a Transformer-based architecture for build.ing contextual representations similar to ELMo, but trained with a different language modeling objective. BERT substantially improves accuracy on tasks requiring sentence-level representations such as question answering and natural language inference. Devlin et al. (2019) report that the BERT base model (110M parameters) was trained on 16 TPU chips for 4 days (96 hours). NVIDIA reports that they can train a BERT model in 3.3 days (79.2 hours) using 4 DGX-2H servers, totaling 64 Tesla V100 GPUs (Forster et al., 2019). 
GPT-2. This model is the latest edition of OpenAI's GPT general-purpose token encoder, also based on Transformer-style self-attention and trained with a language modeling objective (Rad.ford et al., 2019). By training a very large model on massive data, Radford et al. (2019) show high zero-shot performance on question answering and language modeling benchmarks. The large model described in Radford et al. (2019) has 1542M parameters and is reported to require 1 week (168 hours) of training on 32 TPUv3 chips. 6 

3 Related work 

There is some precedent for work characterizing the computational requirements of training and inference in modern neural network architectures in the computer vision community. Li et al. (2016) present a detailed study of the energy use required for training and inference in popular convolutional models for image classification in computer vision, including fine-grained analysis comparing different neural network layer types. Canziani et al. (2016) assess image classification model accuracy as a function of model size and gigaflops required during inference. They also measure average power draw required during inference on GPUs as a function of batch size. Neither work analyzes the recurrent and self-attention models that have become commonplace in NLP, nor do they extrapolate power to estimates of carbon and dol.lar cost of training. 
Analysis of hyperparameter tuning has been performed in the context of improved algorithms for hyperparameter search (Bergstra et al., 2011; Bergstra and Bengio, 2012; Snoek et al., 2012). To our knowledge there exists to date no analysis of the computation required for R&D and hyperparameter tuning of neural network models in NLP. 
6Via the authors on Reddit. 
7GPU lower bound computed using pre-emptible <<P100/V100>> U.S. resources priced at <<FORMULA>>, upper bound uses on-demand U.S. resources priced at <<FORMULA>>. We similarly use pre-emptible (<<FORMULA>>) and on-demand (<<FORMULA>>) pricing as lower and upper bounds for TPU v2/3; cheaper bulk contracts are available. 

<<TABLE>>

Table 3: Estimated cost of training a model in terms of CO2 emissions (lbs) and cloud compute cost (USD).7 Power and carbon footprint are omitted for TPUs due to lack of public information on power draw for this hardware. 

4 Experimental results 

4.1 Cost of training 
Table 3 lists CO2 emissions and estimated cost of training the models described in 2.1. Of note is that TPUs are more cost-efficient than GPUs on workloads that make sense for that hardware (e.g. BERT). We also see that models emit substantial carbon emissions; training BERT on GPU is roughly equivalent to a trans-American fight. So et al. (2019) report that NAS achieves a new state-of-the-art BLEU score of 29.7 for English to Ger.man machine translation, an increase of just 0.1 BLEU at the cost of at least $150k in on-demand compute time and non-trivial carbon emissions. 

4.2 Cost of development: Case study 
To quantify the computational requirements of R&D for a new model we study the logs of all training required to develop Linguistically-Informed Self-Attention (Strubell et al., 2018), a multi-task model that performs part-of-speech tagging, labeled dependency parsing, predicate detection and semantic role labeling. This model makes for an interesting case study as a representative NLP pipeline and as a Best Long Paper at EMNLP. 
Model training associated with the project spanned a period of 172 days (approx. 6 months). During that time 123 small hyperparameter grid searches were performed, resulting in 4789 jobs in total. Jobs varied in length ranging from a minimum of 3 minutes, indicating a crash, to a maximum of 9 days, with an average job length of 52 hours. All training was done on a combination of NVIDIA Titan X (72%) and M40 (28%) GPUs.8 
The sum GPU time required for the project totaled 9998 days (27 years). This averages to 

<<TABLE>>

Table 4: Estimated cost in terms of cloud compute and electricity for training: (1) a single model (2) a single tune and (3) all models trained during R&D. 
about 60 GPUs running constantly throughout the 6 month duration of the project. Table 4 lists upper and lower bounds of the estimated cost in terms of Google Cloud compute and raw electricity re.quired to develop and deploy this model.9 We see that while training a single model is relatively inexpensive, the cost of tuning a model for a new dataset, which we estimate here to require 24 jobs, or performing the full R&D required to develop this model, quickly becomes extremely expensive. 

5 Conclusions 

Authors should report training time and sensitivity to hyperparameters. 
Our experiments suggest that it would be beneficial to directly compare different models to per.form a cost-bene<6E>t (accuracy) analysis. To ad.dress this, when proposing a model that is meant to be re-trained for downstream use, such as re.training on a new domain or fine-tuning on a new task, authors should report training time and computational resources required, as well as model sensitivity to hyperparameters. This will enable direct comparison across models, allowing subsequent consumers of these models to accurately assess whether the required computational resources 

We approximate cloud compute cost using P100 pricing. 9Based on average U.S cost of electricity of $0.12/kWh. 
are compatible with their setting. More explicit characterization of tuning time could also reveal inconsistencies in time spent tuning baseline models compared to proposed contributions. Realizing this will require: (1) a standard, hardware-independent measurement of training time, such as gigaflops required to convergence, and (2) a standard measurement of model sensitivity to data and hyperparameters, such as variance with respect to hyperparameters searched. 
Academic researchers need equitable access to computation resources. 

Recent advances in available compute come at a high price not attainable to all who desire access. Most of the models studied in this paper were developed outside academia; recent improvements in state-of-the-art accuracy are possible thanks to industry access to large-scale compute. 
Limiting this style of research to industry labs hurts the NLP research community in many ways. First, it stifles creativity. Researchers with good ideas but without access to large-scale compute will simply not be able to execute their ideas, instead constrained to focus on different problems. Second, it prohibits certain types of Research on the basis of access to financial resources. This even more deeply promotes the already problematic rich get richer cycle of research funding, where groups that are already successful and thus well-funded tend to receive more funding due to their existing accomplishments. Third, the prohibitive start-up cost of building in-house resources forces resource-poor groups to rely on cloud compute services such as AWS, Google Cloud and Microsoft Azure. 
While these services provide valuable, flexible, and often relatively environmentally friendly compute resources, it is more cost effective for academic researchers, who often work for nonprofit educational institutions and whose research is funded by government entities, to pool resources to build shared compute centers at the level of funding agencies, such as the U.S. National Science Foundation. For example, an off-the-shelf GPU server containing 8 NVIDIA 1080 Ti GPUs and supporting hardware can be purchased for approximately $20,000 USD. At that cost, the hardware required to develop the model in our case study (approximately 58 GPUs for 172 days) would cost $145,000 USD plus electricity, about half the estimated cost to use on-demand cloud GPUs. Unlike money spent on cloud compute, however, that invested in centralized resources would continue to pay off as resources are shared across many projects. A government-funded academic compute cloud would provide equitable access to all researchers. 
Researchers should prioritize computationally efficient hardware and algorithms. 
We recommend a concerted effort by industry and academia to promote research of more computationally efficient algorithms, as well as hardware that requires less energy. An effort can also be made in terms of software. There is already a precedent for NLP software packages prioritizing efficient models. An additional avenue through which NLP and machine learning software developers could aid in reducing the energy associated with model tuning is by providing easy.to-use APIs implementing more efficient alternatives to brute-force grid search for hyperparameter tuning, e.g. random or Bayesian hyperparameter search techniques (Bergstra et al., 2011; Bergstra and Bengio, 2012; Snoek et al., 2012). While software packages implementing these techniques do exist,10 they are rarely employed in practice for tuning NLP models. This is likely because their interoperability with popular deep learning frameworks such as PyTorch and TensorFlow is not optimized, i.e. there are not simple examples of how to tune TensorFlow Estimators using Bayesian search. Integrating these tools into the work<72>ows with which NLP researchers and practitioners are already familiar could have notable im.pact on the cost of developing and tuning in NLP. 

Acknowledgements 

We are grateful to Sherief Farouk and the anonymous reviewers for helpful feedback on earlier drafts. This work was supported in part by the Centers for Data Science and Intelligent Information Retrieval, the Chan-Zuckerberg Initiative under the Scientific Knowledge Base Construction project, the IBM Cognitive Horizons Network agreement no. W1668553, and National Science Foundation grant no. IIS-1514053. Any opinions, findings and conclusions or recommendations ex.pressed in this material are those of the authors and do not necessarily reflect those of the sponsor. 
For example, the Hyperopt Python library. 

References 

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Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. In 31st Conference on Neural Information Processing Systems (NIPS). 
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<<START>> <<START>> <<START>>
Finite-Element Neural Networks for Solving Differential Equations 
Pradeep Ramuhalli, Member, IEEE, Lalita Udpa, Senior Member, IEEE, and Satish S. Udpa, Fellow, IEEE 

Abstract

The solution of partial differential equations (PDE) arises in a wide variety of engineering problems. Solutions to most practical problems use numerical analysis techniques such as finite-element or finite-difference methods. The drawbacks of these approaches include computational costs associated with the modeling of complex geometries. This paper proposes a finite-element neural network (FENN) obtained by embedding a finite-element model in a neural network architecture that enables fast and ac.curate solution of the forward problem. Results of applying the FENN to several simple electromagnetic forward and inverse problems are presented. Initial results indicate that the FENN performance as a forward model is comparable to that of the conventional finite-element method (FEM). The FENN can also be used in an iterative approach to solve inverse problems associated with the PDE. Results showing the ability of the FENN to solve the in.verse problem given the measured signal are also presented. The parallel nature of the FENN also makes it an attractive solution for parallel implementation in hardware and software. 

I. INTRODUCTION 

Solutions of differential equations arise in a wide variety of engineering applications in electromagnetics, signal processing, computational fluid dynamics, etc. These equations are typically solved using either analytical or numerical methods. Analytical solution methods are however feasible only for simple geometries, which limits their applicability. In most practical problems with complex boundary conditions, numerical analysis methods are required in order to obtain a reasonable solution. An example is the solution of Maxwell's equations in electromagnetics. Solutions to Maxwell's equations are used in a variety of applications for calculating the interaction of electromagnetic (EM) fields with different types of media. 
Very often, the solution to differential equations is necessary for solving the corresponding inverse problems. Inverse problems in general are ill-posed, lacking continuous dependence of the measurements on the input. This has resulted in the development of a variety of solution techniques ranging from simple calibration procedures to other direct (analytical) and iterative approaches [1]. Iterative methods typically employ a forward model that simulates the underlying physical process (Fig. 1) [2]. An initial estimate of the solution of the inverse problem (represented by 
in Fig. 1) is applied to the forward model, 
Manuscript received January 17, 2004; revised April 2, 2005. 

<<FIGURE>>

Fig. 1. Iterative inversion method for solving inverse problems. 
resulting in the corresponding solution to the forward problem 

<<ALGORITHM>>

Although finite-element methods (FEMs) [3], [4] are extremely popular for solving differential equations, their major drawback is computational complexity. This problem becomes more acute when three-dimensional (3-D) finite-element models are used in an iterative algorithm for solving the inverse problem. Recently, several authors have suggested the use of neural networks (MLP or RBF networks [5]) for solving differential equations [6][9]. 
In these techniques, a neural network is trained using a large database containing the input data and the solution of the differential equation. The neural network during generalization learns the mapping corresponding to the PDE. Alternatively, in [10], the solution to a differential equation is written as a constant term, and an adjustable term with parameters that need to be determined. A neural network is used to determine the optimal values of the parameters. This approach is applicable only to problems with regular boundaries. An extension of the approach to problems with irregular boundaries is given in [11]. Other neural network based differential equation solvers use multilayer perceptron networks or variations on the MLP to approximate the unknown function in a PDE [12][14]. A combination of the PDE and boundary conditions is used to construct an objective function that is minimized during the training process. 
A major limitation of these approaches is that the network architecture is selected somewhat arbitrarily. A second drawback is that the performance of the neural networks depends on the data used in training and testing. As long the test data is similar to the training data, the network can interpolate between the training data points to obtain a reasonable prediction. However, when the test signal is no longer similar to the training data, the 
network is forced to extrapolate and the performance degrades. One way around this difficulty is to ensure that the training data.base has a diverse set of signals. However, this is difficult to ensure in practice. Alternatively, we have to design neural net.works that are capable of extrapolation. Extrapolation methods are discussed extensively in literature [15][18], but the design of an extrapolation neural network involves several issues particularly for ensuring that the error in the network prediction stays within reasonable bounds during the extrapolation procedure. 
An ideal solution to this problem would be to combine the power of numerical models with the computational speed of neural networks, i.e., to embed a numerical model in a neural network structure. One such finite-element neural network (FENN) formulation has been reported by Takeuchi and Kosugi [19]. This approach, based on error minimization, derives the neural network using the energy functional resulting from the finite-element formulation. Other reports of FENN combinations are either similar to the Takeuchi method [20], [21] or use Hopfield neural networks to solve the forward problem [22], [23]. Kalkkuhl et al. [24] provide a description of a FEM-based approach to NARX modeling that may be interpreted both as a local model network, as well as a single layer feedforward network. A slightly different approach to merging numerical methods and neural networks is given in [25], where the finite-difference time domain (FDTD) method is cast in a neural network framework for the purpose of solving electromagnetic forward problems. The related problem of mesh generation in finite-element models has also been tackled using neural networks (for instance, [26]). Generally, these networks are designed to solve the forward problem, and must be modified to solve inverse problems. 
This paper proposes a new approach that embeds a finite-element model commonly used in the solution of differential equations in a neural network. The network, called the FENN, can solve the forward problem and can also be used in an iterative algorithm to solve inverse problems. The primary advantage of this approach is that the FEM is represented in a parallel form. Thus, it has the potential to alleviate the computational cost associated with using the FEM in an iterative algorithm for solving inverse problems. More importantly, the FENN does not need any training, and the computation of the weights is a one-time process. The proposed approach is also different in that the neural network architecture developed can be used to solve the forward and inverse problems. The structure of the neural network is also simpler than those reported in the literature, making it easier to implement in parallel in both hardware and software. 
The rest of this paper is organized as follows. Section II briefly describes the FEM, and derives the proposed FENN. In this paper, we focus on the problem of solving typical equations encountered in electromagnetic nondestructive evaluation (NDE). However, the same concepts can be easily applied to solve differential equations encountered in other fields. Sections III, IV and V present the application of the FENN to solving forward and inverse problems, along with initial results. A discussion of the advantages and disadvantages of the proposed FENN architecture is given in Section IV. Finally, Section V draws conclusions from the results and presents ideas for future work. 

II. THE FENN 

This section briefly describes the FEM and proposes its reformulation into a parallel neural network structure. Details about the FEM can be found in [3] and [4]. 

A. The FEM 

Consider a typical boundary value problem with the governing differential equation 

<<FORMULA>>        (1) 

where <<FORMULA>> is a differential operator, <<FORMULA>> is the applied source or forcing function, and 
is the unknown quantity. This differential equation can be solved in conjunction with boundary conditions on the boundary 
enclosing the domain 
The variational formulation used in finite-element analysis determines the unknown 

by minimizing the functional [3], [4] (2) with respect to the trial function 

The minimization procedure starts by dividing into small subdomains called elements (Fig. 2) and representing in each element by means of basis functions  defined over the element (3) where 
is the unknown solution in element 

<<FORMULA>>     (3)

is the basis function associated with node in element , is the value of the unknown quantity at node and is the total number of nodes associated with element <<FORMULA>> In general, the basis functions (also referred to as interpolation functions or shape functions) can be linear, quadratic, or of higher order. Typically, finite-element models use either linear or polynomial spline basis functions. 
The functional within an element is expressed as 

<<FORMULA>>         (4) 

By substituting (3) in (4), we obtain the discrete version of the functional within each element  

<<FORMULA>>         (5)  
where is the transpose of a matrix, mental matrix with elements  is the  ele. 

<<FORMULA>>         (6)  

and is an vector with elements 

<<FORMULA>>             (7) 

Combining the values in (5) for each of the elements (8) where is the global matrix derived from the terms of the elemental matrices for different elements, and 
is the total number of nodes, also called the stiffness matrix, is a sparse, banded matrix. Equation (8) is the discrete version of the functional and can be minimized with respect to the nodal parameters 
by taking the derivative of with respect to <<FORMULA>> and setting it equal to zero, which results in the matrix equation 

<<FORMULA>>         (9) 

Boundary conditions for these problems are usually of two types: natural boundary conditions and essential boundary conditions. Essential boundary conditions (also referred to as Dirichlet boundary conditions) impose constraints on the value of the unknown 
at several nodes. Natural boundary  conditions (of which Neumann boundary conditions are a special case) impose constraints on the change in 
across a boundary. Dirichlet boundary conditions are imposed on the functional minimization (9), by deleting the rows and columns of the matrix corresponding to the nodes on the Dirichlet boundary and modifying 
in (9). 


Natural boundary conditions are applied in the FEM by adding an additional term to the functional. These boundary conditions are then incorporated into the functional and are  satisfied automatically during the solution procedure. As an example, consider the natural boundary condition represented by the following equation [3] on 

<<FORMULA>>             (10) 

where <<FORMULA>> represents the Neumann boundary, is its outward normal unit vector, is some constant, and , <<FORMULA>>, and are known parameters associated with the boundary. Assuming that the boundary 
is made up of segments, we can  define boundary matrices and with elements 

<<FORMULA>>         (11) 

where <<FORMULA>>are basis functions  defined over segment and is the length of the segment. The elements of <<FORMULA>> are added to the elements of that correspond to the nodes on the boundary. Similarly, the elements of <<FORMULA>> are added to the corresponding elements of 
<<FORMULA>> The global matrix (9) is thus modified as follows before solving for 


<<FORMULA>>         (12) 


                        <<FIGURE>>

Fig. 3. FEM domain discretization using two elements and four nodes. 

This process ensures that natural boundary conditions are implicitly and automatically  satisfied during the FEM solution procedure. 

B. The FENN 

This section describes how the finite-element model can be converted into a parallel network form. We focus on solving typical inverse problems arising in electromagnetic NDE, but the basic idea is applicable to other areas as well. NDE inverse problems can be formulated as the problem of  finding the material properties (such as the conductivity or the permeability) within the domain of the problem. Since the domain is discretized in the FEM method by a large number of elements, the problem can be posed as one of  finding the material properties in each of these elements. These properties are usually embedded in the differential operator <<FORMULA>> or equivalently, in the global matrix 
<<FORMULA>> Thus, in order to be able to iteratively estimate these properties from the measurements, the material properties need to be separated out from 
<<FORMULA>> This separation is easier to achieve at the element matrix level. For nodes <<FORMULA>> and in element 

<FORMULA>>          (13) 

where <<FORMULA>> is the parameter representing the material property in element <<FORMULA>> and <<FORMULA>> represents the differential operator at the 

<<FIGURE>>

Fig. 4. FENN. 

element level without embedded in it. Substituting (13) into the functional, we get 

<<FORMULA>>         (14) 
If we  define 

<<FORMULA>>          (15) 

where 

<<FORMULA>>         (16) 

<<FORMULA>>             (17) 


Equation (17) expresses the functional explicitly in terms of <<FORMULA>> The assumption that is constant within each element is implicit in this expression. This assumption is usually  satisfied in problems in NDE where each element in the FEM mesh is defined within the confines of a domain, and at no time does a single element cross domain boundaries. Furthermore, each  element is small enough that minor variations in 
within an  element may be ignored. Equation (17) can be easily converted into a parallel network form. The neural network comprises an input, output and hidden layer. In the general case with 
<<FORMULA>> elements and <<FORMULA>> nodes in the FEM mesh, the input layer with network inputs takes the values in each element as input. The hidden layer has 
neurons arranged in groups of neurons, corresponding to the members of the global <<FORMULA>> matrix 

. The output of each group of hidden layer neurons is the corresponding row vector of 
. The weights from the input to the hidden layer are set to the appropriate values of 
. Each neuron in the hidden layer acts as a summation unit, (equivalent to a summation followed by a linear activation function [5]). The outputs of the hidden layer neurons are the elements of the global matrix 

as given in (15). Each group of hidden neurons is connected to one output neuron (giving a total of output neurons) by a set of weights with each element of 
representing the nodal values. Note that the set of weights 
between the first group of hidden neurons and the first output neuron are the same as the set of weights between the second group of hidden neurons and the second output neuron (as well as between successive groups of hidden neurons and the corresponding output neuron). Each output neuron is also a summation unit followed by a linear activation function, and the output of each neuron is equal to 

<<FORMULA>>             (18) 

where the second part of (18) is obtained by using (15). As an example, the FENN architecture for a two-element, four-node FEM mesh (Fig. 3) is shown in Fig. 4. In this case, the FENN has two input neurons, 16 hidden layer neurons and four output neurons. The <20>gure illustrates the grouping of the hidden layer neurons, as well as the similarity inherent in the weights that connect each group of hidden layer neurons to the corresponding output neuron. To simplify the <20>gure, the weights between the network input and hidden layer neurons are depicted by means of vectors 
(for , 2, 3, 4 and , 2), where the individual weight values <<FORMULA>> are  defined as in (16). 
1) Boundary Conditions in the FENN: Note that the elements of <<FORMULA>> and in (11) do not depend on the material properties <<FORMULA>> and need to be added appropriately to the global matrix 
and the source vector as shown in (12). 

<<FIGURE>>

Fig. 5. Geometry of mesh for 1-D FEM. 

<<FIGURE>>

Fig. 6. Flowchart (with example) for designing the FENN for a general PDE. 

Equation (12) thus implies that natural boundary conditions can be ap-layer neurons. These weights will be referred to as the clamped plied in the FENN as bias inputs to the hidden layer neurons weights, while the remaining weights will be referred to as the that are a part of the boundary, and the corresponding output free weights. An example of these weights is presented later. neurons. Dirichlet boundary conditions are applied by clamping The FENN architecture was derived without consideration of the corresponding weights between the hidden layer and output the dimensionality of the problem at hand, and thus can be used for 1-, 2-, 3-, or higher dimensional problems. The number of nodes and elements in the FEM mesh dictates the number of neurons in the different layers. The weights between the input and hidden layer change depending on node-element connectivity information. 
The major drawback of the FENN is the number of neurons and weights necessary. However, the memory requirements can be reduced considerably, since most of the weights between the input and hidden layer are zero. These weights, and the corresponding connections, can be discarded. Similarly, most of the elements of the 
matrix are also zero (is a banded  matrix). The corresponding neurons in the hidden layer can also be discarded, reducing memory and computation requirements considerably. Furthermore, the weights between each group of hidden layer neurons and the output layer are the same 
. Weight-sharing approaches can be used here to further reduce the storage requirements. 

C. A 1-D Example 

Consider the 1-D equation 

<<FORMULA>>         (19) 

on the boundary <<FORMULA>> defined by <<FORMULA>> and 
are constants depending on the material and 
is the applied source. Laplace's equation and Poisson's equation are special cases of this equation. The FENN formulation for this problem starts by discretizing the domain of interest with <<FORMULA>> elements and 
nodes. In one dimension, each element is  defined by two nodes (Fig. 5).  define basis functions <<FORMULA>> and <<FORMULA>> over each element <<FORMULA>> and let 
is the value of <<FORMULA>> on node <<FORMULA>> in element <<FORMULA>> An example of the basis functions is shown in Fig. 5. For these basis functions, i.e., 

<<FORMULA>>         (20) 

the element matrices are given by [3] 

<<FORMULA>>         (21) 

<<FORMULA>>             (22) 

Here, <<FORMULA>> is the length of element <<FORMULA>> The global matrix 
is then constructed by selectively adding the element matrices based on the nodes that form an element.  Specifically, 
is a sparse tridiagonal matrix, and its nonzero elements are given by 

<<FORMULA>>         (23) 

Fig. 7. Shielded microstrip geometry. (a) Complete problem description. (b) Problem description using symmetry considerations. 
The network implementation of (23) can be derived as fol.lows. If <<FORMULA>> and <<FORMULA>> values at each element are the inputs to the network, 
<<FORMULA>> and <<FORMULA>> form the weights between the input and hidden layers. The network thus uses input neurons and 
hidden neurons. The values of <<FORMULA>> at each of the nodes are assigned as weights between the hidden and output layers, and the source 
is the desired output of this network (corresponding to the output neurons). Dirichlet boundary conditions on 
are applied as explained earlier. 

D. General Case 

Fig. 6 shows a  flowchart of the general scheme for converting a differential equation into the FENN structure. An example in two dimensions is also provided next to the  flowchart. We start with the differential equation and the boundary conditions and formulate the FEM using the variational method. This in.volves discretizing the domain of interest with 
elements and 

nodes, selecting basis functions, writing the functional for each element and obtaining the element matrices and the source vector. The example presented uses the FEM mesh shown in Fig. 3, with 
elements, and <<FORMULA>> nodes, and linear basis functions. The unknown solution to the differential equation 
is represented by its values at each of the nodes in the finite-element mesh <<FORMULA>> The element matrices 
are then separated into two parts, with one part dependent on the material properties <<FORMULA>> and 
while the other is independent of them. The FENN is then designed to have input neurons, hidden neurons, and output neurons, where <<FORMULA>> is the number of material property parameters. In the example under consideration, <<FORMULA>>, since we have two 
material property parameters ( and ). The  first group of input neurons takes in the values while the second group takes in the 
values in each element. The weights from the input to the hidden layer are set to the appropriate values of 
<<FORMULA>> In the example, since nodes 1, 2, and 3 are part of element 1 (see Fig. 3), the weights from the  first input node 
to the  first group of four neurons in the hidden layer are given by 

<<FORMULA>>         (24) 

The last weight is zero since node 4 is not a part of element 1. Each group of hidden neurons is connected to one output neuron (giving a total of 
output neurons) by a set of weights <<FORMULA>> with each element of representing the nodal values. The output of each neuron in the output layer is equal to 

<<FIGURE>>

Fig. 8. Forward problem solutions for shielded microstrip problem show the contours of constant potential for: (a) FEM solution and (b) FENN solution. (c) Error between (a) and (b). The x-and y-axes show the nodes in the FEM discretization of the domain, and the z-axis in (c) shows the error at each of these nodes in volts. 

III. FORWARD AND INVERSE PROBLEM FORMULATION USING FENN.

where is the output of the FENN based approach, then, for the gradients of the error with respect to the free hidden layer weights is given by the FENN architecture and algorithm lends itself to solving 

<<FORMULA>>             (27)

both the forward and inverse problems. The forward problem involves determining the weights 
given the material parameters Equation (27) can be used to solve the forward problem. 

Similarly, the applied source to solve the inverse problem,
while the inverse problem the gradients of the error involves determining and (input of the FENN) are necessary, and approach can be used to solve both these problems. Suppose we are given by  define the error at the output of the FENN as 

            <<TABLE>>

TABLE I SUMMARY OF PERFORMANCE OF THE FENN ALGORITHM FOR VARIOUS PDES 

For the forward problem, such an approach is equivalent to the iterative approaches used to solve for the unknown nodal values in the FEM [4]. 

IV. RESULTS 

A. Forward Model Results 
The FENN was tested using both 1-and 2-D versions of Poisson<6F>s equation 
<<FORMULA>>     (30) 
where represents the material property, and is the applied source. For instance, in electromagnetics may represent the permittivity while represents the charge density. 
As the  first example, consider the following 2-D equation 
<<FORMULA>>         (31)  
with boundary conditions and <<FORMULA>> on <<FORMULA>> (32)  
on <<FORMULA>> (33)  

This is the governing equation for the shielded microstrip trans.mission line problem shown in Fig. 7. The forward problem computes the electric potential due to the shielded microstrip shown in Fig. 7(a). The potentials are zero on the shielding con.ductor. Since the geometry is symmetric, we can solve the equiv.alent problem shown in Fig. 7(b), by applying the homogeneous Neumann condition on the plane of symmetry. The inner con.ductor (microstrip) is held at a constant potential of volts. Finally, we also assume that the material inside the shielding conductor has a permittivity , where K is a constant. The permittivity in this case corresponds to the material property .  Specifically, and . The homogeneous Neu.mann boundary condition is equivalent to setting . The microstrip and the shielding conductor correspond to the Dirichlet boundary, with <<FORMULA>> on the microstrip and 
on the outer boundary [Fig. 7(b)]. Finally, there is no source term in this example (the source term would correspond to a charge distribution in the domain of interest), i.e., <<FORMULA>> In this ex.ample, we assume that volts. Further, we assume that the domain of interest is 

The solution to the forward problem is presented in Fig. 8, with the FEM solution using 11 nodes in each direction shown in Fig. 8(a) and the corresponding FENN solution in Fig. 8(b). These <20>gures show contours of constant potential. The error be.tween the FEM and FENN solutions is presented in Fig. 8(c). As seen from the <20>gure, the FENN is seen to match the FEM solu.tion accurately, with the peak error at any node on the order of 
Several other examples were also used to test the FENN and the results are summarized in Table I. Column 1 shows the PDE used to evaluate the FENN performance, while column 2 shows the boundary conditions used. The analytic solution to the problem is indicated in Column 3. The FENN structure and the number of iterations for convergence using a gradient de.scent approach are indicated in Columns 4 and 5, respectively. The FENN structure, as explained earlier, has an 
are the number of elements and nodes in the FEM mesh, respectively, and 
is the number of hidden neurons, and corresponds to the number of nonzero elements in the FEM global matrix 
Finally, Columns 6 and 7 present the sum-squared error (SSE) and the maximum error in the solution, respectively, where the errors are computed with respect to the analytical solution. These results indicate that the FENN is capable of accurately deter.mining the potential 
One advantage of the FENN approach is that the computation of the input-hidden layer weights is a one-time process, as long as the differential equation does not change. The only changes necessary to solve the different problems are changes in the input 
and the desired output.

B. Inverse Model Results 

The FENN was also used to solve several simple inverse problems based on (30). In all cases, the objective was to determine 

<<FIGURE>>

Fig. 9. FENN inversion results for Poisson's equation with initial solutions (a) 
the value of <<FORMULA>> and <<FORMULA>> for given values of <<FORMULA>> and 
The <<FORMULA>> first example is a 1-D problem that involves determining 
given and <<FORMULA>> 
for the differential equation 
<<FORMULA>>     (34) 

with boundary conditions <<FORMULA>> and <<FORMULA>>. The analytical solution to this inverse problem is 
<<FORMULA>> and 

<<FORMULA>>             (35) 

As seen from (35), the problem has an infinite number of solutions and we expect the solution procedure to converge to one of these solutions depending on the initial value. 

Fig. 9(a) and (b) shows two solutions to this inverse problem for two different initializations (shown using triangles). In both cases, the FENN solution (in stars) is seen to match the analytical solution (squares). The SSE in both cases was on the order of 

<<FORMULA>>

In order to obtain a unique solution, we need to constrain the value of at the boundary as well. Consider the same differen.
tial equation as (34), but with  and  specified as follows:  
and  
(36)  


The analytical solution for this equation is .To solve this problem, we set and clamp the value of at and as follows: , . The results of the constrained inversion obtained using 11 nodes and 10 elements in the corresponding finite-element mesh are shown in Fig. 10. Fig. 10(a) shows the comparison between the analytical solution (solid line with squares) and the FENN result (solid line with stars). The initial value of is shown in the figure as a dashed line. Fig. 10(b) shows the comparison between the actual and desired forcing function at the FENN 


output. This result indicates that the SSE in the forcing function, as well as the SSE in the inversion result, is fairly large (0.0148 and 0.0197, respectively). The reason for this was traced back to the mesh discretization. Fig. 11 shows the SSE in the output of the FENN and the SSE in the inverse problem solution as a function of FEM discretization. It is seen that increasing the discretization significantly improves the solution. Similar results were observed for other problems. 

V. DISCUSSION AND CONCLUSION 

The FENN is closely related to the finite-element model used to solve differential equations. The FENN architecture has a weight structure that allows both the forward and inverse problems to be solved using simple gradient-based algorithms. Initial results indicate that the proposed FENN algorithm is capable of accurately solving both the forward and inverse problems. In addition, the forward problem solution from the FENN is seen to exactly match the FEM solution, indicating that the FENN represents the finite-element model exactly in a parallel configuration. 
The major advantage of the FENN is that it represents the finite-element model in a parallel form, enabling parallel implementation in either hardware or software. Further, computing gradients in the FENN is very simple. This is an advantage in solving both forward and inverse problems using gradient-based methods. The gradients can also be computed in parallel and the lack of nonlinearities in the neuron activation functions makes the computation of gradients simpler. A major advantage of this approach for solving inverse problems is that it avoids inverting the global matrix in each iteration. The FENN also does not require any training, since most of its weights can be computed in advance and stored. The weights depend on the governing differential equation and its associated boundary conditions, and as long as these two factors do not change, the weights do not change. This is especially an advantage in solving inverse problems in electromagnetic NDE. This approach also reduces the computational effort associated with the network. 

Future work will concentrate on applying the FENN to 3-D electromagnetic NDE problems. The robustness of the approach will also be tested, since the ability of these approaches to in.vert practical noisy measurements is important. Furthermore, the use of better optimization algorithms, like conjugate gradient methods, is expected to improve the solution speed. In addition, parallel implementation of the FENN in both hardware and software is under investigation. The approach described in this paper is very general in that it can be applied to a variety of inverse problems in fields other than electromagnetic NDE. Some of these other applications will also be investigated to show the general nature of the proposed method. 

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