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2020-08-06 20:53:44 +00:00
Published as a conference paper at ICLR 2019
THE LOTTERY TICKET HYPOTHESIS :
FINDING SPARSE , T RAINABLE NEURAL NETWORKS
Jonathan Frankle Michael Carbin
MIT CSAIL MIT CSAIL
jfrankle@csail.mit.edu mcarbin@csail.mit.edu
ABSTRACT
arXiv:1803.03635v5 [cs.LG] 4 Mar 2019 Neural network pruning techniques can reduce the parameter counts of trained net-
works by over 90%, decreasing storage requirements and improving computational
performance of inference without compromising accuracy. However, contemporary
experience is that the sparse architectures produced by pruning are difficult to train
from the start, which would similarly improve training performance.
We find that a standard pruning technique naturally uncovers subnetworks whose
initializations made them capable of training effectively. Based on these results, we
articulate thelottery ticket hypothesis: dense, randomly-initialized, feed-forward
networks contain subnetworks (winning tickets) that—when trained in isolation—
reach test accuracy comparable to the original network in a similar number of
iterations. The winning tickets we find have won the initialization lottery: their
connections have initial weights that make training particularly effective.
We present an algorithm to identify winning tickets and a series of experiments
that support the lottery ticket hypothesis and the importance of these fortuitous
initializations. We consistently find winning tickets that are less than 10-20% of
the size of several fully-connected and convolutional feed-forward architectures
for MNIST and CIFAR10. Above this size, the winning tickets that we find learn
faster than the original network and reach higher test accuracy.
1 I NTRODUCTION
Techniques for eliminating unnecessary weights from neural networks (pruning) (LeCun et al., 1990;
Hassibi & Stork, 1993; Han et al., 2015; Li et al., 2016) can reduce parameter-counts by more than
90% without harming accuracy. Doing so decreases the size (Han et al., 2015; Hinton et al., 2015)
or energy consumption (Yang et al., 2017; Molchanov et al., 2016; Luo et al., 2017) of the trained
networks, making inference more efficient. However, if a network can be reduced in size, why do we
not train this smaller architecture instead in the interest of making training more efficient as well?
Contemporary experience is that the architectures uncovered by pruning are harder to train from the
start, reaching lower accuracy than the original networks. 1
Consider an example. In Figure 1, we randomly sample and train subnetworks from a fully-connected
network for MNIST and convolutional networks for CIFAR10. Random sampling models the effect
of the unstructured pruning used by LeCun et al. (1990) and Han et al. (2015). Across various levels
of sparsity, dashed lines trace the iteration of minimum validation loss 2 and the test accuracy at that
iteration. The sparser the network, the slower the learning and the lower the eventual test accuracy.
1 “Training a pruned model from scratch performs worse than retraining a pruned model, which may indicate
the difficulty of training a network with a small capacity.” (Li et al., 2016) “During retraining, it is better to retain
the weights from the initial training phase for the connections that survived pruning than it is to re-initialize the
pruned layers...gradient descent is able to find a good solution when the network is initially trained, but not after
re-initializing some layers and retraining them.” (Han et al., 2015)
2 As a proxy for the speed at which a network learns, we use the iteration at which an early-stopping criterion
would end training. The particular early-stopping criterion we employ throughout this paper is the iteration of
minimum validation loss during training. See Appendix C for more details on this choice.
1 Published as a conference paper at ICLR 2019
Lenet random Conv-6 random Conv-4 random Conv-2 random
30K 1.000
Accuracy at Early-Stop (Test)
Accuracy at Early-Stop (Test) Early-Stop Iteration (Val.) Early-Stop Iteration (Val.) 0.8
40K 0.975
20K
0.950 0.7
20K 10K
0.925 0.6
0 0 0.900
10041.116.97.02.91.20.50.2 10041.217.07.1 3.0 1.3 10041.116.97.02.91.20.50.2 10041.217.07.1 3.0 1.3
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
Figure 1: The iteration at which early-stopping would occur (left) and the test accuracy at that iteration
(right) of the Lenet architecture for MNIST and the Conv-2, Conv-4, and Conv-6 architectures for
CIFAR10 (see Figure 2) when trained starting at various sizes. Dashed lines are randomly sampled
sparse networks (average of ten trials). Solid lines are winning tickets (average of five trials).
In this paper, we show that there consistently exist smaller subnetworks that train from the start and
learn at least as fast as their larger counterparts while reaching similar test accuracy. Solid lines in
Figure 1 show networks that we find. Based on these results, we statethe lottery ticket hypothesis.
The Lottery Ticket Hypothesis.A randomly-initialized, dense neural network contains a subnet-
work that is initialized such that—when trained in isolation—it can match the test accuracy of the
original network after training for at most the same number of iterations.
More formally, consider a dense feed-forward neural networkf(x;)with initial parameters=
0 D . When optimizing with stochastic gradient descent (SGD) on a training set,freaches
minimum validation losslat iterationjwith test accuracya. In addition, consider trainingf(x;m)
with a maskm2 f0;1gjj on its parameters such that its initialization ism0 . When optimizing
with SGD on the same training set (withmfixed),freaches minimum validation lossl0 at iterationj0
with test accuracya0 . The lottery ticket hypothesis predicts that9mfor whichj0 j(commensurate
training time),a0 a(commensurate accuracy), andkmk0 jj(fewer parameters).
We find that a standard pruning technique automatically uncovers such trainable subnetworks from
fully-connected and convolutional feed-forward networks. We designate these trainable subnetworks,
f(x;m0 ),winning tickets, since those that we find have won the initialization lottery with a
combination of weights and connections capable of learning. When their parameters are randomly
reinitialized (f(x;m0 )where0 D 0 0 ), our winning tickets no longer match the performance of
the original network, offering evidence that these smaller networks do not train effectively unless
they are appropriately initialized.
Identifying winning tickets.We identify a winning ticket by training a network and pruning its
smallest-magnitude weights. The remaining, unpruned connections constitute the architecture of the
winning ticket. Unique to our work, each unpruned connection’s value is then reset to its initialization
from original networkbeforeit was trained. This forms our central experiment:
1.Randomly initialize a neural networkf(x;0 )(where0 D ).
2.Train the network forjiterations, arriving at parametersj .
3.Prunep%of the parameters inj , creating a maskm.
4.Reset the remaining parameters to their values in0 , creating the winning ticketf(x;m0 ).
As described, this pruning approach isone-shot: the network is trained once,p%of weights are
pruned, and the surviving weights are reset. However, in this paper, we focus oniterative pruning,
which repeatedly trains, prunes, and resets the network overnrounds; each round prunesp1
n %of the
weights that survive the previous round. Our results show that iterative pruning finds winning tickets
that match the accuracy of the original network at smaller sizes than does one-shot pruning.
Results.We identify winning tickets in a fully-connected architecture for MNIST and convolutional
architectures for CIFAR10 across several optimization strategies (SGD, momentum, and Adam) with
techniques like dropout, weight decay, batchnorm, and residual connections. We use an unstructured
pruning technique, so these winning tickets are sparse. In deeper networks, our pruning-based strategy
for finding winning tickets is sensitive to the learning rate: it requires warmup to find winning tickets
at higher learning rates. The winning tickets we find are 10-20% (or less) of the size of the original
2 Published as a conference paper at ICLR 2019
Network Lenet Conv-2 Conv-4 Conv-6 Resnet-18 VGG-19
64, 64, pool 16, 3x[16, 16] 2x64 pool 2x128
64, 64, pool 128, 128, pool 3x[32, 32] pool, 4x256, pool
Convolutions 64, 64, pool 128, 128, pool 256, 256, pool 3x[64, 64] 4x512, pool, 4x512
FC Layers 300, 100, 10 256, 256, 10 256, 256, 10 256, 256, 10 avg-pool, 10 avg-pool, 10
All/Conv Weights 266K 4.3M / 38K 2.4M / 260K 1.7M / 1.1M 274K / 270K 20.0M
Iterations/Batch 50K / 60 20K / 60 25K / 60 30K / 60 30K / 128 112K / 64
Optimizer Adam 1.2e-3 Adam 2e-4 Adam 3e-4 Adam 3e-4 SGD 0.1-0.01-0.001 Momentum 0.9!
Pruning Rate fc20% conv10% fc20% conv10% fc20% conv15% fc20% conv20% fc0% conv20% fc0%
Figure 2: Architectures tested in this paper. Convolutions are 3x3. Lenet is from LeCun et al. (1998).
Conv-2/4/6 are variants of VGG (Simonyan & Zisserman, 2014). Resnet-18 is from He et al. (2016).
VGG-19 for CIFAR10 is adapted from Liu et al. (2019). Initializations are Gaussian Glorot (Glorot
& Bengio, 2010). Brackets denote residual connections around layers.
network (smaller size). Down to that size, they meet or exceed the original network’s test accuracy
(commensurate accuracy) in at most the same number of iterations (commensurate training time).
When randomly reinitialized, winning tickets perform far worse, meaning structure alone cannot
explain a winning ticket’s success.
The Lottery Ticket Conjecture.Returning to our motivating question, we extend our hypothesis
into an untested conjecture that SGD seeks out and trains a subset of well-initialized weights. Dense,
randomly-initialized networks are easier to train than the sparse networks that result from pruning
because there are more possible subnetworks from which training might recover a winning ticket.
Contributions.
We demonstrate that pruning uncovers trainable subnetworks that reach test accuracy compa-
rable to the original networks from which they derived in a comparable number of iterations.
We show that pruning finds winning tickets that learn faster than the original network while
reaching higher test accuracy and generalizing better.
We propose thelottery ticket hypothesisas a new perspective on the composition of neural
networks to explain these findings.
Implications.In this paper, we empirically study the lottery ticket hypothesis. Now that we have
demonstrated the existence of winning tickets, we hope to exploit this knowledge to:
Improve training performance.Since winning tickets can be trained from the start in isolation, a hope
is that we can design training schemes that search for winning tickets and prune as early as possible.
Design better networks.Winning tickets reveal combinations of sparse architectures and initializations
that are particularly adept at learning. We can take inspiration from winning tickets to design new
architectures and initialization schemes with the same properties that are conducive to learning. We
may even be able to transfer winning tickets discovered for one task to many others.
Improve our theoretical understanding of neural networks.We can study why randomly-initialized
feed-forward networks seem to contain winning tickets and potential implications for theoretical
study of optimization (Du et al., 2019) and generalization (Zhou et al., 2018; Arora et al., 2018).
2 W INNING TICKETS IN FULLY -C ONNECTED NETWORKS
In this Section, we assess the lottery ticket hypothesis as applied to fully-connected networks trained
on MNIST. We use the Lenet-300-100 architecture (LeCun et al., 1998) as described in Figure 2.
We follow the outline from Section 1: after randomly initializing and training a network, we prune
the network and reset the remaining connections to their original initializations. We use a simple
layer-wise pruning heuristic: remove a percentage of the weights with the lowest magnitudes within
each layer (as in Han et al. (2015)). Connections to outputs are pruned at half of the rate of the rest of
the network. We explore other hyperparameters in Appendix G, including learning rates, optimization
strategies (SGD, momentum), initialization schemes, and network sizes.
3 Published as a conference paper at ICLR 2019
100.0 51.3 21.1 7.0 3.6 1.9 51.3 (reinit) 21.1 (reinit)
0.99 0.99 0.99
0.98 0.98 0.98
Test Accuracy Test Accuracy Test Accuracy0.97 0.97 0.97
0.96 0.96 0.96
0.95 0.95 0.95
0.94 0.94 0.94
0 5000 10000 15000 0 5000 10000 15000 0 5000 10000 15000
Training Iterations Training Iterations Training Iterations
Figure 3: Test accuracy on Lenet (iterative pruning) as training proceeds. Each curve is the average
of five trials. Labels arePm —the fraction of weights remaining in the network after pruning. Error
bars are the minimum and maximum of any trial.
Notation.Pm =kmk0 is the sparsity of maskm, e.g.,Pjj m = 25%when 75% of weights are pruned.
Iterative pruning.The winning tickets we find learn faster than the original network. Figure 3 plots
the average test accuracy when training winning tickets iteratively pruned to various extents. Error
bars are the minimum and maximum of five runs. For the first pruning rounds, networks learn faster
and reach higher test accuracy the more they are pruned (left graph in Figure 3). A winning ticket
comprising 51.3% of the weights from the original network (i.e.,Pm = 51:3%) reaches higher test
accuracy faster than the original network but slower than whenPm = 21:1%. WhenPm <21:1%,
learning slows (middle graph). WhenPm = 3:6%, a winning ticket regresses to the performance of
the original network. A similar pattern repeats throughout this paper.
Figure 4a summarizes this behavior for all pruning levels when iteratively pruning by 20% per
iteration (blue). On the left is the iteration at which each network reaches minimum validation loss
(i.e., when the early-stopping criterion would halt training) in relation to the percent of weights
remaining after pruning; in the middle is test accuracy at that iteration. We use the iteration at which
the early-stopping criterion is met as a proxy for how quickly the network learns.
The winning tickets learn faster asPm decreases from 100% to 21%, at which point early-stopping
occurs38%earlier than for the original network. Further pruning causes learning to slow, returning
to the early-stopping performance of the original network whenPm = 3:6%. Test accuracy increases
with pruning, improving by more than 0.3 percentage points whenPm = 13:5%; after this point,
accuracy decreases, returning to the level of the original network whenPm = 3:6%.
At early stopping, training accuracy (Figure 4a, right) increases with pruning in a similar pattern to
test accuracy, seemingly implying that winning tickets optimize more effectively but do not generalize
better. However, at iteration 50,000 (Figure 4b), iteratively-pruned winning tickets still see a test
accuracy improvement of up to 0.35 percentage points in spite of the fact that training accuracy
reaches 100% for nearly all networks (Appendix D, Figure 12). This means that the gap between
training accuracy and test accuracy is smaller for winning tickets, pointing to improved generalization.
Random reinitialization.To measure the importance of a winning ticket’s initialization, we retain
the structure of a winning ticket (i.e., the maskm) but randomly sample a new initialization0 D 0 .
We randomly reinitialize each winning ticket three times, making 15 total per point in Figure 4. We
find that initialization is crucial for the efficacy of a winning ticket. The right graph in Figure 3
shows this experiment for iterative pruning. In addition to the original network and winning tickets at
Pm = 51%and21%are the random reinitialization experiments. Where the winning tickets learn
faster as they are pruned, they learn progressively slower when randomly reinitialized.
The broader results of this experiment are orange line in Figure 4a. Unlike winning tickets, the
reinitialized networks learn increasingly slower than the original network and lose test accuracy after
little pruning. The average reinitialized iterative winning ticket’s test accuracy drops off from the
original accuracy whenPm = 21:1%, compared to 2.9% for the winning ticket. WhenPm = 21%,
the winning ticket reaches minimum validation loss 2.51x faster than when reinitialized and is half a
percentage point more accurate. All networks reach 100% training accuracy forPm 5%; Figure
4 Published as a conference paper at ICLR 2019
Random Reinit (Oneshot) Winning Ticket (Oneshot) Random Reinit (Iterative) Winning Ticket (Iterative)
35K 0.99 1.00
30K
Accuracy at Early-Stop (Train)Accuracy at Early-Stop (Test) 0.98 0.99
Early-Stop Iteration (Val.) 25K 0.97 0.98
20K 0.96 0.97
15K 0.95 0.96
10K 0.94 0.95
5K 0.93 0.94
0 0.92 0.93
10051.326.313.57.03.61.91.00.50.3 10051.326.313.57.03.61.91.00.50.3 10051.326.313.57.03.61.91.00.50.3
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
(a) Early-stopping iteration and accuracy for all pruning methods.
0.99 25K 0.990
Accuracy at Iteration 50K (Test)0.98
Accuracy at Early-Stop (Test) 0.983
Early-Stop Iteration (Val.) 20K
0.97 0.976
0.96 15K 0.969
0.95 10K 0.962
0.94 0.955
5K 0.93 0.948
0.92 0 0.941
10051.326.313.57.03.61.91.00.50.3 10087.575.062.650.137.625.112.7 10087.575.062.650.137.625.112.7
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
(b) Accuracy at end of training. (c) Early-stopping iteration and accuracy for one-shot pruning.
Figure 4: Early-stopping iteration and accuracy of Lenet under one-shot and iterative pruning.
Average of five trials; error bars for the minimum and maximum values. At iteration 50,000, training
accuracy100%forPm 2%for iterative winning tickets (see Appendix D, Figure 12).
4b therefore shows that the winning tickets generalize substantially better than when randomly
reinitialized. This experiment supports the lottery ticket hypothesis’ emphasis on initialization:
the original initialization withstands and benefits from pruning, while the random reinitialization’s
performance immediately suffers and diminishes steadily.
One-shot pruning.Although iterative pruning extracts smaller winning tickets, repeated training
means they are costly to find. One-shot pruning makes it possible to identify winning tickets
without this repeated training. Figure 4c shows the results of one-shot pruning (green) and randomly
reinitializing (red); one-shot pruning does indeed find winning tickets. When67:5%Pm 17:6%,
the average winning tickets reach minimum validation accuracy earlier than the original network.
When95:0%Pm 5:17%, test accuracy is higher than the original network. However, iteratively-
pruned winning tickets learn faster and reach higher test accuracy at smaller network sizes. The
green and red lines in Figure 4c are reproduced on the logarithmic axes of Figure 4a, making this
performance gap clear. Since our goal is to identify the smallest possible winning tickets, we focus
on iterative pruning throughout the rest of the paper.
3 W INNING TICKETS IN CONVOLUTIONAL NETWORKS
Here, we apply the lottery ticket hypothesis to convolutional networks on CIFAR10, increasing
both the complexity of the learning problem and the size of the networks. We consider the Conv-2,
Conv-4, and Conv-6 architectures in Figure 2, which are scaled-down variants of the VGG (Simonyan
& Zisserman, 2014) family. The networks have two, four, or six convolutional layers followed by
two fully-connected layers; max-pooling occurs after every two convolutional layers. The networks
cover a range from near-fully-connected to traditional convolutional networks, with less than 1% of
parameters in convolutional layers in Conv-2 to nearly two thirds in Conv-6. 3
Finding winning tickets. The solid lines in Figure 5 (top) show the iterative lottery ticket experiment
on Conv-2 (blue), Conv-4 (orange), and Conv-6 (green) at the per-layer pruning rates from Figure 2.
The pattern from Lenet in Section 2 repeats: as the network is pruned, it learns faster and test accuracy
rises as compared to the original network. In this case, the results are more pronounced. Winning
3 Appendix H explores other hyperparameters, including learning rates, optimization strategies (SGD, mo-
mentum), and the relative rates at which to prune convolutional and fully-connected layers.
5 Published as a conference paper at ICLR 2019
Conv-2 Conv-2 reinit Conv-4 Conv-4 reinit Conv-6 Conv-6 reinit
20K 0.85
Accuracy at Early-Stop (Test) Early-Stop Iteration (Val.)16K 0.80
12K 0.75
8K 0.70
4K 0.65
0 0.60
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining 1.0
Accuracy at Iteration 20/25/30K (Test) Accuracy at Early-Stop (Train) 0.85
0.9 0.80
0.8 0.75
0.70
0.7
0.65
0.6 0.60
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
Figure 5: Early-stopping iteration and test and training accuracy of the Conv-2/4/6 architectures when
iteratively pruned and when randomly reinitialized. Each solid line is the average of five trials; each
dashed line is the average of fifteen reinitializations (three per trial). The bottom right graph plots test
accuracy of winning tickets at iterations corresponding to the last iteration of training for the original
network (20,000 for Conv-2, 25,000 for Conv-4, and 30,000 for Conv-6); at this iteration, training
accuracy100%forPm 2%for winning tickets (see Appendix D).
tickets reach minimum validation loss at best 3.5x faster for Conv-2 (Pm = 8:8%), 3.5x for Conv-4
(Pm = 9:2%), and 2.5x for Conv-6 (Pm = 15:1%). Test accuracy improves at best 3.4 percentage
points for Conv-2 (Pm = 4:6%), 3.5 for Conv-4 (Pm = 11:1%), and 3.3 for Conv-6 (Pm = 26:4%).
All three networks remain above their original average test accuracy whenPm >2%.
As in Section 2, training accuracy at the early-stopping iteration rises with test accuracy. However, at
iteration 20,000 for Conv-2, 25,000 for Conv-4, and 30,000 for Conv-6 (the iterations corresponding
to the final training iteration for the original network), training accuracy reaches 100% for all networks
whenPm 2%(Appendix D, Figure 13) and winning tickets still maintain higher test accuracy
(Figure 5 bottom right). This means that the gap between test and training accuracy is smaller for
winning tickets, indicating they generalize better.
Random reinitialization.We repeat the random reinitialization experiment from Section 2, which
appears as the dashed lines in Figure 5. These networks again take increasingly longer to learn upon
continued pruning. Just as with Lenet on MNIST (Section 2), test accuracy drops off more quickly
for the random reinitialization experiments. However, unlike Lenet, test accuracy at early-stopping
time initially remains steady and even improves for Conv-2 and Conv-4, indicating that—at moderate
levels of pruning—the structure of the winning tickets alone may lead to better accuracy.
Dropout.Dropout (Srivastava et al., 2014; Hinton et al., 2012) improves accuracy by randomly dis-
abling a fraction of the units (i.e., randomly sampling a subnetwork) on each training iteration. Baldi
& Sadowski (2013) characterize dropout as simultaneously training the ensemble of all subnetworks.
Since the lottery ticket hypothesis suggests that one of these subnetworks comprises a winning ticket,
it is natural to ask whether dropout and our strategy for finding winning tickets interact.
Figure 6 shows the results of training Conv-2, Conv-4, and Conv-6 with a dropout rate of 0.5. Dashed
lines are the network performance without dropout (the solid lines in Figure 5). 4 We continue to find
winning tickets when training with dropout. Dropout increases initial test accuracy (2.1, 3.0, and 2.4
percentage points on average for Conv-2, Conv-4, and Conv-6, respectively), and iterative pruning
increases it further (up to an additional 2.3, 4.6, and 4.7 percentage points, respectively, on average).
Learning becomes faster with iterative pruning as before, but less dramatically in the case of Conv-2.
4 We choose new learning rates for the networks as trained with dropout—see Appendix H.5.
6 Published as a conference paper at ICLR 2019
Conv-2 dropout Conv-2 Conv-4 dropout Conv-4 Conv-6 dropout Conv-6
40K 0.85
35K
Accuracy at Early-Stop (Test) Early-Stop Iteration (Val.) 0.81 30K
25K 0.77
20K
15K 0.73
10K 0.69
5K
0 0.65
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
Figure 6: Early-stopping iteration and test accuracy at early-stopping of Conv-2/4/6 when iteratively
pruned and trained with dropout. The dashed lines are the same networks trained without dropout
(the solid lines in Figure 5). Learning rates are 0.0003 for Conv-2 and 0.0002 for Conv-4 and Conv-6.
rate 0.1 rand reinit rate 0.01 rand reinit rate 0.1, warmup 10K rand reinit
0.94 0.94 0.94
0.92 0.92 0.92
0.90 0.90
Test Accuracy (112K)0.90
Test Accuracy (30K)
Test Accuracy (60K) 0.88 0.88 0.88
0.86 0.86 0.86
0.84 0.84 0.84
0.82 0.82 0.82
0.80 0.80 0.80
10041.016.86.9 2.8 1.2 0.50.20.1 10041.016.86.9 2.8 1.2 0.50.20.1 10041.016.86.9 2.8 1.2 0.50.20.1
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
Figure 7: Test accuracy (at 30K, 60K, and 112K iterations) of VGG-19 when iteratively pruned.
These improvements suggest that our iterative pruning strategy interacts with dropout in a comple-
mentary way. Srivastava et al. (2014) observe that dropout induces sparse activations in the final
network; it is possible that dropout-induced sparsity primes a network to be pruned. If so, dropout
techniques that target weights (Wan et al., 2013) or learn per-weight dropout probabilities (Molchanov
et al., 2017; Louizos et al., 2018) could make winning tickets even easier to find.
4 VGG AND RESNET FOR CIFAR10
Here, we study the lottery ticket hypothesis on networks evocative of the architectures and techniques
used in practice. Specifically, we consider VGG-style deep convolutional networks (VGG-19 on
CIFAR10—Simonyan & Zisserman (2014)) and residual networks (Resnet-18 on CIFAR10—He
et al. (2016)). 5 These networks are trained with batchnorm, weight decay, decreasing learning
rate schedules, and augmented training data. We continue to find winning tickets for all of these
architectures; however, our method for finding them, iterative pruning, is sensitive to the particular
learning rate used. In these experiments, rather than measure early-stopping time (which, for these
larger networks, is entangled with learning rate schedules), we plot accuracy at several moments
during training to illustrate the relative rates at which accuracy improves.
Global pruning.On Lenet and Conv-2/4/6, we prune each layer separately at the same rate. For
Resnet-18 and VGG-19, we modify this strategy slightly: we prune these deeper networksglobally,
removing the lowest-magnitude weights collectively across all convolutional layers. In Appendix
I.1, we find that global pruning identifies smaller winning tickets for Resnet-18 and VGG-19. Our
conjectured explanation for this behavior is as follows: For these deeper networks, some layers have
far more parameters than others. For example, the first two convolutional layers of VGG-19 have
1728 and 36864 parameters, while the last has 2.35 million. When all layers are pruned at the same
rate, these smaller layers become bottlenecks, preventing us from identifying the smallest possible
winning tickets. Global pruning makes it possible to avoid this pitfall.
VGG-19.We study the variant VGG-19 adapted for CIFAR10 by Liu et al. (2019); we use the
the same training regime and hyperparameters: 160 epochs (112,480 iterations) with SGD with
5 See Figure 2 and Appendices I for details on the networks, hyperparameters, and training regimes.
7 Published as a conference paper at ICLR 2019
rate 0.1 rand reinit rate 0.01 rand reinit rate 0.03, warmup 20K rand reinit
0.85 0.85 0.90
Test Accuracy (10K) 0.80
Test Accuracy (20K) 0.80
Test Accuracy (30K)0.88
0.86
0.75 0.75
0.84
0.70 0.70
0.82
0.65 0.65
100 64.441.727.117.811.88.05.5 100 64.441.727.117.811.88.05.5 100 64.441.727.117.811.88.05.5
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
Figure 8: Test accuracy (at 10K, 20K, and 30K iterations) of Resnet-18 when iteratively pruned.
momentum (0.9) and decreasing the learning rate by a factor of 10 at 80 and 120 epochs. This
network has 20 million parameters. Figure 7 shows the results of iterative pruning and random
reinitialization on VGG-19 at two initial learning rates: 0.1 (used in Liu et al. (2019)) and 0.01. At the
higher learning rate, iterative pruning does not find winning tickets, and performance is no better than
when the pruned networks are randomly reinitialized. However, at the lower learning rate, the usual
pattern reemerges, with subnetworks that remain within 1 percentage point of the original accuracy
whilePm 3:5%. (They are not winning tickets, since they do not match the original accuracy.)
When randomly reinitialized, the subnetworks lose accuracy as they are pruned in the same manner as
other experiments throughout this paper. Although these subnetworks learn faster than the unpruned
network early in training (Figure 7 left), this accuracy advantage erodes later in training due to the
lower initial learning rate. However, these subnetworks still learn faster than when reinitialized.
To bridge the gap between the lottery ticket behavior of the lower learning rate and the accuracy
advantage of the higher learning rate, we explore the effect of linear learning rate warmup from 0 to
the initial learning rate overkiterations. Training VGG-19 with warmup (k= 10000, green line) at
learning rate 0.1 improves the test accuracy of the unpruned network by about one percentage point.
Warmup makes it possible to find winning tickets, exceeding this initial accuracy whenPm 1:5%.
Resnet-18.Resnet-18 (He et al., 2016) is a 20 layer convolutional network with residual connections
designed for CIFAR10. It has 271,000 parameters. We train the network for 30,000 iterations with
SGD with momentum (0.9), decreasing the learning rate by a factor of 10 at 20,000 and 25,000
iterations. Figure 8 shows the results of iterative pruning and random reinitialization at learning
rates 0.1 (used in He et al. (2016)) and 0.01. These results largely mirror those of VGG: iterative
pruning finds winning tickets at the lower learning rate but not the higher learning rate. The accuracy
of the best winning tickets at the lower learning rate (89.5% when41:7%Pm 21:9%) falls
short of the original network’s accuracy at the higher learning rate (90.5%). At lower learning rate,
the winning ticket again initially learns faster (left plots of Figure 8), but falls behind the unpruned
network at the higher learning rate later in training (right plot). Winning tickets trained with warmup
close the accuracy gap with the unpruned network at the higher learning rate, reaching 90.5% test
accuracy with learning rate 0.03 (warmup,k= 20000) atPm = 27:1%. For these hyperparameters,
we still find winning tickets whenPm 11:8%. Even with warmup, however, we could not find
hyperparameters for which we could identify winning tickets at the original learning rate, 0.1.
5 D ISCUSSION
Existing work on neural network pruning (e.g., Han et al. (2015)) demonstrates that the function
learned by a neural network can often be represented with fewer parameters. Pruning typically
proceeds by training the original network, removing connections, and further fine-tuning. In effect,
the initial training initializes the weights of the pruned network so that it can learn in isolation during
fine-tuning. We seek to determine if similarly sparse networks can learn from the start. We find that
the architectures studied in this paper reliably contain such trainable subnetworks, and the lottery
ticket hypothesis proposes that this property applies in general. Our empirical study of the existence
and nature of winning tickets invites a number of follow-up questions.
The importance of winning ticket initialization.When randomly reinitialized, a winning ticket
learns more slowly and achieves lower test accuracy, suggesting that initialization is important to
its success. One possible explanation for this behavior is these initial weights are close to their final
8 Published as a conference paper at ICLR 2019
values after training—that in the most extreme case, they are already trained. However, experiments
in Appendix F show the opposite—that the winning ticket weights move further than other weights.
This suggests that the benefit of the initialization is connected to the optimization algorithm, dataset,
and model. For example, the winning ticket initialization might land in a region of the loss landscape
that is particularly amenable to optimization by the chosen optimization algorithm.
Liu et al. (2019) find that pruned networks are indeed trainable when randomly reinitialized, seemingly
contradicting conventional wisdom and our random reinitialization experiments. For example, on
VGG-19 (for which we share the same setup), they find that networks pruned by up to 80% and
randomly reinitialized match the accuracy of the original network. Our experiments in Figure 7
confirm these findings at this level of sparsity (below which Liu et al. do not present data). However,
after further pruning, initialization matters: we find winning tickets when VGG-19 is pruned by up
to 98.5%; when reinitialized, these tickets reach much lower accuracy. We hypothesize that—up
to a certain level of sparsity—highly overparameterized networks can be pruned, reinitialized, and
retrained successfully; however, beyond this point, extremely pruned, less severely overparamterized
networks only maintain accuracy with fortuitous initialization.
The importance of winning ticket structure.The initialization that gives rise to a winning ticket
is arranged in a particular sparse architecture. Since we uncover winning tickets through heavy
use of training data, we hypothesize that the structure of our winning tickets encodes an inductive
bias customized to the learning task at hand. Cohen & Shashua (2016) show that the inductive bias
embedded in the structure of a deep network determines the kinds of data that it can separate more
parameter-efficiently than can a shallow network; although Cohen & Shashua (2016) focus on the
pooling geometry of convolutional networks, a similar effect may be at play with the structure of
winning tickets, allowing them to learn even when heavily pruned.
The improved generalization of winning tickets.We reliably find winning tickets that generalize
better, exceeding the test accuracy of the original network while matching its training accuracy.
Test accuracy increases and then decreases as we prune, forming anOccam’s Hill(Rasmussen &
Ghahramani, 2001) where the original, overparameterized model has too much complexity (perhaps
overfitting) and the extremely pruned model has too little. The conventional view of the relationship
between compression and generalization is that compact hypotheses can better generalize (Rissanen,
1986). Recent theoretical work shows a similar link for neural networks, proving tighter generalization
bounds for networks that can be compressed further (Zhou et al. (2018) for pruning/quantization
and Arora et al. (2018) for noise robustness). The lottery ticket hypothesis offers a complementary
perspective on this relationship—that larger networks might explicitly contain simpler representations.
Implications for neural network optimization.Winning tickets can reach accuracy equivalent to
that of the original, unpruned network, but with significantly fewer parameters. This observation
connects to recent work on the role of overparameterization in neural network training. For example,
Du et al. (2019) prove that sufficiently overparameterized two-layer relu networks (with fixed-size
second layers) trained with SGD converge to global optima. A key question, then, is whether the
presence of a winning ticket is necessary or sufficient for SGD to optimize a neural network to a
particular test accuracy. We conjecture (but do not empirically show) that SGD seeks out and trains a
well-initialized subnetwork. By this logic, overparameterized networks are easier to train because
they have more combinations of subnetworks that are potential winning tickets.
6 L IMITATIONS AND FUTURE WORK
We only consider vision-centric classification tasks on smaller datasets (MNIST, CIFAR10). We do
not investigate larger datasets (namely Imagenet (Russakovsky et al., 2015)): iterative pruning is
computationally intensive, requiring training a network 15 or more times consecutively for multiple
trials. In future work, we intend to explore more efficient methods for finding winning tickets that
will make it possible to study the lottery ticket hypothesis in more resource-intensive settings.
Sparse pruning is our only method for finding winning tickets. Although we reduce parameter-counts,
the resulting architectures are not optimized for modern libraries or hardware. In future work, we
intend to study other pruning methods from the extensive contemporary literature, such as structured
pruning (which would produce networks optimized for contemporary hardware) and non-magnitude
pruning methods (which could produce smaller winning tickets or find them earlier).
9 Published as a conference paper at ICLR 2019
The winning tickets we find have initializations that allow them to match the performance of the
unpruned networks at sizes too small for randomly-initialized networks to do the same. In future
work, we intend to study the properties of these initializations that, in concert with the inductive
biases of the pruned network architectures, make these networks particularly adept at learning.
On deeper networks (Resnet-18 and VGG-19), iterative pruning is unable to find winning tickets
unless we train the networks with learning rate warmup. In future work, we plan to explore why
warmup is necessary and whether other improvements to our scheme for identifying winning tickets
could obviate the need for these hyperparameter modifications.
7 R ELATED WORK
In practice, neural networks tend to be dramatically overparameterized. Distillation (Ba & Caruana,
2014; Hinton et al., 2015) and pruning (LeCun et al., 1990; Han et al., 2015) rely on the fact that
parameters can be reduced while preserving accuracy. Even with sufficient capacity to memorize
training data, networks naturally learn simpler functions (Zhang et al., 2016; Neyshabur et al., 2014;
Arpit et al., 2017). Contemporary experience (Bengio et al., 2006; Hinton et al., 2015; Zhang et al.,
2016) and Figure 1 suggest that overparameterized networks are easier to train. We show that dense
networks contain sparse subnetworks capable of learning on their own starting from their original
initializations. Several other research directions aim to train small or sparse networks.
Prior to training.Squeezenet (Iandola et al., 2016) and MobileNets (Howard et al., 2017) are
specifically engineered image-recognition networks that are an order of magnitude smaller than
standard architectures. Denil et al. (2013) represent weight matrices as products of lower-rank factors.
Li et al. (2018) restrict optimization to a small, randomly-sampled subspace of the parameter space
(meaning all parameters can still be updated); they successfully train networks under this restriction.
We show that one need not even update all parameters to optimize a network, and we find winning
tickets through a principled search process involving pruning. Our contribution to this class of
approaches is to demonstrate that sparse, trainable networks exist within larger networks.
After training.Distillation (Ba & Caruana, 2014; Hinton et al., 2015) trains small networks to mimic
the behavior of large networks; small networks are easier to train in this paradigm. Recent pruning
work compresses large models to run with limited resources (e.g., on mobile devices). Although
pruning is central to our experiments, we study why training needs the overparameterized networks
that make pruning possible. LeCun et al. (1990) and Hassibi & Stork (1993) first explored pruning
based on second derivatives. More recently, Han et al. (2015) showed per-weight magnitude-based
pruning substantially reduces the size of image-recognition networks. Guo et al. (2016) restore
pruned connections as they become relevant again. Han et al. (2017) and Jin et al. (2016) restore
pruned connections to increase network capacity after small weights have been pruned and surviving
weights fine-tuned. Other proposed pruning heuristics include pruning based on activations (Hu et al.,
2016), redundancy (Mariet & Sra, 2016; Srinivas & Babu, 2015a), per-layer second derivatives (Dong
et al., 2017), and energy/computation efficiency (Yang et al., 2017) (e.g., pruning convolutional
filters (Li et al., 2016; Molchanov et al., 2016; Luo et al., 2017) or channels (He et al., 2017)). Cohen
et al. (2016) observe that convolutional filters are sensitive to initialization (“The Filter Lottery”);
throughout training, they randomly reinitialize unimportant filters.
During training.Bellec et al. (2018) train with sparse networks and replace weights that reach
zero with new random connections. Srinivas et al. (2017) and Louizos et al. (2018) learn gating
variables that minimize the number of nonzero parameters. Narang et al. (2017) integrate magnitude-
based pruning into training. Gal & Ghahramani (2016) show that dropout approximates Bayesian
inference in Gaussian processes. Bayesian perspectives on dropout learn dropout probabilities during
training (Gal et al., 2017; Kingma et al., 2015; Srinivas & Babu, 2016). Techniques that learn per-
weight, per-unit (Srinivas & Babu, 2016), or structured dropout probabilities naturally (Molchanov
et al., 2017; Neklyudov et al., 2017) or explicitly (Louizos et al., 2017; Srinivas & Babu, 2015b)
prune and sparsify networks during training as dropout probabilities for some weights reach 1. In
contrast, we train networks at least once to find winning tickets. These techniques might also find
winning tickets, or, by inducing sparsity, might beneficially interact with our methods.
10 Published as a conference paper at ICLR 2019
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13 Published as a conference paper at ICLR 2019
A A CKNOWLEDGMENTS
We gratefully acknowledge IBM, which—through the MIT-IBM Watson AI Lab—contributed the
computational resources necessary to conduct the experiments in this paper. We particularly thank
IBM researchers German Goldszmidt, David Cox, Ian Molloy, and Benjamin Edwards for their
generous contributions of infrastructure, technical support, and feedback. We also wish to thank
Aleksander Madry, Shafi Goldwasser, Ed Felten, David Bieber, Karolina Dziugaite, Daniel Weitzner,
and R. David Edelman for support, feedback, and helpful discussions over the course of this project.
This work was supported in part by the Office of Naval Research (ONR N00014-17-1-2699).
B I TERATIVE PRUNING STRATEGIES
In this Appendix, we examine two different ways of structuring the iterative pruning strategy that we
use throughout the main body of the paper to find winning tickets.
Strategy 1: Iterative pruning with resetting.
1.Randomly initialize a neural networkf(x;m)where=0 andm= 1 jj is a mask.
2.Train the network forjiterations, reaching parametersmj .
3.Prunes%of the parameters, creating an updated maskm0 wherePm0 = (Pm s)%.
4.Reset the weights of the remaining portion of the network to their values in0 . That is, let
=0 .
5.Letm=m0 and repeat steps 2 through 4 until a sufficiently pruned network has been
obtained.
Strategy 2: Iterative pruning with continued training.
1.Randomly initialize a neural networkf(x;m)where=0 andm= 1 jj is a mask.
2.Train the network forjiterations.
3.Prunes%of the parameters, creating an updated maskm0 wherePm0 = (Pm s)%.
4.Letm=m0 and repeat steps 2 and 3 until a sufficiently pruned network has been obtained.
5.Reset the weights of the remaining portion of the network to their values in0 . That is, let
=0 .
The difference between these two strategies is that, after each round of pruning, Strategy 2 retrains
using the already-trained weights, whereas Strategy 1 resets the network weights back to their initial
values before retraining. In both cases, after the network has been sufficiently pruned, its weights are
reset back to the original initializations.
Figures 9 and 10 compare the two strategies on the Lenet and Conv-2/4/6 architectures on the
hyperparameters we select in Appendices G and H. In all cases, the Strategy 1 maintains higher
validation accuracy and faster early-stopping times to smaller network sizes.
C E ARLY STOPPING CRITERION
Throughout this paper, we are interested in measuring the speed at which networks learn. As a proxy
for this quantity, we measure the iteration at which an early-stopping criterion would end training.
The specific criterion we employ is the iteration of minimum validation loss. In this Subsection, we
further explain that criterion.
Validation and test loss follow a pattern where they decrease early in the training process, reach a
minimum, and then begin to increase as the model overfits to the training data. Figure 11 shows an
example of the validation loss as training progresses; these graphs use Lenet, iterative pruning, and
Adam with a learning rate of 0.0012 (the learning rate we will select in the following subsection).
This Figure shows the validation loss corresponding to the test accuracies in Figure 3.
14 Published as a conference paper at ICLR 2019
continued training resetting
50K
0.98
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.)40K
0.96
30K
0.94
20K
0.92 10K
0 0.90
100 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3 100 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3
Percent of Weights Remaining Percent of Weights Remaining
Figure 9: The early-stopping iteration and accuracy at early-stopping of the iterative lottery ticket
experiment on the Lenet architecture when iteratively pruned using the resetting and continued
training strategies.
Conv-2 (continued training) Conv-2 (resetting) Conv-4 (continued training) Conv-4 (resetting) Conv-6 (continued training) Conv-6 (resetting)
0.85 30K
Accuracy at Early-Stop (Val.)0.80
Early-Stop Iteration (Val.)20K 0.75
0.70
10K
0.65
0 0.60
100 56.2 31.9 18.2 10.5 6.1 3.6 2.1 1.2 100 56.2 31.9 18.2 10.5 6.1 3.6 2.1 1.2
Percent of Weights Remaining Percent of Weights Remaining
Figure 10: The early-stopping iteration and accuracy at early-stopping of the iterative lottery ticket
experiment on the Conv-2, Conv-4, and Conv-6 architectures when iteratively pruned using the
resetting and continued training strategies.
100.0 51.3 21.1 7.0 3.6 1.9 51.3 (reinit) 21.1 (reinit)
0.20 0.20 0.20
0.15 0.15 0.15
Validation Loss Validation Loss Validation Loss 0.10 0.10 0.10
0.05 0.05 0.05
0.00 0.00 0.00
0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000
Training Iterations Training Iterations Training Iterations
Figure 11: The validation loss data corresponding to Figure 3, i.e., the validation loss as training
progresses for several different levels of pruning in the iterative pruning experiment. Each line is
the average of five training runs at the same level of iterative pruning; the labels are the percentage
of weights from the original network that remain after pruning. Each network was trained with
Adam at a learning rate of 0.0012. The left graph shows winning tickets that learn increasingly faster
than the original network and reach lower loss. The middle graph shows winning tickets that learn
increasingly slower after the fastest early-stopping time has been reached. The right graph contrasts
the loss of winning tickets to the loss of randomly reinitialized networks.
15 Published as a conference paper at ICLR 2019
Random Reinit (Oneshot) Winning Ticket (Oneshot) Random Reinit (Iterative) Winning Ticket (Iterative)
35K 0.99 1.00
30K
Accuracy at Early-Stop (Train)Accuracy at Early-Stop (Test) 0.98 0.99
Early-Stop Iteration (Val.) 25K 0.97 0.98
20K 0.96 0.97
15K 0.95 0.96
10K 0.94 0.95
5K 0.93 0.94
0 0.92 0.93
10051.326.313.57.03.61.91.00.50.3 10051.326.313.57.03.61.91.00.50.3 10051.326.313.57.03.61.91.00.50.3
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
0.99 1.00
Accuracy at Iteration 50K (Test)0.98 0.99
0.97
Accuracy at Iteration 0.98
0.96
50000 (Train) 0.97
0.95 0.96
0.94 0.95
0.93 0.94
0.92 0.93
10051.326.313.57.03.61.91.00.50.3 10051.326.313.57.03.61.91.00.50.3
Percent of Weights Remaining Percent of Weights Remaining
Figure 12: Figure 4 augmented with a graph of the training accuracy at the end of 50,000 iterations.
In all cases, validation loss initially drops, after which it forms a clear bottom and then begins
increasing again. Our early-stopping criterion identifies this bottom. We consider networks that reach
this moment sooner to have learned “faster.” In support of this notion, the ordering in which each
experiment meets our early-stopping criterion in Figure 3 is the same order in which each experiment
reaches a particular test accuracy threshold in Figure 3.
Throughout this paper, in order to contextualize this learning speed, we also present the test accuracy
of the network at the iteration of minimum validation loss. In the main body of the paper, we find
that winning tickets both arrive at early-stopping sooner and reach higher test accuracy at this point.
D T RAINING ACCURACY FOR LOTTERY TICKET EXPERIMENTS
This Appendix accompanies Figure 4 (the accuracy and early-stopping iterations of Lenet on MNIST
from Section 2) and Figure 5 (the accuracy and early-stopping iterations of Conv-2, Conv-4, and
Conv-6 in Section Section 3) in the main body of the paper. Those figures show the iteration of
early-stopping, the test accuracy at early-stopping, the training accuracy at early-stopping, and the
test accuracy at the end of the training process. However, we did not have space to include a graph
of the training accuracy at the end of the training process, which we assert in the main body of the
paper to be 100% for all but the most heavily pruned networks. In this Appendix, we include those
additional graphs in Figure 12 (corresponding to Figure 4) and Figure 13 (corresponding to Figure 5).
As we describe in the main body of the paper, training accuracy reaches 100% in all cases for all but
the most heavily pruned networks. However, training accuracy remains at 100% longer for winning
tickets than for randomly reinitialized networks.
E C OMPARING RANDOM REINITIALIZATION AND RANDOM SPARSITY
In this Appendix, we aim to understand the relative performance of randomly reinitialized winning
tickets and randomly sparse networks.
1.Networks found via iterative pruning with the original initializations (blue in Figure 14).
2.Networks found via iterative pruning that are randomly reinitialized (orange in Figure 14).
3.Random sparse subnetworks with the same number of parameters as those found via iterative
pruning (green in Figure 14).
16 Published as a conference paper at ICLR 2019
Conv-2 Conv-2 reinit Conv-4 Conv-4 reinit Conv-6 Conv-6 reinit
20K 0.85
Accuracy at Early-Stop (Test) Early-Stop Iteration (Val.) 16K 0.80
12K 0.75
8K 0.70
4K 0.65
0 0.60
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
1.0
Accuracy at Early-Stop (Train)0.9
0.8
0.7
0.6
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining
Accuracy at Iteration 20/25/30K (Train) Accuracy at Iteration 20/25/30K (Test) 0.85 1.0
0.80 0.9
0.75
0.8
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0.7 0.65
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100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
Figure 13: Figure 5 augmented with a graph of the training accuracy at the end of the training process.
17 Published as a conference paper at ICLR 2019
Figure 14 shows this comparison for all of the major experiments in this paper. For the fully-connected
Lenet architecture for MNIST, we find that the randomly reinitialized networks outperform random
sparsity. However, for all of the other, convolutional networks studied in this paper, there is no
significant difference in performance between the two. We hypothesize that the fully-connected
network for MNIST sees these benefits because only certain parts of the MNIST images contain
useful information for classification, meaning connections in some parts of the network will be more
valuable than others. This is less true with convolutions, which are not constrained to any one part of
the input image.
F E XAMINING WINNING TICKETS
In this Appendix, we examine the structure of winning tickets to gain insight into why winning tickets
are able to learn effectively even when so heavily pruned. Throughout this Appendix, we study the
winning tickets from the Lenet architecture trained on MNIST. Unless otherwise stated, we use the
same hyperparameters as in Section 2: glorot initialization and adam optimization.
F.1 W INNING TICKET INITIALIZATION (A DAM )
Figure 15 shows the distributions of winning ticket initializations for four different levels ofPm . To
clarify, these are the distributions of the initial weights of the connections that have survived the
pruning process. The blue, orange, and green lines show the distribution of weights for the first
hidden layer, second hidden layer, and output layer, respectively. The weights are collected from five
different trials of the lottery ticket experiment, but the distributions for each individual trial closely
mirror those aggregated from across all of the trials. The histograms have been normalized so that the
area under each curve is 1.
The left-most graph in Figure 15 shows the initialization distributions for the unpruned networks. We
use glorot initialization, so each of the layers has a different standard deviation. As the network is
pruned, the first hidden layer maintains its distribution. However, the second hidden layer and the
output layer become increasingly bimodal, with peaks on either side of 0. Interestingly, the peaks
are asymmetric: the second hidden layer has more positive initializations remaining than negative
initializations, and the reverse is true for the output layer.
The connections in the second hidden layer and output layer that survive the pruning process tend
to have higher magnitude-initializations. Since we find winning tickets by pruning the connections
with the lowest magnitudes in each layer at theend, the connections with the lowest-magnitude
initializations must still have the lowest-magnitude weights at the end of training. A different trend
holds for the input layer: it maintains its distribution, meaning a connection’s initialization has less
relation to its final weight.
F.2 W INNING TICKET INITIALIZATIONS (SGD)
We also consider the winning tickets obtained when training the network with SGD learning rate 0.8
(selected as described in Appendix G). The bimodal distributions from Figure 15 are present across
all layers (see Figure 16. The connections with the highest-magnitude initializations are more likely
to survive the pruning process, meaning winning ticket initializations have a bimodal distribution
with peaks on opposite sides of 0. Just as with the adam-optimized winning tickets, these peaks are
of different sizes, with the first hidden layer favoring negative initializations and the second hidden
layer and output layer favoring positive initializations. Just as with the adam results, we confirm that
each individual trial evidences the same asymmetry as the aggregate graphs in Figure 16.
F.3 R EINITIALIZING FROM WINNING TICKET INITIALIZATIONS
Considering that the initialization distributions of winning ticketsDm are so different from the
Gaussian distributionDused to initialize the unpruned network, it is natural to ask whether randomly
reinitializing winning tickets fromDm rather thanDwill improve winning ticket performance. We do
not find this to be the case. Figure 17 shows the performance of winning tickets whose initializations
are randomly sampled from the distribution of initializations contained in the winning tickets for
18 Published as a conference paper at ICLR 2019
original initialization random reinitialization random sparsity
Lenet Conv-2
0.75
Test Accuracy at Final Iteration 0.98
Test Accuracy at Final Iteration0.70
0.96
0.650.94
0.92 0.60
0.90 0.55
100 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
Conv-4 Conv-6
0.85 0.85
Test Accuracy at Final Iteration Test Accuracy at Final Iteration0.80 0.80
0.75 0.75
0.70 0.70
0.65 0.65
100 53.5 29.1 16.2 9.2 5.4 3.2 2.0 1.3 100 56.2 31.9 18.2 10.5 6.1 3.6 2.1 1.2
Percent of Weights Remaining Percent of Weights Remaining
Resnet-18 (0.03, warmup 20000) VGG-19 (0.1, warmup 10000)
0.94
Test Accuracy at Final Iteration 0.90
Test Accuracy at Final Iteration0.92
0.88 0.90
0.880.86 0.86
0.84 0.84
0.820.82
0.80
100 64.4 41.7 27.1 17.8 11.8 8.0 5.5 100 41.0 16.8 6.9 2.8 1.2 0.5 0.2 0.1
Percent of Weights Remaining Percent of Weights Remaining
Resnet-18 (0.1) VGG-19 (0.1)
0.94
Test Accuracy at Final Iteration 0.90
Test Accuracy at Final Iteration0.92
0.88 0.90
0.880.86 0.86
0.84 0.84
0.820.82
0.80
100 64.4 41.7 27.1 17.8 11.8 8.0 5.5 100 41.0 16.8 6.9 2.8 1.2 0.5 0.2 0.1
Percent of Weights Remaining Percent of Weights Remaining
Resnet-18 (0.01) VGG-19 (0.01)
0.94
Test Accuracy at Final Iteration0.90
Test Accuracy at Final Iteration 0.92
0.88 0.90
0.880.86 0.86
0.84 0.84
0.820.82
0.80
100 64.4 41.7 27.1 17.8 11.8 8.0 5.5 100 41.0 16.8 6.9 2.8 1.2 0.5 0.2 0.1
Percent of Weights Remaining Percent of Weights Remaining
Figure 14: The test accuracy at the final iteration for each of the networks studied in this paper.
19 Published as a conference paper at ICLR 2019
41.05% Remaining 16.88% Remaining 6.95% Remaining 100.00% Remaining
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Initial Weight Initial Weight Initial Weight Initial Weight
Figure 15: The distribution of initializations in winning tickets pruned to the levels specified in the
titles of each plot. The blue, orange, and green lines show the distributions for the first hidden layer,
second hidden layer, and output layer of the Lenet architecture for MNIST when trained with the
adam optimizer and the hyperparameters used in 2. The distributions have been normalized so that
the area under each curve is 1.
6.95% Remaining 100.00% Remaining 41.05% Remaining 16.88% Remaining
10 12.5 8
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Figure 16: Same as Figure 15 where the network is trained with SGD at rate 0.8.
adam. More concretely, letDm =f(i) jm(i) = 1gbe the set of initializations found in the winning 0 ticket with maskm. We sample a new set of parameters0 D x;m0 ).0 m and train the networkf( 0 We perform this sampling on a per-layer basis. The results of this experiment are in Figure 17.
Winning tickets reinitialized fromDm perform little better than when randomly reinitialized fromD.
We attempted the same experiment with the SGD-trained winning tickets and found similar results.
F.4 P RUNING AT ITERATION 0
One other way of interpreting the graphs of winning ticket initialization distributions is as follows:
weights that begin small stay small, get pruned, and never become part of the winning ticket. (The
only exception to this characterization is the first hidden layer for the adam-trained winning tickets.)
If this is the case, then perhaps low-magnitude weights were never important to the network and can
be pruned from the very beginning. Figure 18 shows the result of attempting this pruning strategy.
Winning tickets selected in this fashion perform even worse than when they are found by iterative
winning ticket reinit sampled from ticket
20K 0.99
Accuracy at Early-Stop (Test) Early-Stop Iteration (Test) 0.98
15K 0.97
0.96 10K 0.95
5K 0.94
0.93
0 0.92
100 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3 100 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3
Percent of Weights Remaining Percent of Weights Remaining
Figure 17: The performance of the winning tickets of the Lenet architecture for MNIST when the
layers are randomly reinitialized from the distribution of initializations contained in the winning
ticket of the corresponding size.
20 Published as a conference paper at ICLR 2019
Iterative Pruning Reinit Prune at Iteration 0
50K 0.990
Accuracy at Early-Stop (Test) Early-Stop Iteration (Test) 45K 0.983
40K 0.976
35K 0.969
30K 0.962
25K 0.955
20K 0.948
15K 0.941
10K 0.934
5K 0.927
0 0.920
100 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3 100 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3
Percent of Weights Remaining Percent of Weights Remaining
Figure 18: The performance of the winning tickets of the Lenet architecture for MNIST when
magnitude pruning is performed before the network is ever trained. The network is subsequently
trained with adam.
(not in ticket) (in ticket)
16.88% Remaining (layer2) 16.88% Remaining (output) 16.88% Remaining (layer1)
5
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|Final Weight - Initial Weight| |Final Weight - Initial Weight| |Final Weight - Initial Weight|
Figure 19: Between the first and last training iteration of the unpruned network, the magnitude by
which weights in the network change. The blue line shows the distribution of magnitudes for weights
that are not in the eventual winning ticket; the orange line shows the distribution of magnitudes for
weights that are in the eventual winning ticket.
pruning and randomly reinitialized. We attempted the same experiment with the SGD-trained winning
tickets and found similar results.
F.5 C OMPARING INITIAL AND FINAL WEIGHTS IN WINNING TICKETS
In this subsection, we consider winning tickets in the context of the larger optimization process. To
do so, we examine the initial and final weights of the unpruned network from which a winning ticket
derives to determine whether weights that will eventually comprise a winning ticket exhibit properties
that distinguish them from the rest of the network.
We consider the magnitude of the difference between initial and final weights. One possible rationale
for the success of winning tickets is that they already happen to be close to the optimum that gradient
descent eventually finds, meaning that winning ticket weights should change by a smaller amount
than the rest of the network. Another possible rationale is that winning tickets are well placed in the
optimization landscape for gradient descent to optimize productively, meaning that winning ticket
weights should change by a larger amount than the rest of the network. Figure 19 shows that winning
ticket weights tend to change by a larger amount then weights in the rest of the network, evidence
that does not support the rationale that winning tickets are already close to the optimum.
It is notable that such a distinction exists between the two distributions. One possible explanation for
this distinction is that the notion of a winning ticket may indeed be a natural part of neural network
optimization. Another is that magnitude-pruning biases the winning tickets we find toward those
containing weights that change in the direction of higher magnitude. Regardless, it offers hope that
winning tickets may be discernible earlier in the training process (or after a single training run),
meaning that there may be more efficient methods for finding winning tickets than iterative pruning.
Figure 20 shows the directions of these changes. It plots the difference between the magnitude of the
final weight and the magnitude of the initial weight, i.e., whether the weight moved toward or away
21 Published as a conference paper at ICLR 2019
(not in ticket) (in ticket)
16.88% Remaining (layer1) 16.88% Remaining (layer2) 16.88% Remaining (output)
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|Final Weight| - |Initial Weight| |Final Weight| - |Initial Weight| |Final Weight| - |Initial Weight|
Figure 20: Between the first and last training iteration of the unpruned network, the magnitude by
which weights move away from 0. The blue line shows the distribution of magnitudes for weights
that are not in the eventual winning ticket; the orange line shows the distribution of magnitudes for
weights that are in the eventual winning ticket.
6.95% Remaining 41.05% Remaining 16.88% Remaining
12.512.5 20
10.010.0 15
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Fraction of Incoming Connections Remaining Fraction of Incoming Connections Remaining Fraction of Incoming Connections Remaining
Figure 21: The fraction of incoming connections that survive the pruning process for each node in
each layer of the Lenet architecture for MNIST as trained with adam.
from 0. In general, winning ticket weights are more likely to increase in magnitude (that is, move
away from 0) than are weights that do not participate in the eventual winning ticket.
F.6 W INNING TICKET CONNECTIVITY
In this Subsection, we study the connectivity of winning tickets. Do some hidden units retain a
large number of incoming connections while others fade away, or does the network retain relatively
even sparsity among all units as it is pruned? We find the latter to be the case when examining the
incoming connectivity of network units: for both adam and SGD, each unit retains a number of
incoming connections approximately in proportion to the amount by which the overall layer has
been pruned. Figures 21 and 22 show the fraction of incoming connections that survive the pruning
process for each node in each layer. Recall that we prune the output layer at half the rate as the rest of
the network, which explains why it has more connectivity than the other layers of the network.
41.05% Remaining 16.88% Remaining 6.95% Remaining
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Fraction of Incoming Connections Remaining Fraction of Incoming Connections Remaining Fraction of Incoming Connections Remaining
Figure 22: Same as Figure 21 where the network is trained with SGD at rate 0.8.
22 Published as a conference paper at ICLR 2019
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6 8 20
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Fraction of Outgoing Connections Remaining Fraction of Outgoing Connections Remaining Fraction of Outgoing Connections Remaining
Figure 23: The fraction of outgoing connections that survive the pruning process for each node in
each layer of the Lenet architecture for MNIST as trained with adam. The blue, orange, and green
lines are the outgoing connections from the input layer, first hidden layer, and second hidden layer,
respectively.
41.05% Remaining 6.95% Remaining 16.88% Remaining
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Figure 24: Same as Figure 23 where the network is trained with SGD at rate 0.8.
However, this is not the case for the outgoing connections. To the contrary, for the adam-trained
networks, certain units retain far more outgoing connections than others (Figure 23). The distributions
are far less smooth than those for the incoming connections, suggesting that certain features are far
more useful to the network than others. This is not unexpected for a fully-connected network on a
task like MNIST, particularly for the input layer: MNIST images contain centered digits, so the pixels
around the edges are not likely to be informative for the network. Indeed, the input layer has two
peaks, one larger peak for input units with a high number of outgoing connections and one smaller
peak for input units with a low number of outgoing connections. Interestingly, the adam-trained
winning tickets develop a much more uneven distribution of outgoing connectivity for the input layer
than does the SGD-trained network (Figure 24).
F.7 A DDING NOISE TO WINNING TICKETS
In this Subsection, we explore the extent to which winning tickets are robust to Gaussian noise added
to their initializations. In the main body of the paper, we find that randomly reinitializing a winning
ticket substantially slows its learning and reduces its eventual test accuracy. In this Subsection,
we study a less extreme way of perturbing a winning ticket. Figure 25 shows the effect of adding
Gaussian noise to the winning ticket initializations. The standard deviation of the noise distribution
of each layer is a multiple of the standard deviation of the layer’s initialization Figure 25 shows noise
distributions with standard deviation0:5,,2, and3. Adding Gaussian noise reduces the test
accuracy of a winning ticket and slows its ability to learn, again demonstrating the importance of
the original initialization. As more noise is added, accuracy decreases. However, winning tickets
are surprisingly robust to noise. Adding noise of0:5barely changes winning ticket accuracy. Even
after adding noise of3, the winning tickets continue to outperform the random reinitialization
experiment.
23 Published as a conference paper at ICLR 2019
winning ticket reinit noise 0.5 noise 1.0 noise 2.0 noise 3.0
20K 0.99
Accuracy at Early-Stop (Test) Early-Stop Iteration (Test) 0.98
15K 0.97
0.96 10K 0.95
5K 0.94
0.93
0 0.92
100 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3 100 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3
Percent of Weights Remaining Percent of Weights Remaining
Figure 25: The performance of the winning tickets of the Lenet architecture for MNIST when
Gaussian noise is added to the initializations. The standard deviations of the noise distributions for
each layer are a multiple of the standard deviations of the initialization distributions; in this Figure,
we consider multiples 0.5, 1, 2, and 3.
G H YPERPARAMETER EXPLORATION FOR FULLY -C ONNECTED NETWORKS
This Appendix accompanies Section 2 of the main paper. It explores the space of hyperparameters
for the Lenet architecture evaluated in Section 2 with two purposes in mind:
1.To explain the hyperparameters selected in the main body of the paper.
2.To evaluate the extent to which the lottery ticket experiment patterns extend to other choices
of hyperparameters.
G.1 E XPERIMENTAL METHODOLOGY
This Section considers the fully-connected Lenet architecture (LeCun et al., 1998), which comprises
two fully-connected hidden layers and a ten unit output layer, on the MNIST dataset. Unless otherwise
stated, the hidden layers have 300 and 100 units each.
The MNIST dataset consists of 60,000 training examples and 10,000 test examples. We randomly
sampled a 5,000-example validation set from the training set and used the remaining 55,000 training
examples as our training set for the rest of the paper (including Section 2). The hyperparameter
selection experiments throughout this Appendix are evaluated using the validation set for determining
both the iteration of early-stopping and the accuracy at early-stopping; the networks in the main body
of this paper (which make use of these hyperparameters) have their accuracy evaluated on the test set.
The training set is presented to the network in mini-batches of 60 examples; at each epoch, the entire
training set is shuffled.
Unless otherwise noted, each line in each graph comprises data from three separate experiments. The
line itself traces the average performance of the experiments and the error bars indicate the minimum
and maximum performance of any one experiment.
Throughout this Appendix, we perform the lottery ticket experiment iteratively with a pruning rate of
20% per iteration (10% for the output layer); we justify the choice of this pruning rate later in this
Appendix. Each layer of the network is pruned independently. On each iteration of the lottery ticket
experiment, the network is trained for 50,000 training iterations regardless of when early-stopping
occurs; in other words, no validation or test data is taken into account during the training process, and
early-stopping times are determined retroactively by examining validation performance. We evaluate
validation and test performance every 100 iterations.
For the main body of the paper, we opt to use the Adam optimizer (Kingma & Ba, 2014) and Gaussian
Glorot initialization (Glorot & Bengio, 2010). Although we can achieve more impressive results on
the lottery ticket experiment with other hyperparameters, we intend these choices to be as generic
as possible in an effort to minimize the extent to which our main results depend on hand-chosen
hyperparameters. In this Appendix, we select the learning rate for Adam that we use in the main body
of the paper.
In addition, we consider a wide range of other hyperparameters, including other optimization
algorithms (SGD with and without momentum), initialization strategies (Gaussian distributions
24 Published as a conference paper at ICLR 2019
2.5e-05 5e-05 0.0002 0.0008 0.0012 0.002 0.0032 0.0064
50K 0.99000
45K 0.98286
40K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 35K 0.97571
30K 0.96857
25K
20K 0.96143
15K 0.95429
10K
0.94714 5K
0 0.94000
100.0 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3 100.0 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3
Percent of Weights Remaining Percent of Weights Remaining
Figure 26: The early-stopping iteration and validation accuracy at that iteration of the iterative lottery
ticket experiment on the Lenet architecture trained with MNIST using the Adam optimizer at various
learning rates. Each line represents a different learning rate.
0.003125 0.00625 0.0125 0.025 0.1 0.4 0.8 1.2
50K 0.99000
45K 0.98286
40K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 35K 0.97571
30K 0.96857
25K
20K 0.96143
15K 0.95429
10K
0.94714 5K
0 0.94000
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Percent of Weights Remaining Percent of Weights Remaining
Figure 27: The early-stopping iteration and validation accuracy at that iteration of the iterative lottery
ticket experiment on the Lenet architecture trained with MNIST using stochastic gradient descent at
various learning rates.
with various standard deviations), network sizes (larger and smaller hidden layers), and pruning
strategies (faster and slower pruning rates). In each experiment, we vary the chosen hyperparameter
while keeping all others at their default values (Adam with the chosen learning rate, Gaussian Glorot
initialization, hidden layers with 300 and 100 units). The data presented in this appendix was collected
by training variations of the Lenet architecture more than 3,000 times.
G.2 L EARNING RATE
In this Subsection, we perform the lottery ticket experiment on the Lenet architecture as optimized
with Adam, SGD, and SGD with momentum at various learning rates.
Here, we select the learning rate that we use for Adam in the main body of the paper. Our criteria for
selecting the learning rate are as follows:
1.On the unpruned network, it should minimize training iterations necessary to reach early-
stopping and maximize validation accuracy at that iteration. That is, it should be a reasonable
hyperparameter for optimizing the unpruned network even if we are not running the lottery
ticket experiment.
2. When running the iterative lottery ticket experiment, it should make it possible to match
the early-stopping iteration and accuracy of the original network with as few parameters as
possible.
3.Of those options that meet (1) and (2), it should be on the conservative (slow) side so that it is
more likely to productively optimize heavily pruned networks under a variety of conditions
with a variety of hyperparameters.
25 Published as a conference paper at ICLR 2019
0.003125 0.0125 0.025 0.05 0.1 0.2 0.4
50K 0.99000
45K 0.98286
40K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 35K 0.97571
30K 0.96857
25K
20K 0.96143
15K 0.95429
10K
0.94714 5K
0 0.94000
100.0 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3 100.0 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3
Percent of Weights Remaining Percent of Weights Remaining
Figure 28: The early-stopping iteration and validation accuracy at that iteration of the iterative lottery
ticket experiment on the Lenet architecture trained with MNIST using stochastic gradient descent
with momentum (0.9) at various learning rates.
Figure 26 shows the early-stopping iteration and validation accuracy at that iteration of performing
the iterative lottery ticket experiment with the Lenet architecture optimized with Adam at various
learning rates. According to the graph on the right of Figure 26, several learning rates between 0.0002
and 0.002 achieve similar levels of validation accuracy on the original network and maintain that
performance to similar levels as the network is pruned. Of those learning rates, 0.0012 and 0.002
produce the fastest early-stopping times and maintain them to the smallest network sizes. We choose
0.0012 due to its higher validation accuracy on the unpruned network and in consideration of criterion
(3) above.
We note that, across all of these learning rates, the lottery ticket pattern (in which learning becomes
faster and validation accuracy increases with iterative pruning) remains present. Even for those
learning rates that did not satisfy the early-stopping criterion within 50,000 iterations (2.5e-05 and
0.0064) still showed accuracy improvements with pruning.
G.3 O THER OPTIMIZATION ALGORITHMS
G.3.1 SGD
Here, we explore the behavior of the lottery ticket experiment when the network is optimized with
stochastic gradient descent (SGD) at various learning rates. The results of doing so appear in Figure
27. The lottery ticket pattern appears across all learning rates, including those that fail to satisfy the
early-stopping criterion within 50,000 iterations. SGD learning rates 0.4 and 0.8 reach early-stopping
in a similar number of iterations as the best Adam learning rates (0.0012 and 0.002) but maintain
this performance when the network has been pruned further (to less than 1% of its original size for
SGD vs. about 3.6% of the original size for Adam). Likewise, on pruned networks, these SGD
learning rates achieve equivalent accuracy to the best Adam learning rates, and they maintain that
high accuracy when the network is pruned as much as the Adam learning rates.
G.3.2 M OMENTUM
Here, we explore the behavior of the lottery ticket experiment when the network is optimized with
SGD with momentum (0.9) at various learning rates. The results of doing so appear in Figure 28.
Once again, the lottery ticket pattern appears across all learning rates, with learning rates between
0.025 and 0.1 maintaining high validation accuracy and faster learning for the longest number of
pruning iterations. Learning rate 0.025 achieves the highest validation accuracy on the unpruned
network; however, its validation accuracy never increases as it is pruned, instead decreasing gradually,
and higher learning rates reach early-stopping faster.
G.4 I TERATIVE PRUNING RATE
When running the iterative lottery ticket experiment on Lenet, we prune each layer of the network
separately at a particular rate. That is, after training the network, we prunek%of the weights in
26 Published as a conference paper at ICLR 2019
0.1 0.4 0.6 0.8 0.2
50K 0.99000
45K 0.98286
40K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 35K 0.97571
30K 0.96857
25K
20K 0.96143
15K 0.95429
10K
0.94714 5K
0 0.94000
100.0 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3 100.0 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3
Percent of Weights Remaining Percent of Weights Remaining
Figure 29: The early-stopping iteration and validation accuracy at that iteration of the iterative lottery
ticket experiment when pruned at different rates. Each line represents a differentpruning rate—the
percentage of lowest-magnitude weights that are pruned from each layer after each training iteration.
0.0125 0.025 0.05 0.1 0.2 0.4 glorot
50K 0.99000
45K 0.98286
40K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 35K 0.97571
30K 0.96857
25K
20K 0.96143
15K 0.95429
10K
0.94714 5K
0 0.94000
100.0 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3 100.0 51.3 26.3 13.5 7.0 3.6 1.9 1.0 0.5 0.3
Percent of Weights Remaining Percent of Weights Remaining
Figure 30: The early-stopping iteration and validation accuracy at that iteration of the iterative lottery
ticket experiment initialized with Gaussian distributions with various standard deviations. Each line
is a different standard deviation for a Gaussian distribution centered at 0.
each layer ( k %of the weights in the output layer) before resetting the weights to their original
initializations and training again. In the main body of the paper, we find that iterative pruning finds 2
smaller winning tickets than one-shot pruning, indicating that pruning too much of the network at
once diminishes performance. Here, we explore different values ofk.
Figure 29 shows the effect of the amount of the network pruned on each pruning iteration on early-
stopping time and validation accuracy. There is a tangible difference in learning speed and validation
accuracy at early-stopping between the lowest pruning rates (0.1 and 0.2) and higher pruning rates (0.4
and above). The lowest pruning rates reach higher validation accuracy and maintain that validation
accuracy to smaller network sizes; they also maintain fast early-stopping times to smaller network
sizes. For the experiments throughout the main body of the paper and this Appendix, we use a
pruning rate of 0.2, which maintains much of the accuracy and learning speed of 0.1 while reducing
the number of training iterations necessary to get to smaller network sizes.
In all of the Lenet experiments, we prune the output layer at half the rate of the rest of the network.
Since the output layer is so small (1,000 weights out of 266,000 for the overall Lenet architecture),
we found that pruning it reaches a point of diminishing returns much earlier the other layers.
G.5 I NITIALIZATION DISTRIBUTION
To this point, we have considered only a Gaussian Glorot (Glorot & Bengio, 2010) initialization
scheme for the network. Figure 30 performs the lottery ticket experiment while initializing the Lenet
architecture from Gaussian distributions with a variety of standard deviations. The networks were
optimized with Adam at the learning rate chosen earlier. The lottery ticket pattern continues to appear
across all standard deviations. When initialized from a Gaussian distribution with standard deviation
27 Published as a conference paper at ICLR 2019
0.1, the Lenet architecture maintained high validation accuracy and low early-stopping times for the
longest, approximately matching the performance of the Glorot-initialized network.
G.6 N ETWORK SIZE
25,9 50,17 100,34 150,50 200,67 300,100 400,134 500,167 700,233 900,300
50K
45K
40K
Early-Stop Iteration (Val.) 35K
30K
25K
20K
15K
10K
5K
0
978.6 501.7 257.3 132.1 67.9 34.9 18.0 9.3 4.8 2.5 1.3
Thousands of Weights Remaining
0.99000
0.98286
Accuracy at Early-Stop (Val.)0.97571
0.96857
0.96143
0.95429
0.94714
0.94000
978.6 501.7 257.3 132.1 67.9 34.9 18.0 9.3 4.8 2.5 1.3
Thousands of Weights Remaining
Figure 31: The early-stopping iteration and validation accuracy at at that iteration of the iterative
lottery ticket experiment on the Lenet architecture with various layer sizes. The label for each line
is the size of the first and second hidden layers of the network. All networks had Gaussian Glorot
initialization and were optimized with Adam (learning rate 0.0012). Note that the x-axis of this plot
charts the number ofweightsremaining, while all other graphs in this section have charted thepercent
of weights remaining.
Throughout this section, we have considered the Lenet architecture with 300 units in the first hidden
layer and 100 units in the second hidden layer. Figure 31 shows the early-stopping iterations and
validation accuracy at that iteration of the Lenet architecture with several other layer sizes. All
networks we tested maintain the 3:1 ratio between units in the first hidden layer and units in the
second hidden layer.
The lottery ticket hypothesis naturally invites a collection of questions related to network size. Gener-
alizing, those questions tend to take the following form: according to the lottery ticket hypothesis, do
larger networks, which contain more subnetworks, find “better” winning tickets? In line with the
generality of this question, there are several different answers.
If we evaluate a winning ticket by the accuracy it achieves, then larger networks do find better
winning tickets. The right graph in Figure 31 shows that, for any particular number of weights (that
is, any particular point on the x-axis), winning tickets derived from initially larger networks reach
higher accuracy. Put another way, in terms of accuracy, the lines are approximately arranged from
bottom to top in increasing order of network size. It is possible that, since larger networks have
more subnetworks, gradient descent found a better winning ticket. Alternatively, the initially larger
networks have more units even when pruned to the same number of weights as smaller networks,
meaning they are able to contain sparse subnetwork configurations that cannot be expressed by
initially smaller networks.
28 Published as a conference paper at ICLR 2019
If we evaluate a winning ticket by the time necessary for it to reach early-stopping, then larger
networks have less of an advantage. The left graph in Figure 31 shows that, in general, early-stopping
iterations do not vary greatly between networks of different initial sizes that have been pruned to the
same number of weights. Upon exceedingly close inspection, winning tickets derived from initially
larger networks tend to learn marginally faster than winning tickets derived from initially smaller
networks, but these differences are slight.
If we evaluate a winning ticket by the size at which it returns to the same accuracy as the original
network, the large networks do not have an advantage. Regardless of the initial network size, the
right graph in Figure 31 shows that winning tickets return to the accuracy of the original network
when they are pruned to between about 9,000 and 15,000 weights.
H H YPERPARAMETER EXPLORATION FOR CONVOLUTIONAL NETWORKS
This Appendix accompanies Sections 3 of the main paper. It explores the space of optimization
algorithms and hyperparameters for the Conv-2, Conv-4, and Conv-6 architectures evaluated in
Section 3 with the same two purposes as Appendix G: explaining the hyperparameters used in the main
body of the paper and evaluating the lottery ticket experiment on other choices of hyperparameters.
H.1 E XPERIMENTAL METHODOLOGY
The Conv-2, Conv-4, and Conv-6 architectures are variants of the VGG (Simonyan & Zisserman,
2014) network architecture scaled down for the CIFAR10 (Krizhevsky & Hinton, 2009) dataset. Like
VGG, the networks consist of a series of modules. Each module has two layers of 3x3 convolutional
filters followed by a maxpool layer with stride 2. After all of the modules are two fully-connected
layers of size 256 followed by an output layer of size 10; in VGG, the fully-connected layers are of
size 4096 and the output layer is of size 1000. Like VGG, the first module has 64 convolutions in
each layer, the second has 128, the third has 256, etc. The Conv-2, Conv-4, and Conv-6 architectures
have 1, 2, and 3 modules, respectively.
The CIFAR10 dataset consists of 50,000 32x32 color (three-channel) training examples and 10,000
test examples. We randomly sampled a 5,000-example validation set from the training set and used the
remaining 45,000 training examples as our training set for the rest of the paper. The hyperparameter
selection experiments throughout this Appendix are evaluated on the validation set, and the examples
in the main body of this paper (which make use of these hyperparameters) are evaluated on test set.
The training set is presented to the network in mini-batches of 60 examples; at each epoch, the entire
training set is shuffled.
The Conv-2, Conv-4, and Conv-6 networks are initialized with Gaussian Glorot initialization (Glorot
& Bengio, 2010) and are trained for the number of iterations specified in Figure 2. The number
of training iterations was selected such that heavily-pruned networks could still train in the time
provided. On dropout experiments, the number of training iterations is tripled to provide enough time
for the dropout-regularized networks to train. We optimize these networks with Adam, and select the
learning rate for each network in this Appendix.
As with the MNIST experiments, validation and test performance is only considered retroactively
and has no effect on the progression of the lottery ticket experiments. We measure validation and test
loss and accuracy every 100 training iterations.
Each line in each graph of this section represents the average of three separate experiments, with
error bars indicating the minimum and maximum value that any experiment took on at that point.
(Experiments in the main body of the paper are conducted five times.)
We allow convolutional layers and fully-connected layers to be pruned at different rates; we select
those rates for each network in this Appendix. The output layer is pruned at half of the rate of the
fully-connected layers for the reasons described in Appendix G.
H.2 L EARNING RATE
In this Subsection, we perform the lottery ticket experiment on the the Conv-2, Conv-4, and Conv-6
architectures as optimized with Adam at various learning rates.
29 Published as a conference paper at ICLR 2019
0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008
10K
0.7
8K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 6K 0.6
4K
0.5
2K
0
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
15K 0.80
12K
Accuracy at Early-Stop (Val.)0.75
Early-Stop Iteration (Val.) 10K
0.70
7K
0.65
5K
0.60
2K
0.55
0
100 53.5 29.1 16.2 9.2 5.4 3.2 2.0 1.3 100 53.5 29.1 16.2 9.2 5.4 3.2 2.0 1.3
Percent of Weights Remaining Percent of Weights Remaining
20K
0.80
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 15K
0.75
10K
0.70
5K 0.65
0 0.60
100 61.8 39.4 25.8 17.3 11.9 8.3 5.8 4.1 3.0 2.1 100 61.8 39.4 25.8 17.3 11.9 8.3 5.8 4.1 3.0 2.1
Percent of Weights Remaining Percent of Weights Remaining
Figure 32: The early-stopping iteration and validation accuracy at that iteration of the iterative lottery
ticket experiment on the Conv-2 (top), Conv-4 (middle), and Conv-6 (bottom) architectures trained
using the Adam optimizer at various learning rates. Each line represents a different learning rate.
30 Published as a conference paper at ICLR 2019
Here, we select the learning rate that we use for Adam in the main body of the paper. Our criteria
for selecting the learning rate are the same as in Appendix G: minimizing training iterations and
maximizing accuracy at early-stopping, finding winning tickets containing as few parameters as
possible, and remaining conservative enough to apply to a range of other experiments.
Figure 32 shows the results of performing the iterative lottery ticket experiment on the Conv-2 (top),
Conv-4 (middle), and Conv-6 (bottom) architectures. Since we have not yet selected the pruning rates
for each network, we temporarily pruned fully-connected layers at 20% per iteration, convolutional
layers at 10% per iteration, and the output layer at 10% per iteration; we explore this part of the
hyperparameter space in a later subsection.
For Conv-2, we select a learning rate of 0.0002, which has the highest initial validation accuracy,
maintains both high validation accuracy and low early-stopping times for the among the longest,
and reaches the fastest early-stopping times. This learning rate also leads to a 3.3 percentage point
improvement in validation accuracy when the network is pruned to 3% of its original size. Other
learning rates, such 0.0004, have lower initial validation accuracy (65.2% vs 67.6%) but eventually
reach higher absolute levels of validation accuracy (71.7%, a 6.5 percentage point increase, vs. 70.9%,
a 3.3 percentage point increase). However, learning rate 0.0002 shows the highest proportional
decrease in early-stopping times: 4.8x (when pruned to 8.8% of the original network size).
For Conv-4, we select learning rate 0.0003, which has among the highest initial validation accuracy,
maintains high validation accuracy and fast early-stopping times when pruned by among the most,
and balances improvements in validation accuracy (3.7 percentage point improvement to 78.6%
when 5.4% of weights remain) and improvements in early-stopping time (4.27x when 11.1% of
weights remain). Other learning rates reach higher validation accuracy (0.0004—3.6 percentage point
improvement to 79.1% accuracy when 5.4% of weights remain) or show better improvements in
early-stopping times (0.0002—5.1x faster when 9.2% of weights remain) but not both.
For Conv-6, we also select learning rate 0.0003 for similar reasons to those provided for Conv-4.
Validation accuracy improves by 2.4 percentage points to 81.5% when 9.31% of weights remain
and early-stopping times improve by 2.61x when pruned to 11.9%. Learning rate 0.0004 reaches
high final validation accuracy (81.9%, an increase of 2.7 percentage points, when 15.2% of weights
remain) but with smaller improvements in early-stopping times, and learning rate 0.0002 shows
greater improvements in early-stopping times (6.26x when 19.7% of weights remain) but reaches
lower overall validation accuracy.
We note that, across nearly all combinations of learning rates, the lottery ticket pattern—where
early-stopping times were maintain or decreased and validation accuracy was maintained or increased
during the course of the lottery ticket experiment—continues to hold. This pattern fails to hold at
the very highest learning rates: early-stopping times decreased only briefly (in the case of Conv-2 or
Conv-4) or not at all (in the case of Conv-6), and accuracy increased only briefly (in the case of all
three networks). This pattern is similar to that which we observe in Section 4: at the highest learning
rates, our iterative pruning algorithm fails to find winning tickets.
H.3 O THER OPTIMIZATION ALGORITHMS
H.3.1 SGD
Here, we explore the behavior of the lottery ticket experiment when the Conv-2, Conv-4, and Conv-6
networks are optimized with stochastic gradient descent (SGD) at various learning rates. The results
of doing so appear in Figure 33. In general, these networks—particularly Conv-2 and Conv-4—
proved challenging to train with SGD and Glorot initialization. As Figure 33 reflects, we could not
find SGD learning rates for which the unpruned networks matched the validation accuracy of the
same networks when trained with Adam; at best, the SGD-trained unpruned networks were typically
2-3 percentage points less accurate. At higher learning rates than those in Figure 32, gradients tended
to explode when training the unpruned network; at lower learning rates, the networks often failed to
learn at all.
At all of the learning rates depicted, we found winning tickets. In all cases, early-stopping times
initially decreased with pruning before eventually increasing again, just as in other lottery ticket
experiments. The Conv-6 network also exhibited the same accuracy patterns as other experiments,
with validation accuracy initially increasing with pruning before eventually decreasing again.
31 Published as a conference paper at ICLR 2019
0.0001 0.0005 0.0008 0.001 0.0015 0.0025
0.70 20K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 0.65 15K
10K 0.60
5K 0.55
0 0.50
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
0.001 0.002 0.003 0.005 0.01
25K
0.74
20K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 0.72
15K
0.70
10K
0.68
5K
0.66
0
100 53.5 29.1 16.2 9.2 5.4 3.2 2.0 1.3 100 53.5 29.1 16.2 9.2 5.4 3.2 2.0 1.3
Percent of Weights Remaining Percent of Weights Remaining
0.0025 0.005 0.01 0.02 0.025 0.03 0.035
0.80 30K
0.78
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 20K 0.76
0.74
10K
0.72
0 0.70
100 61.8 39.4 25.8 17.3 11.9 8.3 5.8 4.1 3.0 2.1 100 61.8 39.4 25.8 17.3 11.9 8.3 5.8 4.1 3.0 2.1
Percent of Weights Remaining Percent of Weights Remaining
Figure 33: The early-stopping iteration and validation accuracy at that iteration of the iterative lottery
ticket experiment on the Conv-2 (top), Conv-4 (middle), and Conv-6 (bottom) architectures trained
using SGD at various learning rates. Each line represents a different learning rate. The legend for
each pair of graphs is above the graphs.
32 Published as a conference paper at ICLR 2019
However, the Conv-2 and Conv-4 architectures exhibited a different validation accuracy pattern
from other experiments in this paper. Accuracy initially declined with pruning before rising as
the network was further pruned; it eventually matched or surpassed the accuracy of the unpruned
network. When they eventually did surpass the accuracy of the original network, the pruned networks
reached early-stopping in about the same or fewer iterations than the original network, constituting
a winning ticket by our definition. Interestingly, this pattern also appeared for Conv-6 networks at
slower SGD learning rates, suggesting that faster learning rates for Conv-2 and Conv-4 than those in
Figure 32 might cause the usual lottery ticket accuracy pattern to reemerge. Unfortunately, at these
higher learning rates, gradients exploded on the unpruned networks, preventing us from running these
experiments.
H.3.2 M OMENTUM
Here, we explore the behavior of the lottery ticket experiment when the network is optimized with
SGD with momentum (0.9) at various learning rates. The results of doing so appear in Figure 34.
In general, the lottery ticket pattern continues to apply, with early-stopping times decreasing and
accuracy increasing as the networks are pruned. However, there were two exceptions to this pattern:
1.At the very lowest learning rates (e.g., learning rate 0.001 for Conv-4 and all but the highest
learning rate for Conv-2), accuracy initially decreased before increasing to higher levels
than reached by the unpruned network; this is the same pattern we observed when training
these networks with SGD.
2.At the very highest learning rates (e.g., learning rates 0.005 and 0.008 for Conv-2 and Conv-
4), early-stopping times never decreased and instead remained stable before increasing; this
is the same pattern we observed for the highest learning rates when training with Adam.
H.4 I TERATIVE PRUNING RATE
For the convolutional network architectures, we select different pruning rates for convolutional and
fully-connected layers. In the Conv-2 and Conv-4 architectures, convolutional parameters make up a
relatively small portion of the overall number of parameters in the models. By pruning convolutions
more slowly, we are likely to be able to prune the model further while maintaining performance.
In other words, we hypothesize that, if all layers were pruned evenly, convolutional layers would
become a bottleneck that would make it more difficult to find lower parameter-count models that are
still able to learn. For Conv-6, the opposite may be true: since nearly two thirds of its parameters are
in convolutional layers, pruning fully-connected layers could become the bottleneck.
Our criterion for selecting hyperparameters in this section is to find a combination of pruning rates
that allows networks to reach the lowest possible parameter-counts while maintaining validation
accuracy at or above the original accuracy and early-stopping times at or below that for the original
network.
Figure 35 shows the results of performing the iterative lottery ticket experiment on Conv-2 (top),
Conv-4 (middle), and Conv-6 (bottom) with different combinations of pruning rates.
According to our criteria, we select an iterative convolutional pruning rate of 10% for Conv-2, 10% for
Conv-4, and 15% for Conv-6. For each network, any rate between 10% and 20% seemed reasonable.
Across all convolutional pruning rates, the lottery ticket pattern continued to appear.
H.5 L EARNING RATES (D ROPOUT )
In order to train the Conv-2, Conv-4, and Conv-6 architectures with dropout, we repeated the exercise
from Section H.2 to select appropriate learning rates. Figure 32 shows the results of performing
the iterative lottery ticket experiment on Conv-2 (top), Conv-4 (middle), and Conv-6 (bottom) with
dropout and Adam at various learning rates. A network trained with dropout takes longer to learn, so
we trained each architecture for three times as many iterations as in the experiments without dropout:
60,000 iterations for Conv-2, 75,000 iterations for Conv-4, and 90,000 iterations for Conv-6. We
iteratively pruned these networks at the rates determined in Section H.4.
33 Published as a conference paper at ICLR 2019
0.0001 0.0003 0.0005 0.0007 0.001 0.0015
0.70
0.68 15K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 0.66
10K
0.64
5K
0.62
0 0.60
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
0.001 0.002 0.003 0.004 0.005 0.008
25K 0.80
20K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 0.75
15K
10K
0.70
5K
0 0.65
100 53.5 29.1 16.2 9.2 5.4 3.2 2.0 1.3 100 53.5 29.1 16.2 9.2 5.4 3.2 2.0 1.3
Percent of Weights Remaining Percent of Weights Remaining
0.001 0.002 0.003 0.004 0.005 0.008
0.85 30K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 0.80 20K
10K 0.75
0 0.70
100 61.8 39.4 25.8 17.3 11.9 8.3 5.8 4.1 3.0 2.1 100 61.8 39.4 25.8 17.3 11.9 8.3 5.8 4.1 3.0 2.1
Percent of Weights Remaining Percent of Weights Remaining
Figure 34: The early-stopping iteration and validation accuracy at that iteration of the iterative lottery
ticket experiment on the Conv-2 (top), Conv-4 (middle), and Conv-6 (bottom) architectures trained
using SGD with momentum (0.9) at various learning rates. Each line represents a different learning
rate. The legend for each pair of graphs is above the graphs. Lines that are unstable and contain large
error bars (large vertical lines) indicate that some experiments failed to learn effectively, leading to
very low accuracy and very high early-stopping times; these experiments reduce the averages that the
lines trace and lead to much wider error bars.
34 Published as a conference paper at ICLR 2019
0.05 0.1 0.15 0.2 0.25 0.3
6K 0.72
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 0.70 4K
0.68
2K
0.66
0
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
25K 0.80
20K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 0.75
15K
10K 0.70
5K
0.65
0
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
0.85 30K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 0.80
20K
0.75
10K
0.70
0 0.65
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
Figure 35: The early-stopping iteration and validation accuracy at that iteration of the iterative lottery
ticket experiment on the Conv-2 (top), Conv-4 (middle), and Conv-6 (bottom) architectures with an
iterative pruning rate of 20% for fully-connected layers. Each line represents a different iterative
pruning rate for convolutional layers.
35 Published as a conference paper at ICLR 2019
The Conv-2 network proved to be difficult to consistently train with dropout. The top right graph
in Figure 36 contains wide error bars and low average accuracy for many learning rates, especially
early in the lottery ticket experiments. This indicates that some or all of the training runs failed to
learn; when they were averaged into the other results, they produced the aforementioned pattern
in the graphs. At learning rate 0.0001, none of the three trials learned productively until pruned to
more than 26.5%, at which point all three trials started learning. At learning rate 0.0002, some of the
trials failed to learn productively until several rounds of iterative pruning had passed. At learning
rate 0.0003, all three networks learned productively at every pruning level. At learning rate 0.0004,
one network occasionally failed to learn. We selected learning rate 0.0003, which seemed to allow
networks to learn productively most often while achieving among the highest initial accuracy.
It is interesting to note that networks that were unable to learn at a particular learning rate (for
example, 0.0001) eventually began learning after several rounds of the lottery ticket experiment (that
is, training, pruning, and resetting repeatedly). It is worth investigating whether this phenomenon
was entirely due to pruning (that is, removing any random collection of weights would put the
network in a configuration more amenable to learning) or whether training the network provided
useful information for pruning, even if the network did not show improved accuracy.
For both the Conv-4 and Conv-6 architectures, a slightly slower learning rate (0.0002 as opposed to
0.0003) leads to the highest accuracy on the unpruned networks in addition to the highest sustained
accuracy and fastest sustained learning as the networks are pruned during the lottery ticket experiment.
With dropout, the unpruned Conv-4 architecture reaches an average validation accuracy of 77.6%, a
2.7 percentage point improvement over the unpruned Conv-4 network trained without dropout and
one percentage point lower than the highest average validation accuracy attained by a winning ticket.
The dropout-trained winning tickets reach 82.6% average validation accuracy when pruned to 7.6%.
Early-stopping times improve by up to 1.58x (when pruned to 7.6%), a smaller improvement than
then 4.27x achieved by a winning ticket obtained without dropout.
With dropout, the unpruned Conv-6 architecture reaches an average validation accuracy of 81.3%,
an improvement of 2.2 percentage points over the accuracy without dropout; this nearly matches
the 81.5% average accuracy obtained by Conv-6 trained without dropout and pruned to 9.31%.
The dropout-trained winning tickets further improve upon these numbers, reaching 84.8% average
validation accuracy when pruned to 10.5%. Improvements in early-stopping times are less dramatic
than without dropout: a 1.5x average improvement when the network is pruned to 15.1%.
At all learning rates we tested, the lottery ticket pattern generally holds for accuracy, with improve-
ments as the networks are pruned. However, not all learning rates show the decreases in early-stopping
times. To the contrary, none of the learning rates for Conv-2 show clear improvements in early-
stopping times as seen in the other lottery ticket experiments. Likewise, the faster learning rates for
Conv-4 and Conv-6 maintain the original early-stopping times until pruned to about 40%, at which
point early-stopping times steadily increase.
H.6 P RUNING CONVOLUTIONS VS . P RUNING FULLY -C ONNECTED LAYERS
Figure 37 shows the effect of pruning convolutions alone (green), fully-connected layers alone
(orange) and pruning both (blue). The x-axis measures the number of parameters remaining to
emphasize the relative contributions made by pruning convolutions and fully-connected layers to
the overall network. In all three cases, pruning convolutions alone leads to higher test accuracy
and faster learning; pruning fully-connected layers alone generally causes test accuracy to worsen
and learning to slow. However, pruning convolutions alone has limited ability to reduce the overall
parameter-count of the network, since fully-connected layers comprise 99%, 89%, and 35% of the
parameters in Conv-2, Conv-4, and Conv-6.
36 Published as a conference paper at ICLR 2019
0.0001 0.0002 0.0003 0.0004 0.0005
60K
Accuracy at Early-Stop (Val.)0.6
Early-Stop Iteration (Val.) 40K
0.4
20K
0.2
0
100 51.4 26.5 13.7 7.1 3.7 1.9 1.0 100 51.4 26.5 13.7 7.1 3.7 1.9 1.0
Percent of Weights Remaining Percent of Weights Remaining
0.85
60K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 0.80
40K
0.75
20K
0 0.70
100 53.5 29.1 16.2 9.2 5.4 3.2 2.0 1.3 100 53.5 29.1 16.2 9.2 5.4 3.2 2.0 1.3
Percent of Weights Remaining Percent of Weights Remaining
0.90
80K
Accuracy at Early-Stop (Val.) Early-Stop Iteration (Val.) 0.85
60K
0.80
40K
0.75 20K
0 0.70
100 56.2 31.9 18.2 10.5 6.1 3.6 2.1 1.2 100 56.2 31.9 18.2 10.5 6.1 3.6 2.1 1.2
Percent of Weights Remaining Percent of Weights Remaining
Figure 36: The early-stopping iteration and validation accuracy at that iteration of the iterative lottery
ticket experiment on the Conv-2 (top), Conv-4 (middle), and Conv-6 (bottom) architectures trained
using dropout and the Adam optimizer at various learning rates. Each line represents a different
learning rate.
37 Published as a conference paper at ICLR 2019
prune both prune fc layers only prune conv layers only
10K 0.70
Accuracy at Early-Stop (Test) Early-Stop Iteration (Val.) 8K 0.68
6K 0.66
4K 0.64
2K 0.62
0 0.60
4000 3000 2000 1000 0 4000 3000 2000 1000 0
Thousands of Weights Remaining Thousands of Weights Remaining
10K 0.80
Accuracy at Early-Stop (Test) Early-Stop Iteration (Val.) 8K 0.78
6K 0.76
4K 0.74
2K 0.72
0 0.70
2000 1000 0 2000 1000 0
Thousands of Weights Remaining Thousands of Weights Remaining
10K 0.85
Accuracy at Early-Stop (Test) Early-Stop Iteration (Val.) 8K 0.82
6K 0.79
4K 0.76
2K 0.73
0 0.70
1500 1000 500 0 1500 1000 500 0
Thousands of Weights Remaining Thousands of Weights Remaining
Figure 37: Early-stopping iteration and accuracy of the Conv-2 (top), Conv-4 (middle), and Conv-6
(bottom) networks when only convolutions are pruned, only fully-connected layers are pruned, and
both are pruned. The x-axis measures the number of parameters remaining, making it possible to
see the relative contributions to the overall network made by pruning FC layers and convolutions
individually.
38 Published as a conference paper at ICLR 2019
I H YPERPARAMETER EXPLORATION FOR VGG-19 AND RESNET -18 ON
CIFAR10
This Appendix accompanies the VGG-19 and Resnet-18 experiments in Section 4. It details the
pruning scheme, training regimes, and hyperparameters that we use for these networks.
I.1 G LOBAL PRUNING
In our experiments with the Lenet and Conv-2/4/6 architectures, we separately prune a fraction of
the parameters in each layer (layer-wise pruning). In our experiments with VGG-19 and Resnet-18,
we instead pruneglobally; that is, we prune all of the weights in convolutional layers collectively
without regard for the specific layer from which any weight originated.
Figures 38 (VGG-19) and 39 (Resnet-18) compare the winning tickets found by global pruning
(solid lines) and layer-wise pruning (dashed lines) for the hyperparameters from Section 4. When
training VGG-19 with learning rate 0.1 and warmup to iteration 10,000, we find winning tickets when
Pm 6:9%for layer-wise pruning vs.Pm 1:5%for global pruning. For other hyperparameters,
accuracy similarly drops off when sooner for layer-wise pruning than for global pruning. Global
pruning also finds smaller winning tickets than layer-wise pruning for Resnet-18, but the difference is
less extreme than for VGG-19.
In Section 4, we discuss the rationale for the efficacy of global pruning on deeper networks. In
summary, the layers in these deep networks have vastly different numbers of parameters (particularly
severely so for VGG-19); if we prune layer-wise, we conjecture that layers with fewer parameters
become bottlenecks on our ability to find smaller winning tickets.
Regardless of whether we use layer-wise or global pruning, the patterns from Section 4 hold: at
learning rate 0.1, iterative pruning finds winning tickets for neither network; at learning rate 0.01, the
lottery ticket pattern reemerges; and when training with warmup to a higher learning rate, iterative
pruning finds winning tickets. Figures 40 (VGG-19) and 41 (Resnet-18) present the same data as
Figures 7 (VGG-19) and 8 (Resnet-18) from Section 4 with layer-wise pruning rather than global
pruning. The graphs follow the same trends as in Section 4, but the smallest winning tickets are larger
than those found by global pruning.
I.2 VGG-19 D ETAILS
The VGG19 architecture was first designed by Simonyan & Zisserman (2014) for Imagenet. The
version that we use here was adapted by Liu et al. (2019) for CIFAR10. The network is structured
as described in Figure 2: it has five groups of 3x3 convolutional layers, the first four of which are
followed by max-pooling (stride 2) and the last of which is followed by average pooling. The network
has one final dense layer connecting the result of the average-pooling to the output.
We largely follow the training procedure for resnet18 described in Appendix I:
We use the same train/test/validation split.
We use the same data augmentation procedure.
We use a batch size of 64.
We use batch normalization.
We use a weight decay of 0.0001.
We use three stages of training at decreasing learning rates. We train for 160 epochs (112,480
iterations), decreasing the learning rate by a factor of ten after 80 and 120 epochs.
We use Gaussian Glorot initialization.
We globally prune the convolutional layers of the network at a rate of 20% per iteration, and we do
not prune the 5120 parameters in the output layer.
Liu et al. (2019) uses an initial pruning rate of 0.1. We train VGG19 with both this learning rate and
a learning rate of 0.01.
39 Published as a conference paper at ICLR 2019
I.3 R ESNET -18 D ETAILS
The Resnet-18 architecture was first introduced by He et al. (2016). The architecture comprises 20
total layers as described in Figure 2: a convolutional layer followed by nine pairs of convolutional
layers (with residual connections around the pairs), average pooling, and a fully-connected output
layer.
We follow the experimental design of He et al. (2016):
We divide the training set into 45,000 training examples and 5,000 validation examples. We
use the validation set to select hyperparameters in this appendix and the test set to evaluate
in Section 4.
We augment training data using random flips and random four pixel pads and crops.
We use a batch size of 128.
We use batch normalization.
We use weight decay of 0.0001.
We train using SGD with momentum (0.9).
We use three stages of training at decreasing learning rates. Our stages last for 20,000,
5,000, and 5,000 iterations each, shorter than the 32,000, 16,000, and 16,000 used in He
et al. (2016). Since each of our iterative pruning experiments requires training the network
15-30 times consecutively, we select this abbreviated training schedule to make it possible
to explore a wider range of hyperparameters.
We use Gaussian Glorot initialization.
We globally prune convolutions at a rate of 20% per iteration. We do not prune the 2560 parameters
used to downsample residual connections or the 640 parameters in the fully-connected output layer,
as they comprise such a small portion of the overall network.
I.4 L EARNING RATE
In Section 4, we observe that iterative pruning is unable to find winning tickets for VGG-19 and
Resnet-18 at the typical, high learning rate used to train the network (0.1) but it is able to do so at a
lower learning rate (0.01). Figures 42 and 43 explore several other learning rates. In general, iterative
pruning cannot find winning tickets at any rate above 0.01 for either network; for higher learning
rates, the pruned networks with the original initialization perform no better than when randomly
reinitialized.
I.5 W ARMUP ITERATION
In Section 4, we describe how adding linear warmup to the initial learning rate makes it possible to
find winning tickets for VGG-19 and Resnet-18 at higher learning rates (and, thereby, winning tickets
that reach higher accuracy). In Figures 44 and 45, we explore the number of iterationskover which
warmup should occur.
For VGG-19, we were able to find values ofkfor which iterative pruning could identify winning
tickets when the network was trained at the original learning rate (0.1). For Resnet-18, warmup made
it possible to increase the learning rate from 0.01 to 0.03, but no further. When exploring values ofk,
we therefore us learning rate 0.1 for VGG-19 and 0.03 for Resnet-18.
In general, the greater the value ofk, the higher the accuracy of the eventual winning tickets.
Resnet-18. For values ofkbelow 5000, accuracy improves rapidly askincreases. This relationship
reaches a point of diminishing returns abovek= 5000. For the experiments in Section 4, we select
k= 20000, which achieves the highest validation accuracy.
VGG-19. For values ofkbelow 5000, accuracy improves rapidly askincreases. This relationship
reaches a point of diminishing returns abovek= 5000. For the experiments in Section 4, we select
k= 10000, as there is little benefit to larger values ofk.
40 Published as a conference paper at ICLR 2019
rate 0.1 (global) rate 0.1 (layerwise) rate 0.01 (global) rate 0.01 (layerwise) rate 0.03, warmup 10K (global) rate 0.03, warmup 10K (layerwise)
0.94 0.94
0.900
0.92 0.92
0.875 0.90
Test Accuracy (112K)0.90
Test Accuracy (60K)
Test Accuracy (30K) 0.850 0.88 0.88
0.825 0.86 0.86
0.84 0.84 0.800
0.82 0.82 0.775
0.80 0.80
100 41.016.8 6.9 2.8 1.2 0.5 0.20.1 100 41.016.8 6.9 2.8 1.2 0.5 0.20.1 100 41.016.8 6.9 2.8 1.2 0.5 0.20.1
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
Figure 38: Validation accuracy (at 30K, 60K, and 112K iterations) of VGG-19 when iteratively
pruned with global (solid) and layer-wise (dashed) pruning.
rate 0.1 (global) rate 0.1 (layerwise) rate 0.01 (global) rate 0.01 (layerwise) rate 0.03, warmup 20K (global) rate 0.03, warmup 20K (layerwise)
0.85 0.85 0.90
Test Accuracy (10K) 0.80
Test Accuracy (20K) 0.80
Test Accuracy (30K)0.88
0.86
0.75 0.75
0.84
0.70 0.70
0.82
0.65 0.65
100 64.4 41.7 27.1 17.8 11.8 8.0 5.5 100 64.4 41.7 27.1 17.8 11.8 8.0 5.5 100 64.4 41.7 27.1 17.8 11.8 8.0 5.5
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
Figure 39: Validation accuracy (at 10K, 20K, and 30K iterations) of Resnet-18 when iteratively
pruned with global (solid) and layer-wise (dashed) pruning.
rate 0.1 rand reinit rate 0.01 rand reinit rate 0.1, warmup 10K rand reinit
0.94 0.94 0.94
0.92 0.92 0.92
0.90 0.90
Test Accuracy (112K)0.90
Test Accuracy (30K)
Test Accuracy (60K) 0.88 0.88 0.88
0.86 0.86 0.86
0.84 0.84 0.84
0.82 0.82 0.82
0.80 0.80 0.80
100 41.0 16.8 6.9 2.8 1.2 0.5 0.2 0.1 100 41.0 16.8 6.9 2.8 1.2 0.5 0.2 0.1 100 41.0 16.8 6.9 2.8 1.2 0.5 0.2 0.1
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
Figure 40: Test accuracy (at 30K, 60K, and 112K iterations) of VGG-19 when iteratively pruned with
layer-wise pruning. This is the same as Figure 7, except with layer-wise pruning rather than global
pruning.
rate 0.1 rate 0.01 rand reinit rate 0.03, warmup 20K rand reinit
0.85 0.85 0.90
Test Accuracy (10K)0.80
Test Accuracy (20K)0.80
Test Accuracy (30K) 0.88
0.86
0.75 0.75
0.84
0.70 0.70
0.82
0.65 0.65
100 64.4 41.7 27.1 17.8 11.8 8.0 5.5 100 64.4 41.7 27.1 17.8 11.8 8.0 5.5 100 64.4 41.7 27.1 17.8 11.8 8.0 5.5
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
Figure 41: Test accuracy (at 10K, 20K, and 30K iterations) of Resnet-18 when iteratively pruned with
layer-wise pruning. This is the same as Figure 8 except with layer-wise pruning rather than global
pruning.
41 Published as a conference paper at ICLR 2019
rate 0.01 rate 0.02 rate 0.03 rate 0.05 rate 0.1 rand reinit
rand reinit rand reinit rand reinit rand reinit
0.90 0.90
0.92
0.85 0.85 0.90
Val. Accuracy (10K) Val. Accuracy (20K) Val. Accuracy (30K)0.80 0.80 0.88
0.75 0.75 0.86
0.840.70 0.70
0.82
0.65 0.65
100 64.4 41.7 27.1 17.8 11.8 8.0 5.5 100 64.4 41.7 27.1 17.8 11.8 8.0 5.5 100 64.4 41.7 27.1 17.8 11.8 8.0 5.5
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
Figure 42: Validation accuracy (at 10K, 20K, and 30K iterations) of Resnet-18 when iteratively
pruned and trained with various learning rates.
rate 0.01 rate 0.02 rate 0.03 rate 0.05 rate 0.1 rand reinit
rand reinit rand reinit rand reinit
0.94 0.94
0.900
0.92 0.92
0.875 0.90
Val. Accuracy (112K)0.90
Val. Accuracy (60K)
Val. Accuracy (30K)0.850 0.88 0.88
0.825 0.86 0.86
0.84 0.84 0.800
0.82 0.82 0.775
0.80 0.80
100 41.016.8 6.9 2.8 1.2 0.5 0.20.1 100 41.016.8 6.9 2.8 1.2 0.5 0.20.1 100 41.016.8 6.9 2.8 1.2 0.5 0.20.1
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
Figure 43: Validation accuracy (at 30K, 60K, and 112K iterations) of VGG-19 when iteratively
pruned and trained with various learning rates.
rate 0.03, warmup 0 rate 0.03, warmup 500 rate 0.03, warmup 1000 rate 0.03, warmup 5000 rate 0.03, warmup 10000 rate 0.03, warmup 20000
0.90 0.90
0.92
0.85 0.85 0.90
Val. Accuracy (10K) Val. Accuracy (20K) Val. Accuracy (30K)0.80 0.80 0.88
0.75 0.75 0.86
0.840.70 0.70
0.82
0.65 0.65
100 64.4 41.7 27.1 17.8 11.8 8.0 5.5 100 64.4 41.7 27.1 17.8 11.8 8.0 5.5 100 64.4 41.7 27.1 17.8 11.8 8.0 5.5
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
Figure 44: Validation accuracy (at 10K, 20K, and 30K iterations) of Resnet-18 when iteratively
pruned and trained with varying amounts of warmup at learning rate 0.03.
rate 0.1, warmup 0 rate 0.1, warmup 1000 rate 0.1, warmup 5000 rate 0.1, warmup 10000 rate 0.1, warmup 20000 rate 0.1, warmup 50000
0.94 0.94 0.94
0.92 0.92 0.92
0.90 0.90
Val. Accuracy (112K) 0.90
Val. Accuracy (30K) Val. Accuracy (60K)0.88 0.88 0.88
0.86 0.86 0.86
0.84 0.84 0.84
0.82 0.82 0.82
0.80 0.80 0.80
100 41.016.8 6.9 2.8 1.2 0.5 0.20.1 100 41.016.8 6.9 2.8 1.2 0.5 0.20.1 100 41.016.8 6.9 2.8 1.2 0.5 0.20.1
Percent of Weights Remaining Percent of Weights Remaining Percent of Weights Remaining
Figure 45: Validation accuracy (at 30K, 60K, and 112K iterations) of VGG-19 when iteratively
pruned and trained with varying amounts of warmup at learning rate 0.1.
42