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2020-08-06 20:53:44 +00:00
Movement Pruning:
Adaptive Sparsity by Fine-Tuning
Victor Sanh 1 , Thomas Wolf 1 , Alexander M. Rush 1,2
1 Hugging Face, 2 Cornell University
{victor,thomas}@huggingface.co;arush@cornell.edu
arXiv:2005.07683v1 [cs.CL] 15 May 2020 Abstract
Magnitude pruning is a widely used strategy for reducing model size in pure
supervised learning; however, it is less effective in the transfer learning regime that
has become standard for state-of-the-art natural language processing applications.
We propose the use ofmovement pruning, a simple, deterministic first-order weight
pruning method that is more adaptive to pretrained model fine-tuning. We give
mathematical foundations to the method and compare it to existing zeroth- and
first-order pruning methods. Experiments show that when pruning large pretrained
language models, movement pruning shows significant improvements in high-
sparsity regimes. When combined with distillation, the approach achieves minimal
accuracy loss with down to only 3% of the model parameters.
1 Introduction
Large-scale transfer learning has become ubiquitous in deep learning and achieves state-of-the-art
performance in applications in natural language processing and related fields. In this setup, a large
model pretrained on a massive generic dataset is then fine-tuned on a smaller annotated dataset to
perform a specific end-task. Model accuracy has been shown to scale with the pretrained model and
dataset size [Raffel et al., 2019]. However, significant resources are required to ship and deploy these
large models, and training the models have high environmental costs [Strubell et al., 2019].
Sparsity induction is a widely used approach to reduce the memory footprint of neural networks at
only a small cost of accuracy. Pruning methods, which remove weights based on their importance,
are a particularly simple and effective method for compressing models to be sent to edge devices such
as mobile phones. Magnitude pruning [Han et al., 2015, 2016], which preserves weights with high
absolute values, is the most widely used method for weight pruning. It has been applied to a large
variety of architectures in computer vision [Guo et al., 2016], in language processing [Gale et al.,
2019], and more recently has been leveraged as a core component in thelottery ticket hypothesis
[Frankle et al., 2019].
While magnitude pruning is highly effective for standard supervised learning, it is inherently less
useful in the transfer learning regime. In supervised learning, weight values are primarily determined
by the end-task training data. In transfer learning, weight values are mostly predetermined by the
original model and are only fine-tuned on the end task. This prevents these methods from learning to
prune based on the fine-tuning step, or “fine-pruning.”
In this work, we argue that to effectively reduce the size of models for transfer learning, one should
instead usemovement pruning, i.e., pruning approaches that consider the changes in weights during
fine-tuning. Movement pruning differs from magnitude pruning in that both weights with low and
high values can be pruned if they shrink during training. This strategy moves the selection criteria
from the 0th to the 1st-order and facilitates greater pruning based on the fine-tuning objective. To
Preprint. Under review. test this approach, we introduce a particularly simple, deterministic version of movement pruning
utilizing the straight-through estimator [Bengio et al., 2013].
We apply movement pruning to pretrained language representations (BERT) [Devlin et al., 2019,
Vaswani et al., 2017] on a diverse set of fine-tuning tasks. In highly sparse regimes (less than 15% of
remaining weights), we observe significant improvements over magnitude pruning and other 1st-order
methods such asL0 regularization [Louizos et al., 2017]. Our models reach 95% of the original
BERT performance with only 5% of the encoder’s weight on natural language inference (MNLI)
[Williams et al., 2018] and question answering (SQuAD v1.1) [Rajpurkar et al., 2016]. Analysis of
the differences between magnitude pruning and movement pruning shows that the two methods lead
to radically different pruned models with movement pruning showing greater ability to adapt to the
end-task.
2 Related Work
In addition to magnitude pruning, there are many other approaches for generic model weight pruning.
Most similar to our approach are methods for using parallel score matrices to augment the weight
matrices [Mallya and Lazebnik, 2018, Ramanujan et al., 2020], which have been applied for convo-
lutional networks. Differing from our methods, these methods keep the weights of the model fixed
(either from a randomly initialized network or a pre-trained network) and the scores are updated to
find a good sparse subnetwork.
Many previous works have also explored using higher-order information to select prunable weights.
LeCun et al. [1989] and Hassibi et al. [1993] leverage the Hessian of the loss to select weights for
deletion. Our method does not require the (possibly costly) computation of second-order derivatives
since the importance scores are obtained simply as the by-product of the standard fine-tuning. Theis
et al. [2018] and Ding et al. [2019] use the absolute value or the square value of the gradient. In
contrast, we found it useful to preserve the direction of movement in our algorithm.
Compressing pretrained language models for transfer learning is also a popular area of study. Other
approaches include knowledge distillation [Sanh et al., 2019, Tang et al., 2019] and structured pruning
[Fan et al., 2020a, Michel et al., 2019]. Our core method does not require an external teacher model
and targets individual weight. We also show that having a teacher can further improve our approach.
Recent work also builds upon iterative magnitude pruning with rewinding [Yu et al., 2020] to train
sparse language models from scratch. This differs from our approach which focuses on the fine-tuning
stage. Finally, another popular compression approach is quantization. Quantization has been applied
to a variety of modern large architectures [Fan et al., 2020b, Zafrir et al., 2019, Gong et al., 2014]
providing high memory compression rates at the cost of no or little performance. As shown in
previous works [Li et al., 2020, Han et al., 2016] quantization and pruning are complimentary and
can be combined to further improve the performance/size ratio.
3 Background: Score-Based Pruning
We first establish shared notation for discussing different neural network pruning strategies. Let
W2Rnn refer to a generic weight matrix in the model (we consider square matrices, but they
could be of any shape). To determine which weights are pruned, we introduce a parallel matrix of
associated importance scoresS2Rnn . Given importance scores, each pruning strategy computes a
maskM2 f0;1gnn . Inference for an inputxbecomesa= (WM)x, whereis the Hadamard
product. A common strategy is to keep the top-vpercent of weights by importance. We define Top v as a function which selects thev%highest values inS:1; STop(S) (1) v i;j = i;j in topv%
0; o.w.
Magnitude-based weight pruning determines the mask based on the absolute value of each weight as a measure of importance. Formally, we have importance scoresS= jWi;j j , and masks 1i;jn M=Top(S)(Eq(1)). There are several extensions to this base setup. Han et al. [2015] use v iterative magnitude pruning: the model is first trained until convergence and weights with the lowest
magnitudes are removed afterward. The sparsified model is then re-trained with the removed weights
fixed to 0. This loop is repeated until the desired sparsity level is reached.
2 Magnitude pruning L0 regularization Movement pruning Soft movement pruning
Pruning Decision 0th order 1st order 1st order 1st order
Masking Function Top v Continuous Hard-Concrete Top v Thresholding
Pruning Structure Local or Global Global Local or Global Global
Learning Objective L L+l0 E(L0 ) L L+mvp R(S)
Gradient Form Gumbel-Softmax Straight-Through Straight-ThroughP P PScoresS jW ) )i;j j (@L )(t W(t) f(S(t) ) (@L )(t) W(t) (@L )(t) W(t
t@W i;j i;j i;j i;j t@W i;j i;j t@W i;j
Table 1: Summary of the pruning methods considered in this work and their specificities. The
expression offofL0 regularization is detailed in Eq (3).
In this study, we focus onautomated gradual pruning[Zhu and Gupta, 2018]. It supplements
magnitude pruning by allowing masked weights to be updated such that they are not fixed for the
entire duration of the training. Automated gradual pruning enables the model to recover from previous
masking choices [Guo et al., 2016]. In addition, one can gradually increases the sparsity levelv during training using a cubic sparsity scheduler:v(t) =vf + (v t 3
i vf ) 1ti . The sparsity nt level at time stept,v(t) is increased from an initial valuevi (usually 0) to a final valuevf innpruning
steps afterti steps of warm-up. The model is thus pruned and trained jointly.
4 Movement Pruning
Magnitude pruning can be seen as utilizing zeroth-order information (absolute value) of the running
model. In this work, we focus on movement pruning methods where importance is derived from
first-order information. Intuitively, instead of selecting weights that are far from zero, we retain
connections that are moving away from zero during the training process. We consider two versions of
movement pruning: hard and soft.
For (hard) movement pruning, masks are computed using the Top v function:M=Top(S). Unlike v magnitude pruning, during training, we learn both the weightsP Wand their importance scoresS.
During the forward pass, we compute for alli,a ni = Wk=1 i;k Mi;k xk .
Since the gradient of Top v is 0 everywhere it is defined, we follow Ramanujan et al. [2020], Mallya
and Lazebnik [2018] and approximate its value with thestraight-through estimator[Bengio et al.,
2013]. In the backward pass, Top v is ignored and the gradient goes "straight-through" toS. The
approximation of gradient of the lossLwith respect toSi;j is given by
@L @L @a= i @L= W x@S j (2)
i;j @a i @S i;j @a i;ji
This implies that the scores of weights are updated, even if these weights are masked in the forward
pass. We prove in Appendix A.1 that movement pruning as an optimization problem will converge.
We also consider a relaxed (soft) version of movement pruning based on the binary mask function
described by Mallya and Lazebnik [2018]. Here we replace hyperparametervwith a fixed global
threshold valuethat controls the binary mask. The mask is calculated asP M= (S> ). In order to
control the sparsity level, we add a regularization termR(S) =mvp (Si;j i;j )which encourages
the importance scores to decrease over time 1 . The coefficientmvp controls the penalty intensity and
thus the sparsity level.
Finally we note that these approaches yield a similar updateL0 regularization based pruning, another
movement based pruning approach [Louizos et al., 2017]. Instead of straight-through,L0 uses the
hard-concretedistribution, where the maskMis sampled for alli;jwith hyperparametersb >0,
l <0, andr >1: u U(0;1) Si;j =(log(u)log(1u) +Si;j )=b
Zi;j = (rl)Si;j +l M i;j = min(1;ReLU(Zi;j ))
The expectedP L0 norm has a closed form involving the parameters of the hard-concrete: E(L0 ) =
logSi;j i;j blog(l=r). Thus, the weights and scores of the model can be optimized in
P1 We also experimented with jSi;j i;j jbut it turned out to be harder to tune while giving similar results.
3 (a) Magnitude pruning (b) Movement pruning
Figure 1: During fine-tuning (on MNLI), the weights stay close to their pre-trained values which
limits the adaptivity of magnitude pruning. We plot the identity line in black. Pruned weights are
plotted in grey. Magnitude pruning selects weights that are far from 0 while movement pruning
selects weights that are moving away from 0.
an end-to-end fashion to minimize the sum of the training lossLand the expectedL0 penalty. A
coefficientl0 controls theL0 penalty and indirectly the sparsity level. Gradients take a similar form:
@L @L rl= W@S i;j xj f(Si;j )wheref(Si;j ) = S Zi;j 1g (3)
i;j @a i b i;j (1Si;j )1f0
At test time, a non-stochastic estimation of the mask is used:M^= min 1;ReLU (rl)(S)+l
and weights multiplied by 0 can simply be discarded.
Table 1 highlights the characteristics of each pruning method. The main differences are in the masking
functions, pruning structure, and the final gradient form.
Method Interpretation In movement pruning, the gradient ofLwith respect toWi;j is given
by the standard gradient derivation: @L = @L M@W i;j @a i;j xj . By combining it to Eq(2), we have i @L = @L W@S i;j @W i;j (we omit the binary mask termMi;j for simplicity). From the gradient update in i;j
Eq (2),S @Li;j is increasing when <0, which happens in two cases: @S i;j
(a) @L <0andW@W i;j >0i;j
(b) @L >0andW@W i;j <0i;j
It means that during trainingWi;j is increasing while being positive or is decreasing while being
negative. It is equivalent to saying thatSi;j is increasing whenWi;j is moving away from 0. Inversely,
Si;j is decreasing when @L >0which means thatW@S i;j is shrinking towards 0. i;j
While magnitude pruning selects the most important weights as the ones which maximize their
distance to 0 (jWi;j j), movement pruning selects the weights which are moving the most away from
0 (Si;j ). For this reason, magnitude pruning can be seen as a 0th order method, whereas movement
pruning is based on a 1st order signal. In fact,Scan be seen as an accumulator of movement: from
equation (2), afterTgradient updates, we have
XS(T) @L= )(t) W(t) (4) i;j S (@W i;j i;jt<T
Figure 1 shows this difference empirically by comparing weight values during fine-tuning against
their pre-trained value. As observed by Gordon et al. [2020], fine-tuned weights stay close in absolute
value to their initial pre-trained values. For magnitude pruning, this stability around the pre-trained
4 values implies that we know with high confidence before even fine-tuning which weights will be
pruned as the weights with the smallest absolute value at pre-training will likely stay small and be
pruned. In contrast, in movement pruning, the pre-trained weights do not have such an awareness of
the pruning decision since the selection is made during fine-tuning (moving away from 0), and both
low and high values can be pruned. We posit that this is critical for the success of the approach as it
is able to prune based on the task-specific data, not only the pre-trained value.
5 Experimental Setup
Transfer learning for NLP uses large pre-trained language models that are fine-tuned on target tasks
[Ruder et al., 2019, Devlin et al., 2019, Radford et al., 2019, Liu et al., 2019]. We experiment with task-
specific pruning ofBERT-base-uncased, a pre-trained model that contains roughly 84M parameters.
We freeze the embedding modules and fine-tune the transformer layers and the task-specific head.
We perform experiments on three monolingual (English) tasks, which are common benchmarks for
the recent progress in transfer learning for NLP: question answering (SQuAD v1.1) [Rajpurkar et al.,
2016], natural language inference (MNLI) [Williams et al., 2018], and sentence similarity (QQP)
[Iyer et al., 2017]. The datasets respectively contain 8K, 393K, and 364K training examples. SQuAD
is formulated as a span extraction task, MNLI and QQP are paired sentence classification tasks.
For a given task, we fine-tune the pre-trained model for the same number of updates (between 6
and 10 epochs) across pruning methods 2 . We follow Zhu and Gupta [2018] and use a cubic sparsity
scheduling for Magnitude Pruning (MaP), Movement Pruning (MvP), and Soft Movement Pruning
(SMvP). Adding a few steps of cool-down at the end of pruning empirically improves the performance
especially in high sparsity regimes. The schedule forvis:
8<vi 0t < t i
v v )3 t (5):f + (vi f )(1tti tf
nt i t < Ttf
vf o.w.
wheretf is the number of cool-down steps.
We compare our results against several state-of-the-art pruning baselines: Reweighted Proximal
Pruning (RPP) [Guo et al., 2019] combines re-weightedL1 minimization and Proximal Projection
[Parikh and Boyd, 2014] to perform unstructured pruning. LayerDrop [Fan et al., 2020a] leverages
structured dropout to prune models at test time. For RPP and LayerDrop, we report results from
authors. We also compare our method against the mini-BERT models, a collection of smaller BERT
models with varying hyper-parameters [Turc et al., 2019].
6 Results
Figure 2 displays the results for the main pruning methods at different levels of pruning on each
dataset. First, we observe the consistency of the comparison between magnitude and movement
pruning: at low sparsity (more than 70% of remaining weights), magnitude pruning outperforms
all methods with little or no loss with respect to the dense model whereas the performance of
movement pruning methods quickly decreases even for low sparsity levels. However, magnitude
pruning performs poorly with high sparsity, and the performance drops extremely quickly. In contrast,
first-order methods show strong performances with less than 15% of remaining weights.
Table 2 shows the specific model scores for different methods at high sparsity levels. Magnitude
pruning on SQuAD achieves 54.5 F1 with 3% of the weights compared to 73.6 F1 withL0 regular-
ization, 76.3 F1 for movement pruning, and 79.9 F1 with soft movement pruning. These experiments
indicate that in high sparsity regimes, importance scores derived from the movement accumulated
during fine-tuning induce significantly better pruned models compared to absolute values.
Next, we compare the difference in performance between first-order methods. We see that straight-
through based hard movement pruning (MvP) is comparable withL0 regularization (with a significant
gap in favor of movement pruning on QQP). Soft movement pruning (SMvP) consistently outperforms
2 Preliminary experiments showed that increasing the number of pruning steps tended to improve the end
performance
5 Figure 2: Comparisons between different pruning methods in high sparsity regimes.Soft move-
ment pruning consistently outperforms other methods in high sparsity regimes.We plot the
performance of the standard fine-tuned BERT along with 95% of its performance.
Table 2: Performance at high sparsity levels. (Soft) movement pruning outperforms current
state-of-the art pruning methods at different high sparsity levels.
BERT base Remaining
fine-tuned Weights (%) MaP L0 Regu MvP soft MvP
SQuAD - Dev 10% 67.7/78.5 69.9/80.1 71.9/81.7 71.3/81.580.4/88.1EM/F1 3% 40.1/54.5 61.6/73.6 65.2/76.3 69.6/79.9
MNLI - Dev 10% 77.8/79.0 77.9/78.5 79.3/79.5 80.7/81.2acc/MM acc 84.5/84.9 3% 68.9/69.8 75.2/75.6 76.1/76.7 79.0/79.7
QQP - Dev 10% 78.8/75.1 87.6/81.9 89.1/85.5 90.2/86.891.4/88.4acc/F1 3% 72.1/58.4 86.5/81.1 85.6/81.0 89.2/85.5
hard movement pruning andL0 regularization by a strong margin and yields the strongest performance
among all pruning methods in high sparsity regimes. These comparisons support the fact that even if
movement pruning (and its relaxed version soft movement pruning) is simpler thanL0 regularization,
it yet yields stronger performances for the same compute budget.
Finally, movement pruning and soft movement pruning compare favorably to the other baselines, ex-
cept for QQP where RPP is on par with soft movement pruning. Movement pruning also outperforms
the fine-tuned mini-BERT models. This is coherent with [Li et al., 2020]: it is both more efficient and
more effective to train a large model and compress it afterward than training a smaller model from
scratch. We do note though that current hardware does not support optimized inference for sparse
models: from an inference speed perspective, it might often desirable to use a small dense model
such as mini-BERT over a sparse alternative of the same size.
Distillation further boosts performance Following previous work, we can further leverage knowl-
edge distillation [Bucila et al., 2006, Hinton et al., 2014] to boost performance for free in the pruned
domain [Jiao et al., 2019, Sanh et al., 2019] using our baseline fine-tunedBERT-basemodel as
teacher. The training objective is a linear combination of the training loss and a knowledge distillation
Figure 3: Comparisons between different pruning methods augmented with distillation.Distillation
improves the performance across all pruning methods and sparsity levels.
6 Table 3: Distillation-augmented performances for selected high sparsity levels.All pruning methods
benefit from distillation signal further enhancing the ratio Performance VS Model Size.
BERT base Remaining
fine-tuned Weights (%) MaP L0 Regu MvP soft MvP
SQuAD - Dev 10% 70.2/80.1 72.4/81.9 75.6/84.3 76.6/84.980.4/88.1EM/F1 3% 45.5/59.6 65.5/75.9 67.5/78.0 72.9/82.4
MNLI - Dev 10% 78.3/79.3 78.7/79.8 80.1/80.4 81.2/81.8acc/MM acc 84.5/84.9 3% 69.4/70.6 76.2/76.5 76.5/77.4 79.6/80.2
QQP - Dev 10% 79.8/65.0 88.1/82.8 89.7/86.2 90.5/87.191.4/88.4acc/F1 3% 72.4/57.8 87.1/82.0 86.1/81.5 89.3/85.6
(a) Distribution of remaining weights (b) Scores and weights learned by
movement pruning
Figure 4: Magnitude pruning and movement pruning leads to pruned models with radically different
weight distribution.
loss on the output distributions. Figure 3 shows the results on SQuAD, MNLI, and QQP for the three
pruning methods boosted with distillation. Overall, we observe that the relative comparisons of the
pruning methods remain unchanged while the performances are strictly increased. Table 3 shows for
instance that on SQuAD, movement pruning at 10% goes from 81.7 F1 to 84.3 F1. When combined
with distillation, soft movement pruning yields the strongest performances across all pruning methods
and studied datasets: it reaches 95% ofBERT-basewith only a fraction of the weights in the encoder
(5% on SQuAD and MNLI).
7 Analysis
Movement pruning is adaptive Figure 4a compares the distribution of the remaining weights for
the same matrix of a model pruned at the same sparsity using magnitude and movement pruning. We
observe that by definition, magnitude pruning removes all the weights that are close to zero, ending
up with two clusters. In contrast, movement pruning leads to a smoother distribution, which covers
the whole interval except for values close to 0.
Figure 4b displays each individual weight against its associated importance score in movement
pruning. We plot pruned weights in grey. We observe that movement pruning induces no simple
relationship between the scores and the weights. Both weights with high absolute value or low
absolute value can be considered important. However, high scores are systematically associated with
non-zero weights (and thus the “v-shape”). This is coherent with the interpretation we gave to the
scores (section 4): a high scoreSindicates that during fine-tuning, the associated weight moved away
from 0 and is thus non-null.
Local and global masks perform similarly We study the influence of the locality of the pruning
decision. While local Top v selects thev% most important weights matrix by matrix, global Top v uncovers non-uniform sparsity patterns in the network by selecting thev% most important weights in
7 Figure 5: Comparison of local and global selec- Figure 6:Remaining weights per layer in thetions of weights on SQuAD at different sparsity Transformer.Global magnitude pruning tends tolevels.For magnitude and movement pruning, prune uniformly layers. Global 1st order meth-local and global Top v performs similarly at all ods allocate the weight to the lower layers whilelevels of sparsity. heavily pruning the highest layers.
the whole network. Previous work has shown that a non-uniform sparsity across layers is crucial to
the performance in high sparsity regimes [He et al., 2018]. In particular, Mallya and Lazebnik [2018]
found that the sparsity tends to increase with the depth of the network layer.
Figure 5 compares the performance of local selection (matrix by matrix) against global selection
(all the matrices) for magnitude pruning and movement pruning. Despite being able to find a
global sparsity structure, we found that global did not significantly outperform local, except in high
sparsity regimes (2.3 F1 points of difference with 3% of remaining weights for movement pruning).
Even though the distillation signal boosts the performance of pruned models, the end performance
difference between local and global selections remains marginal.
Figure 6 shows the remaining weights percentage obtained per layer when the model is pruned until
10% with global pruning methods. Global weight pruning is able to allocate sparsity non-uniformly
through the network, and it has been shown to be crucial for the performance in high sparsity regimes
[He et al., 2018]. We notice that except for global magnitude pruning, all the global pruning methods
tend to allocate a significant part of the weights to the lowest layers while heavily pruning in the
highest layers. Global magnitude pruning tends to prune similarly to local magnitude pruning, i.e.,
uniformly across layers.
8 Conclusion
We consider the case of pruning of pretrained models for task-specific fine-tuning and compare
zeroth- and first-order pruning methods. We show that a simple method for weight pruning based on
straight-through gradients is effective for this task and that it adapts using a first-order importance
score. We apply this movement pruning to a transformer-based architecture and empirically show that
our method consistently yields strong improvements over existing methods in high-sparsity regimes.
The analysis demonstrates how this approach adapts to the fine-tuning regime in a way that magnitude
pruning cannot. In future work, it would also be interesting to leverage group-sparsity inducing
penalties [Bach et al., 2011] to remove entire columns or filters. In this setup, we would associate a
score to a group of weights (a column or a row for instance). In the transformer architecture, it would
give a systematic way to perform feature selection and remove entire columns of the embedding
matrix.
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10 A Appendices
A.1 Guarantees on the decrease of the training loss
As the scores are updated, the relative order of the importances is likely shuffled, and some connections
will be replaced by more important ones. Under certain conditions, we are able to formally prove that
as these replacements happen, the training loss is guaranteed to decrease. Our proof is adapted from
[Ramanujan et al., 2020] to consider the case of fine-tuableW.
We suppose that (a) the training lossLis smooth and admits a first-order Taylor development
everywhere it is defined and (b) the learning rate ofW(W >0) is small. We define the TopK
function as the analog of the Top v function, wherekis an integer instead of a proportion. We first
consider the case wherek= 1in the TopK masking, meaning that only one connection is remaining
(and the other weights are deactivated/masked). Let’s denoteWi;j this sole remaining connection at
stept. Following Eq (1), it means that81u;vn;S (t)u;v S(t) .i;j
We suppose that at stept+ 1, connections are swapped and the only remaining connection at step
t+ 1is(k;l). We have:
(
Att; 81u;vn;S (t)u;v S(t)
i;j (6) Att+ 1; 81u;vn;S (t+1)u;v S(t+1)
k;l
Eq(6)gives the following inequality:S(t+1) S(t) S(t+1) S(t) . After re-injecting the gradient k;l k;l i;j i;j update in Eq (2), we have:
@L ) @L
S W(t x (7)@a k;l l S W(t) x
k @a i;j ji
Moreover, the conditions in Eq(6)lead to the following inferences:a(t) =W(t) x a(t+1) =i i;j j and k
W(t+1) xk;l l .
Since t)W is small,jj(a(t+1) ;a (t+1) )(a( ;a (t) )jj i k i k 2 is also small. Because the training lossLis
smooth, we can write the 1st order Taylor development ofLin point(a(t) ;a (t) ):i k
L(a(t+1) ;a (t+1) ) L(a(t) ;a (t) )i k i k
@L (a(t+1) a(t) @L) + (a(t+1) a(t) )@a k kk @a i ii
@L= W(t+1) @Lx W(t) x@a k;l l
k @a i;j ji
@L @L @L (8) = W(t+1) x W(t) @Lx (t) x@a k;l l + ( W(t) x
k @a k;l l +
k @a k;l l ) Wi;j jk @a i
@L= (W(t+1) x ) @L
@a k;l l W(t) @Lx xk;l l ) + ( W(t
k @a k;l l W(t) x
k @a i;j j )
i
@L @L @L @L= x (S(t) ) x@a l (W x k;l ) + ( W(t) l W(t) x
k @a l m
k @a k;lk @a i;j j )
i
The first term is null because of inequalities(6)and the second term is negative because of inequality
(7). ThusL(a(t+1) ;a (t+1) ) L(a(t) ;a (t) ): when connection(k;l)becomes more important than i k i (i;j), the connections are swapped and the training loss decreases between step k tandt+ 1. Similarly, we can generalize the proof to a setE=f(ai ;b i );(ci ;d i );iNg ofNswapping
connections.
We note that this proof is not specific to theTopKmasking function. In fact, we can extend the proof
using theThresholdmasking functionM:= (S>=)[Mallya and Lazebnik, 2018]. Inequalities
(6) are still valid and the proof stays unchanged.
11 Last, we note these guarantees do not hold if we consider the absolute value of the scoresjSi;j j(as
it is done in Ding et al. [2019] for instance). We prove it by contradiction. If it was the case, it
would also be true one specific case: thenegative thresholdmasking function (M:= (S< )where
<0).
We suppose that at stept+ 1, the only remaining connection(i;j)is replaced by(k;l):
(
Att; 81u;vn;S (t) S(t)
i;j u;v (9)Att+ 1; 81u;vn;S (t+1) S(t+1)
k;l u;v
The inequality on the gradient update becomes: @LS W(t) x@a k k;l l < @LS W@a i;j xj and following i
the same development as in Eq(8), we haveL(a(t+1) ;a (t+1) ) L(a(t) ;a (t) )0: the loss increases. i k i We proved by contradiction that the guarantees on the decrease of the loss do not hold if we consider k
the absolute value of the score as a proxy for importance.
12