> <> <> Where <<[·; ·]>> denotes concatenation of a scalar onto a column vector. The parameters are <>, <> and <> , <> . Here we use 100 hidden units, i.e.<< h = 100>>. We have <>, 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 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 11: MNIST Classification <

> 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 <> and an estimate of <> Table 3: Classification on MNIST <> These are combined with a weighted average and integrated along the solution trajectory. These terms are motivated by the expansion <> 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: <> 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) <> <> <> <> < <> <> 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 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. <> (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 <> (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) <> (ODE) <> (3) The Euler discretization of this ODE with step-size <<τ>> is where <> is the divergence operator, <>, 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 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 <> 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 .  <> (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.. <> (6) 2.2. The need for regularity <> (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 <> (10) <> (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 <> (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 <> (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- <> (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 <> <

> 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 <> estimate of the trace is given by <> (14) where <> 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 <> 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 <> 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 <> (15) analagous to gradient regularization, which has been shown to improve generalization (Drucker & LeCun, 1992; Novak <> (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 <> 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 <> (12) norm regularization is available with essentially no extra computational cost. Equivalently it may be computed as 5. Algorithm description <> (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 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 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 <> implementation, the Jacobian Frobenius estamate reuses the computatian T ∇ f from the divergence estimate for efficiency. We remark that the kinetic energy term only <> requires the computation of a dot product. Thus just as in FFJORD, our implementation scales linearly with the <> (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 <> (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 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 <> 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 <>; 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 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 tural properties of residual networks to efficiently compute the Laplacian. Avron, H. and Toledo, S. Randomized algorithms for esti- mating the trace of an implicit symmetric positive semi- definite matrix. J. ACM, 58(2):8:1–8:34, 2011. doi: 9. Discussion 10.1145/1944345.1944349. URL https://doi.org/ In practice, RNODE is simple to implement, and only re- 10.1145/1944345.1944349. quires augmenting the dynamics (ODE) with two extra scalar equations (one for the kinetic energy term, and an- Behrmann, J., Grathwohl, W., Chen, R. T. Q., Duve- other for the Jacobian penalty). In the setting of FFJORD, naud, D., and Jacobsen, J. Invertible residual networks. because we may recycle intermediary terms used in the In Chaudhuri, K. and Salakhutdinov, R. (eds.), Pro- divergence estimate, the computational cost of evaluating ceedings of the 36th International Conference on Ma- these two extra equations is minimal. RNODE introduces chine Learning, ICML 2019, 9-15 June 2019, Long two extra hyperparameters related to the strength of the reg- Beach, California, USA, volume 97 of Proceedings ularizers; we have found these required almost no tuning. of Machine Learning Research, pp. 573–582. PMLR, 2019. URL http://proceedings.mlr.press/ Although the problem of classification was not considered v97/behrmann19a.html. in this work, we believe RNODE may offer similar im- provements both in training time and the regularity of the Benamou, J.-D. and Brenier, Y. A computational fluid me- classifier learned. In the classification setting we expect the chanics solution to the Monge-Kantorovich mass transfer computional overhead of calculating the two extra terms problem. Numerische Mathematik, 84(3):375–393, 2000. should be marginal relative to gains made in training time. Chen, T. Q., Rubanova, Y., Bettencourt, J., and Duve- naud, D. Neural Ordinary Differential Equations. In 10. Conclusion Advances in Neural Information Processing Systems 31: We have presented RNODE, a regularized method for neu- Annual Conference on Neural Information Processing ral ODEs. This regularization approach is theoretically Systems 2018, NeurIPS 2018, 3-8 December 2018, well-motivated, and encourages neural ODEs to learn well- Montréal, Canada, pp. 6572–6583, 2018. URL http: behaved dynamics. As a consequence, numerical integration //papers.nips.cc/paper/7892-neural- of the learned dynamics is straight forward and relatively ordinary-differential-equations. easy, which means fewer discretizations are needed to solve Chen, T. Q., Behrmann, J., Duvenaud, D., and Jacobsen, the dynamics. In many circumstances, this allows for the re- J. Residual flows for invertible generative modeling. placement of adaptive solvers with fixed grid solvers, which In Wallach, H. M., Larochelle, H., Beygelzimer, can be more efficient during training. This leads to a sub- A., d’Alché-Buc, F., Fox, E. B., and Garnett, R. stantial speed up in training time, while still maintaining (eds.), Advances in Neural Information Processing the same empirical performance, opening the use of neural Systems 32: Annual Conference on Neural Information ODEs to large-scale applications. Processing Systems 2019, NeurIPS 2019, 8-14 Decem- ber 2019, Vancouver, BC, Canada, pp. 9913–9923, 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 AFOSR Grant No. FA9550-18-1-0167, and ONR Grant No. Conference on Learning Representations, ICLR 2017, N00014-18-1-2527. Toulon, France, April 24-26, 2017, Conference Track Pro- ceedings, 2017. 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M., Larochelle, H., Beygelz- imer, A., d’Alché-Buc, F., Fox, E. B., and Garnett, R. (eds.), Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information 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 <> (21) <> (18a) Finally, if we assume that {xi }N i=1 are iid sampled from p, <> (18b) we obtain the empirical objective function <<ρ0 (x) = p>>, (18c) <<ρT (z) = q>>. (18d) <> (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. <> 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 <> (19a) <>, (19b) <>, (19c) <>. (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 <> (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 <> <

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

> 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. <

> <> <> <> <> <> <> 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 1.1: Computing the output values of a discrete convolution. <

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

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

> 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, <> 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, <> 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, <> 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, <> <

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

> 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 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 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 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 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, <> 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): <> 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 <> 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 <> 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 <> <

> 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 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 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 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 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 <> 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 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 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 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 <> <

> 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 <> <

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

> 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 <> 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, <> <

> 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. Bibliography Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G. S., Davis, A., Dean, J., Devin, M.,et al.(2015). Tensorflow: Large- scale machine learning on heterogeneous systems. Software available from tensorflow.org. Bastien, F., Lamblin, P., Pascanu, R., Bergstra, J., Goodfellow, I., Bergeron, A., Bouchard, N., Warde-Farley, D., and Bengio, Y. (2012). Theano: new features and speed improvements.arXiv preprint arXiv:1211.5590. Bergstra, J., Breuleux, O., Bastien, F., Lamblin, P., Pascanu, R., Desjardins, G., Turian, J., Warde-Farley, D., and Bengio, Y. (2010). Theano: A cpu and gpu math compiler in python. InProc. 9th Python in Science Conf, pages 1–7. Boureau, Y., Bach, F., LeCun, Y., and Ponce, J. (2010a). Learning mid-level features for recognition. InProc. 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Visualizing and underst and ing convolu- tional networks. InComputer vision–ECCV 2014, pages 818–833. Springer. Zeiler, M. D., Taylor, G. W., and Fergus, R. (2011). Adaptive deconvolutional networks for mid and high level feature learning. InComputer Vision (ICCV), 2011 IEEE International Conference on, pages 2018–2025. IEEE. <> <> <> <> <> <> 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 <

> 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. <

> 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 <>, a circulant matrix R^2 R^dxd is defined as: <> scratch manner. A recent trend in this direction is to prune redundant, <> non-informative weights in a pre-trained CNN model. For <> example, Srinivas and Babu [21] explored the redundancy <> among neurons, and proposed a data-free pruning method to remove redundant neurons. Hanet al.[22] proposed to reduce <> 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. <> (2) There is also growing interest in training compact CNNs whereS,GandBare random diagonal matrices. 2 with sparsity constraints. Those sparsity constraints are typ- <> 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. <

> <

> 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 <> (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 <> and <> 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 <

> 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- <> (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: <> (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 <> (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 <> (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] <> 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 <> (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 <> is defined as: cific applications, like object detection, which requires s both convolutional and fully connected layers, you can <> (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]. This framework provides a mechanism allowing the algorithm REFERENCES to automatically learn how to exploit structure in the problem [1]A. Krizhevsky, I. Sutskever, and G. Hinton, “Imagenet classification with of interest. Very recently, leveraging reinforcement learning deep convolutional neural networks,” inNIPS, 2012. to efficiently sample the design space and improve the model [2]Y. Taigman, M. Yang, M. Ranzato, and L. 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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. <> <> <> <> <> <> 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 <> , of internal units are <> , and of output units are <> . 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 <> for connections from PEs to the ij Input Layer Dynamical Reservoir Read-out <

> 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 <> 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 <>, (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 <>, (2.2) where <> are the output unit’s nonlinear functions <> (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 <>. 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 <>. 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, <> bij ’s are the input layer weights, andgis the PE nonlinearity. The approxi- mation produced by the TDNN is then <>, (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 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 <>) is to use 2 the Wiener solution given by Haykin (2001): <> 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 <>. Linearizing the system around the current statex(n), one obtains the Jacobian matrix, <>, defined by <> 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 <> is <> Adjoint <<(zI−A)>>. The poles of the transfer function can be obtained by solving <>. 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 <>,Pis the eigenvectormatrixandDisthediagonalmatrixofeigenvalues <> of W.SinceF(n)andD are diagonal, <> is the eigendecomposition of <>. Here, each entry of <>, is an eigenvalue of J. Therefore, <> since <>. <

> 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 <> is given by Renyi (1970): <> 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 <> whereKσ is the kernel function with the kernel sizeσ. Then the Renyi’s quadratic entropy can be estimated by (Principe et al., 2000) <> 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 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) <> The characteristic polynomial of W_i's <>, (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 <> ,<<(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 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 <>. The update equation forbis given by <> 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, <>, optimally trained with the desired signal <>, 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 <> (Jaeger, 2002a). The STM capacity, MC, of the network is defined as <>. 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,<>, that is uniformly distributed over [−0.5, 0.5]. The goal is to train the ESN to generate the delayed versions of the input, <>. 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 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 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 <>. The total numbers of wrong decisions for <> 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 <>, where <>. The input to the system is chosen to be <>. 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. 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(1964).Optimum seeking methods. Upper Saddle River, NJ: Prentice Hall. Williams, R. J., & Zipser, D. (1989). A learning algorithm for continually running fully recurrent neural networks.Neural Computation, 1, 270–280. <> <> <> <> <> <> 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 <>. Let <> 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 <>. In this scenario, we wish to approximate the intractable posterior distribution <> with a fixed form approximate posterior <> by optimizing the variational parameters according to: <> where <> denotes the entropy and <> is known as the evidence-lower-bound (ELBO) or negative variational free energy. As indicated in eq.1, <> naturally decomposes into a minimum cost for communicating the targets <> under the assumption that the sender and receiver agreed on a n=1 prior <> and that the receiver knows the inputs <> 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 <> 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 <>. 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: <>; (2) where <>. 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 <>: <>; (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 <> 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 <>. 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] <>. It turns out that we can recover the log-uniform prior over the weightswif we marginalize over the scales z: <> (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 : <>; (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: <>; (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 <>, hence optimize w.r.t._2 . The <> lower bound under this prior and approximate posterior becomes: <> (7) Under this particular variational posterior parametrization the negative KL-divergence from the conditional prior <> to the approximate posterior <> is independent of z: <> (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 <>, where <>. Now if <> and <> we perform variational inference under the p(W~)p(z)prior with an approximate posterior that has Q the form of <>, with <>, 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, <>, and zi z takes the following form [51]: <>; (9) where <> 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.<>. 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, <>; 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. <> 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: <>; (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: <>; (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: <>; 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): <>; (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: <>; (13) where <> 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: <>; (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 <> 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: <>. 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 <> 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 <> i i and sa sb by exploiting properties of the log-normal distribution 7 and thus forming the implied: <> (17) As a threshold rule for group pruning we will use the negative log-mode 8 of the local log-normal r.v. <> , i.e. prune when <>, with <>. This ignores <> and <>, but nonetheless we found <> 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: <>; (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 <>. Furthermore, similarly to the group normal-Jeffreys approach, we will use the variational posterior marginal variances: <>; (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 <>. 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. <

> 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. <> 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 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. 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> 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 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 <> (21) <> (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 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, <> (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 <>, i.e. no shrinkage/regularization forw, when <<= 1>> we have that <>, i.e.wis exactly zero, and when <<=1>> we have that <>. 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, <>, whereas the normal-Jeffreys / log-uniform prior onzcorresponds <> with <>. 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 <>. 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 <> 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. <>. The negative KL-divergence can be expressed as follows: <> (24) The second term is the entropy of the log-normal distribution which has the following form: <> (25) The first term is the negative cross-entropy of the log-normal approximate posterior from the inverse- Gamma prior: <> (26) <> (27) Since the natural logarithm of a log-normal distribution <> follows a normal distribution <> we have that <>. Furthermore we have that <> then <>, therefore <>. Putting everything together we have that: <> (28) Therefore the negative KL-divergence is: <> (29) Now let p(z) be a Gamma prior, i.e. <>. We have that the negative cross-entropy changes to: <> (30) <> (31) <> (32)2 Therefore the negative KL-divergence is: <> (33) Now, by employing the aforementioned we can express the negative KL-divergence from <> to <> as follows: <> with the KL-divergence for the weight distribution <> given by eq.8 in the main paper. E. Visualizations <

> 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 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 2Convolutional BC-GNJ layerh. <> Algorithm 3 Fully connected BC-GHS layerh. <> Algorithm 4Convolutional BC-GHS layerh. <> <> <> <> <> <> <> 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 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 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 <> 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 <>, while minimizing reconstruction error, we formulate our problem as follow: <> (1) F is Frobenius norm. <> 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: <> (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. <> (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: <>. (4) Here <> (size N.). W is n reshaped W, <>. After obtained result W, it is reshaped back to W. Then we assign <>. Constrain <> satisfies. We alternatively optimize (i) and (ii). In the beginning, W is initialized from the trained model, <>, namely no penalty, and <>. We gradually increase <> For each change of <>, we iterate these two steps until k is stable. After <> satisfies, we obtain the final solution W from <> In practice, we found that the two steps iteration is time consuming. So we apply (i) multiple times, <> until <> satisfies. Then apply (ii) just once, to obtain <> 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: <> (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 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 <> 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 <> our optimization. Here, Y . is the current feature map after <> 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 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 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 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: <>. 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 3. GPU acceleration comparison. We measure forward-pass time per image. Our approach generalizes well on GPU (smaller is better). <> Table 4. Comparisons with training from scratch, under 4. acceleration. Our fine-tuned model outperforms scratch trained counterparts (smaller is better). <> Table 5.Acceleration for Faster R-CNN detection. <> 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 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 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. 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IEEE transactions on pattern analysis and machine intelli.gence, 38(10):1943fi1955, 2016. 1, 2, 3, 5, 6, 7 <> <> <> <> <> <> 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 <> where x is an input vector, <> is obtained from a linear dis- criminant function hi <> with e.g. <>, or <> or <>. A learning algorithm must specify how to select m, the <> 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 <> (used in SVMs), the cross-entropy loss <> (used in logistic regression), and the exponential loss <> (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 <>, 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 <> is added, only <> 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 <> (that would be the residue <> 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 <>. An element of W can be understood as the output weights vector in a neural network. Let < R>> the function that maps any element <> of <>. <> 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 <>. Let < 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 <> be the training set. Let <> be a convex regularization functional that penalizes for the choice of more “complex” parameters (e.g., <> 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: <> (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 <> is a convex function of w. Proof. <> is convex in w and <<>> is convex in w, by the above construction. C is additive in <> 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 <>, with <>, 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 <>, 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 <> the hinge loss, and <> 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 <>. Let us rewrite t=1 this cost function as the constrained optimization problem: <> (C1) <> (C2) Using a standard technique, the above program can be recast as a linear program. Defin- ing <> 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): <> In the case of a finite number Jof base learners, <>. 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: <> it is compact (as a closed bounded interval of <> is a concave function it is even a linear function); <> is convex in <<>> (it is actually linear in <<>>); <> (therefore finite) ( (P)is the largest value of F satisfying the constraints); for every set of n+1 points <>, there exists ~such that <> for <> (one can take <> 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 <>, with <> 1 and <> is the set of soft or hard linear classifiers (depending on choice of s()). Algorithm ConvexNN( D, Q, , s) <> Theorem 4.1.AlgorithmConvexNN Pstops when it reaches the global optimum of <>. 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 <>, then, since Cis convex in the output weights, for any 2(0;1) , we have <>. 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 <> is positive. With o() denoting a quantity such that o()!0 when !0, the difference (w) =XC(w )C(w)can now be written: <> since for i2Ip , thanks to step (7) of the algorithm, we have @C (w) = 0 . Thus the @w inequality <> rewrites into <> which, when !0, yields (note that <> does not depend on ! since j is linear in ): <> (2) i being the optimal classifier chosen in step (5a) or (5c), all hidden units <> verify <> <> <> , 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 <> with respect to u and b, the c’s being in <> 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 <>, <>. It is obvious that those sums only depend on the order of the products <>, <>. 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 <>. Therefore, there are at most as many order changes as there are pairs of different points, i.e., <>. In the case of d=2, we can enumerate all the different angles for which there is a change, namely a1 ;:::;a z with <>. We then need to test at least one <> for each interval a2 i < <>, and also one u for <>, which makes a total of <> possibilities. 2 It is possible to generalize this result in higher dimensions, and as shown in (Marcotte and Savard, 1992), one can achieve <> time. Algorithm 1 Optimal linear classifier search <> 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 <>. Consider a new hidden neuron whose output is hi when the input is xi . Let us also denote <> the vector of output weights between the new hidden neuron and the <> output neurons. The gradient with respect to j is <> with <> the value of the j-th output neuron with input <>. This means that if, for a given j, we have <>, moving Pj away from 0 can only increase the cost. Therefore, the right quantity to consider is <>. We must therefore find <>. 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 <>, 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, <>. 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 <> 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 Chvatal, V. (1983).´ Linear Programming. W.H. Freeman. Delalleau, O., Bengio, Y., and Le Roux, N. (2005). Efficient non-parametric function induction in semi-supervised learning. In Cowell, R. and Ghahramani, Z., editors,Proceedings of AIS- TATS’2005, pages 96–103. Freund, Y. and Schapire, R. E. (1997). A decision theoretic generalization of on-line learning and an application to boosting.Journal of Computer and System Science, 55(1):119–139. Friedman, J. (2001). Greedy function approximation: a gradient boosting machine.Annals of Statis- tics, 29:1180. Hettich, R. and Kortanek, K. (1993). Semi-infinite programming: theory, methods, and applications. SIAM Review, 35(3):380–429. Marcotte, P. and Savard, G. (1992). Novel approaches to the discrimination problem.Zeitschrift fr 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 <> <> <> <> <> <> 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 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 2: Representing the matrix sparsity with relative index. Padding filler zero to prevent overflow. <

> 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: <> (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 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 <> 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 <> into k clusters <>, n%k, so as to minimize the within-cluster sum of squares (WCSS): <> (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 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: <> (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 2: Compression statistics for LeNet-300-100. P: pruning, Q:quantization, H:Huffman coding. <> Table 3: Compression statistics for LeNet-5. P: pruning, Q:quantization, H:Huffman coding. <> 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 5: Compression statistics for VGG-16. P: pruning, Q:quantization, H:Huffman coding. <> 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 6: Accuracy v.s. compression rate under different compression methods. Pruning and quantization works best when combined. <

> 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 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 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 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. <

> 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 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. <

> 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. REFERENCES Anwar, Sajid, Hwang, Kyuyeon, and Sung, Wonyong. Fixed point optimization of deep convolutional neural networks for object recognition. InAcoustics, Speech and Signal Processing (ICASSP), 2015 IEEE International Conference on, pp. 1131–1135. IEEE, 2015. Arora, Sanjeev, Bhaskara, Aditya, Ge, Rong, and Ma, Tengyu. Provable bounds for learning some deep representations. InProceedings of the 31th International Conference on Machine Learning, ICML 2014, pp. 584–592, 2014. BVLC. Caffe model zoo. URLhttp://caffe.berkeleyvision.org/model_zoo. Chen, Wenlin, Wilson, James T., Tyree, Stephen, Weinberger, Kilian Q., and Chen, Yixin. Compress- ing neural networks with the hashing trick.arXiv preprint arXiv:1504.04788, 2015. Collins, Maxwell D and Kohli, Pushmeet. Memory bounded deep convolutional networks.arXiv preprint arXiv:1412.1442, 2014. Denil, Misha, Shakibi, Babak, Dinh, Laurent, de Freitas, Nando, et al. Predicting parameters in deep learning. InAdvances in Neural Information Processing Systems, pp. 2148–2156, 2013. Denton, Emily L, Zaremba, Wojciech, Bruna, Joan, LeCun, Yann, and Fergus, Rob. Exploiting linear structure within convolutional networks for efficient evaluation. InAdvances in Neural Information Processing Systems, pp. 1269–1277, 2014. Girshick, Ross. Fast r-cnn.arXiv preprint arXiv:1504.08083, 2015. Gong, Yunchao, Liu, Liu, Yang, Ming, and Bourdev, Lubomir. Compressing deep convolutional networks using vector quantization.arXiv preprint arXiv:1412.6115, 2014. Han, Song, Pool, Jeff, Tran, John, and Dally, William J. Learning both weights and connections for efficient neural networks. InAdvances in Neural Information Processing Systems, 2015. Han, Song, Liu, Xingyu, Mao, Huizi, Pu, Jing, Pedram, Ardavan, Horowitz, Mark A, and Dally, William J. EIE: Efficient inference engine on compressed deep neural network.arXiv preprint arXiv:1602.01528, 2016. Hanson, Stephen Jose and Pratt, Lorien Y. Comparing biases for minimal network construction with´ back-propagation. InAdvances in neural information processing systems, pp. 177–185, 1989. Hassibi, Babak, Stork, David G, et al. Second order derivatives for network pruning: Optimal brain surgeon.Advances in neural information processing systems, pp. 164–164, 1993. Hwang, Kyuyeon and Sung, Wonyong. Fixed-point feedforward deep neural network design using weights+ 1, 0, and- 1. InSignal Processing Systems (SiPS), 2014 IEEE Workshop on, pp. 1–6. IEEE, 2014. Jia, Yangqing, Shelhamer, Evan, Donahue, Jeff, Karayev, Sergey, Long, Jonathan, Girshick, Ross, Guadarrama, Sergio, and Darrell, Trevor. Caffe: Convolutional architecture for fast feature embedding.arXiv preprint arXiv:1408.5093, 2014. Krizhevsky, Alex, Sutskever, Ilya, and Hinton, Geoffrey E. Imagenet classification with deep convolutional neural networks. InNIPS, pp. 1097–1105, 2012. LeCun, Yann, Denker, John S, Solla, Sara A, Howard, Richard E, and Jackel, Lawrence D. Optimal brain damage. InNIPs, volume 89, 1989. LeCun, Yann, Bottou, Leon, Bengio, Yoshua, and Haffner, Patrick. Gradient-based learning applied to document recognition.Proceedings of the IEEE, 86(11):2278–2324, 1998. Lin, Min, Chen, Qiang, and Yan, Shuicheng. Network in network.arXiv:1312.4400, 2013. NVIDIA. Technical brief: NVIDIA jetson TK1 development kit bringing GPU-accelerated computing to embedded systems, a. URLhttp://www.nvidia.com. NVIDIA. Whitepaper: GPU-based deep learning inference: A performance and power analysis, b. URLhttp://www.nvidia.com/object/white-papers.html. Simonyan, Karen and Zisserman, Andrew. Very deep convolutional networks for large-scale image recognition.arXiv preprint arXiv:1409.1556, 2014. Strom, Nikko. Phoneme probability estimation with dynamic sparsely connected artificial neural¨ networks.The Free Speech Journal, 1(5):1–41, 1997. Szegedy, Christian, Liu, Wei, Jia, Yangqing, Sermanet, Pierre, Reed, Scott, Anguelov, Dragomir, Erhan, Dumitru, Vanhoucke, Vincent, and Rabinovich, Andrew. Going deeper with convolutions. arXiv preprint arXiv:1409.4842, 2014. Van Leeuwen, Jan. On the construction of huffman trees. InICALP, pp. 382–410, 1976. Van Nguyen, Hien, Zhou, Kevin, and Vemulapalli, Raviteja. Cross-domain synthesis of medical images using efficient location-sensitive deep network. InMedical Image Computing and Computer- Assisted Intervention–MICCAI 2015, pp. 677–684. Springer, 2015. Vanhoucke, Vincent, Senior, Andrew, and Mao, Mark Z. Improving the speed of neural networks on cpus. InProc. Deep Learning and Unsupervised Feature Learning NIPS Workshop, 2011. Yang, Zichao, Moczulski, Marcin, Denil, Misha, de Freitas, Nando, Smola, Alex, Song, Le, and Wang, Ziyu. Deep fried convnets.arXiv preprint arXiv:1412.7149, 2014. 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 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. <> <> <> <> <> <> <> 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 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 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- <

> 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 <> of labeled training samples and outputs a classifier <> 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 <> training error. Definition 1 (Effective Model Complexity)TheEffective Model Complexity(EMC) of a training procedureT, with respect to distribution D and parameter <>, is defined as: <> whereError <> 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 <>, if we consider the task of predicting labels based on n samples from D then: Under-parametrized regime.If <> is sufficiently smaller than n, any perturbation of T that increases its effective complexity will decrease the test error. Over-parameterized regime.If <> is sufficiently larger than n, any perturbation of T that increases its effective complexity will decrease the test error. Critically parameterized regime.If <>, 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 <> (and currently heuristically use <>). 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 <>: 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 <> 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 (<>). 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 (<>), 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 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 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 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 <

> 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 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 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 <>, 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 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 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. REFERENCES Madhu S Advani and Andrew M Saxe. High-dimensional dynamics of generalization error in neural networks.arXiv preprint arXiv:1710.03667, 2017. Peter L Bartlett, Philip M Long, Gabor Lugosi, and Alexander Tsigler. Benign overfitting in linear´ regression.arXiv preprint arXiv:1906.11300, 2019. 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> 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 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 <> 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 (<>): At gradient stept, the learning rate is set to <>. We set learning-rate with respect to number of gradient steps, and not epochs, <> in order to allow comparison between experiments with varying train-set sizes. Dynamic drop (<>, 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 <> 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 <> is distributed uniformly on <>. A standard probabilistic model defines an autoregressive factorization of the likelihood: <> Given a set of training samplesS, we define <> In practice,S is not constructed from independent samples from D, but rather by first sampling <<(x,y)>> and then including all <> 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 <> 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 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 <> 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 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 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 17:Epoch-wise double descentfor ResNet18 trained with SGD and multiple learning rate schedules <

> 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 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 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 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 <> 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 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 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 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 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 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 <> and <>, 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 27 E.3 ENSEMBLING <

> 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 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). <> <> <> <> <> <> 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 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 <>, we let the stacked nonlinear layers fit another mapping of <>. The original mapping is recast into <>. 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 <> 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 <> 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., <> (assuming that the input and output are of the same dimensions). So rather than expect stacked layers to approximate <>, we explicitly let these layers approximate a residual function <>. The original function thus becomes <>. 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: <>. (1) Here x and y are the input and output vectors of the lay.ers considered. The function <> represents the residual mapping to be learned. For the example in Fig. 2 that has two layers, <> in which <> denotes ReLU [29] and the biases are omitted for simplifying notations. The operation <> is performed by a shortcut connection and element-wise addition. We adopt the second nonlinearity after the addition (i.e., <>, 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: <>. (2) We can also use a square matrix <> 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: <>, 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 <> 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 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. <

> 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 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 4. Error rates (%) of single-model results on the ImageNet validation set (except fi reported on the test set). <> Table 5. Error rates (%) of ensembles. The top-5 error is on the test set of ImageNet and reported by the test server. <> 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. <

> 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: <

> 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 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 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 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 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. References [1] Y. Bengio, P. Simard, and P. Frasconi. Learning long-term dependencies with gradient descent is difficult. IEEE Transactions on Neural Networks, 5(2):157fi166, 1994. [2] C. M. 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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 9. Object detection improvements on MS COCO using Faster R-CNN and ResNet-101. <> 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 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 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 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. <> <> <> <> <> <> 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: <> (1) <> is the input data, and <> are the predictions. A task-specific cost function <> 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: <> (2) Backward pass with DFA The gradient signal <> of the (i+1)-th layer violates synaptic asymmetry. DFA replaces it with a random projection of the topmost derivative of the loss, <>. 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 <>. With B_i, a fixed random matrix of appropriate shape drawn at initialization for each layers: <> (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 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]. <

> 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. <> 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 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 <>. With the scheduler, the initial learning rate is <> 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 <> 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. 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[87]Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, high- performance deep learning library. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché- Buc, E. Fox, and R. Garnett, editors,Advances in Neural Information Processing Systems 32, pages 8024–8035. Curran Associates, Inc., 2019. URLhttp://papers.neurips.cc/paper/ 9015-pytorch-an-imperative-style-high-performance-deep-learning-library. pdf. 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 <>. 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. <> 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. <> 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. <

> 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. <

> 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. <> 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 <> rather than <> 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 <> to <> in <> steps, as well as over the dropout probability, from <> to in <> steps (where applicable). On DeepFM, this search leads to reduce the learning rate from <> with BP to <> with DFA, but to keep the 0.5 dropout rate. On Deep & Cross, we reduce learning rate from <> to <>, with no dropout in both cases. In AFN, we reduce dropout from <> to <> 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 <> instead of <>, 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 <> instead of <> 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 <>, double the learning rate from <> to <>, 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 <> to <>, although it could be set back to <> with a scheduler. We tried two values of 0.98 and 0.999. We also tried to change <> 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 <>, and <> is the dimension of the queries and keys. The layer performs: <> (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. <> 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 A.2: Sample renders for every scene of the NeRF-Synthetic dataset, for NeRF and NeRF-Dual trained with DFA. <

> Figure A.3: Sample renders for every scene of the LLFF-Real dataset, for NeRF and NeRF-Dual trained with DFA. <> <> <> <> <> <> 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 <>. Once {<>} 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 <>, and C is a local property defined as <>. 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 <> and a high clustering coefficient <>, while for a random graph <> and <> [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 {<>} of physical distances. The number <> can be the space distance between the two vertices or the strength of their possible interaction: we suppose <> can be the geographical distance between stations in transportation systems (in such a case <> 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<>ity of chemical reactions along a direct connection in a biological system. Of course, in the particular case of an unweighted graph <>. 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 {<>} is therefore calculated by using the information contained both in matrix {a_ij} and in matrix {<>}. We have <>, 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 <> in the communication between vertex i and j can be then defined to be inversely proportional to the shortest distance: <>. When there is no path in the graph between i and j, <> and consistently <>. The average efficiency of G can be defined as: <> To normalize E we consider the ideal case G_id in which the graph G has all the <> 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- <>. The efficiency <> <> considered in the following of the paper is always divided by <> and therefore <>. 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 <>, the local efficiency <> 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 <>: small-world networks have high <> and <>, 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). <> is a reasonable approximation of <>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 <>, 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 <> and <> is rewired 2 with probability p and <> and <> are reported in <> as functions of p [16]. For <

> 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 (<>). <

> FIG. 1. FIG.1 Global and local efficiency for the graph example considered in [2]. A regular lattice with <> and <> 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 <>. At p � 0.1, E_glob has almost reached the value of the random graph, though <> has only diminished by very little from the value of 0.82 of the regular lattice. Small worlds have high E_glob and <>. 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 <>. At p . 0.1, E_glob has almost reached the maximum value of 0.4, though <> 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 <> and <>. Of course in such an example the short cuts connect at almost no cost vertices that would otherwise be much farther apart (because <>). 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 <>, 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 <>. 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 <>. 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�s are almost equal). At a local scale, the bigger <> 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 <> 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 (<>). 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�el�emy, and H. E. Stanley, Proc. Natl. Acad. Sci. 97, 11149 (2000). [12] R. Albert, H. Jeong, and A.-L. Barab�asi, Nature 401, 130 (1999). [13] A.-L. Barab�asi and R. Albert, Science 286, 509 (1999). [14] B. Bollob�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�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�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 II. Communication networks. Data on the World Wide Web from http://www.nd.edu/�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 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. <> <> <> <>