The Combinatorial Brain Surgeon: Pruning Weights That Cancel One ...

The Combinatorial Brain Surgeon: Pruning Weights That Cancel One Another in Neural Networks

Xin Yu * 1 Thiago Serra * 2 Srikumar Ramalingam 3 Shandian Zhe 1

Abstract

Neural networks tend to achieve better accuracy with training if they are larger -- even if the resulting models are overparameterized. Nevertheless, carefully removing such excess of parameters before, during, or after training may also produce models with similar or even improved accuracy. In many cases, that can be curiously achieved by heuristics as simple as removing a percentage of the weights with the smallest absolute value -- even though absolute value is not a perfect proxy for weight relevance. With the premise that obtaining significantly better performance from pruning depends on accounting for the combined effect of removing multiple weights, we revisit one of the classic approaches for impactbased pruning: the Optimal Brain Surgeon (OBS). We propose a tractable heuristic for solving the combinatorial extension of OBS, in which we select weights for simultaneous removal, and we combine it with a single-pass systematic update of unpruned weights. Our selection method outperforms other methods for high sparsity, and the single-pass weight update is also advantageous if applied after those methods. Source code: yuxwind/CBS.

1. Introduction

In a world where large and overparameterized neural networks keep GPUs burning hot because machine learning researchers rethought generalization (Zhang et al., 2017; Belkin et al., 2019) and started tuning neural architectures with a hope for globally convergent loss landscapes (Li et al.,

*Equal contribution 1University of Utah, Salt Lake City, UT, United States 2Bucknell University, Lewisburg, PA, United States 3Google Research, New York, NY, United States. Correspondence to: Shandian Zhe , Thiago Serra .

Proceedings of the 39 th International Conference on Machine Learning, Baltimore, Maryland, USA, PMLR 162, 2022. Copyright 2022 by the author(s).

2018; Sun et al., 2020), network pruning can perhaps save us from parameter redundancy (Denil et al., 2013).

Network pruning can lead to more parameter-efficient networks, with which we can save on model deploying and storage costs, and even to models with better accuracy. Although the extent to which we can prune depends on the task (Liebenwein et al., 2021) and pruning may have an uneven impact across classes if not done properly (Hooker et al., 2019; Paganini, 2020; Hooker et al., 2020; Good et al., 2022), pruning can also make neural networks more robust to adversarial manipulation (Wu & Wang, 2021).

From a perspective of model expressiveness using linear regions, we may see network pruning as a means to close the gap between the highly complex models that can be theoretically learned with an architecture (Pascanu et al., 2014; Montu?far et al., 2014; Telgarsky, 2015; Montu?far, 2017; Arora et al., 2018a; Serra et al., 2018; Serra & Ramalingam, 2020; Xiong et al., 2020; Montu?far et al., 2021) and the relatively less complex models obtained in practice (Hanin & Rolnick, 2019a;b; Tseran & Montu?far, 2021).

Curiously, however, the long-standing magnitude-based pruning approach (Hanson & Pratt, 1988; Mozer & Smolensky, 1989; Janowsky, 1989) remains remarkably competitive (Blalock et al., 2020) despite having equally longstanding evidence that it does not offer a good proxy for parameter relevance (Hassibi & Stork, 1992). But why?

We conjecture that focusing on the impact of removing each parameter alone prevents impact-based methods from being more effective. In other words, we believe that the decision about pruning each parameter should be based on which other parameters are also removed. Ideally, we want the effect of such removals to cancel one another to the largest extent possible while achieving the aimed sparsity.

In this work, we revisit the functional Taylor expansion of the loss function L on a choice of weights w RN around the learned weights w? RN of the neural network as

L(w) - L(w?) =(w - w?)T L(w?)

+

1 2

(w

-

w? )T

2 L(w? )(w

-

w? )

+ O( w - w? 3),

The Combinatorial Brain Surgeon: Pruning Weights That Cancel One Another in Neural Networks

which is the basis for classic methods such as Optimal Brain Damage (OBD) by LeCun et al. (1989) and Optimal Brain Surgeon (OBS) by Hassibi & Stork (1992).

Like in OBD and OBS, we assume that (i) the training

converged to a local minimum, so L(w?) = 0; and (ii) w is sufficiently close to w?, so O( w - w? 3) 0. Hence,

L(w)

-

L(w? )

1 2

(w

-

w?)T 2L(w?)(w

-

w? ).

(1)

Under such conditions, we can assess the impact of pruning the i-th weight with wi = 0 and wj = w?j j = i.

Local quadratic models of the loss function have been used in varied ways, then and now. In OBD, the Hessian matrix H := 2L(w~) is further assumed to be diagonal, hence implying that pruning one weight has no impact on pruning the remaining weights. In OBS, the Hessian matrix H is no longer assumed to be diagonal, and the optimality conditions are used to update the remaining weights of the network based on removing the weight which causes the least approximate increase to the loss function. The first modern revival of this approach is the Layerwise OBS by Dong et al. (2017), in which each layer is pruned independently from the others. More recently, WoodFisher by Singh & Alistarh (2020) operates over all the layers by introducing a new method that more efficiently approximates the Hessian inverse H-1, which is necessary for the weight updates of OBS. In a sense, those modern revivals focused on keeping the calculation of H-1 manageable.

However, current OBS approaches do not account for the effect of one pruning decision on other pruning decisions. Whereas OBS does take into account the impact of pruning a given weight on the unpruned weights, only the weight with smallest approximated impact on the loss function is pruned at each step. In the most recent work, Singh & Alistarh (2020) described the updates involved on choosing two weights for simultaneous removal, but nevertheless observed that considering the removal of multiple weights together would be impractical if performed in such a way.

While not disagreeing with Singh & Alistarh's stance, we nevertheless proceed to formulate this problem and then consider how to approach it in a tractable way. In a nutshell, the contributions of this paper are the following:

(i) We propose a formulation of the Combinatorial Brain Surgeon (CBS) problem using Mixed-Integer Quadratic Programming (MIQP), which for tractability is decomposed into the problems of pruned weight selection and unpruned weight update (Section 3);

(ii) We propose a local search algorithm to improve the selection of the weights to be pruned (Section 4);

(iii) We propose a randomized extension of magnitudebased pruning to obtain a diverse pool of starting

points for the local search algorithm (Section 5); and

(iv) We decouple the systematic weight update from weight selection to circumvent the scalability issue anticipated by Singh & Alistarh (2020) (Section 6).

We focus on single-shot pruning after training to make before and after comparisons easier; as well as to facilitate comparing the results of our method with existing work, in particular that of Singh & Alistarh (2020). We discuss additional related work in Section 2, evaluate the proposed algorithms in Section 7, and draw conclusions in Section 8.

2. Related Work

Blalock et al. (2020) observes that most work in network pruning relies on either magnitude-based or impact-based methods for selecting which weights to remove from the neural network. In fact, the list of key references for each is so long that we will use different paragraphs for each.

Magnitude-based methods select the weights with smallest absolute value (Hanson & Pratt, 1988; Mozer & Smolensky, 1989; Janowsky, 1989; Han et al., 2015; 2016; Li et al., 2017; Frankle & Carbin, 2019; Elesedy et al., 2020; Gordon et al., 2020; Tanaka et al., 2020; Liu et al., 2021b).

Impact-based methods aim to select weights which would have less impact on the model if removed. That includes gradient-based approaches such as ours but we also regard other approaches (LeCun et al., 1989; Hassibi & Stork, 1992; Hassibi et al., 1993; Lebedev & Lempitsky, 2016; Molchanov et al., 2017; Dong et al., 2017; Yu et al., 2018; Zeng & Urtasun, 2018; Baykal et al., 2019; Lee et al., 2019; Wang et al., 2019; Liebenwein et al., 2020; Wang et al., 2020; Xing et al., 2020; Singh & Alistarh, 2020).

To those we can add the recent stream of exact methods, which aim to preserve the model intact while pruning the network (Serra et al., 2020; Sourek & Zelezny, 2021; Serra et al., 2021; Chen et al., 2021; Ganev & Walters, 2022). To the best of our understanding, these methods are currently only beneficial under specific conditions.

In addition to the discussion of pruning two parameters under OBS by Singh & Alistarh (2020), other recent works consider joint parameter pruning in neural networks. Chen et al. (2021) partition the parameters into zero-invariant groups for removal. Liu et al. (2021a) identify coupled channels that should be either kept or pruned together.

Across all these types of approaches, most work has been done on pruning trained neural networks and then fine tuning them afterwards. However, there is an growing body of work on pruning during training or at initialization (Frankle & Carbin, 2019; Lee et al., 2019; Liu et al., 2019; Lee et al., 2020; Wang et al., 2020; Renda et al., 2020; Tanaka et al.,

The Combinatorial Brain Surgeon: Pruning Weights That Cancel One Another in Neural Networks

2020; Frankle et al., 2021; Zhang et al., 2021).

The interest for the latter topic is in part attributed to the Lottery Ticket Hypothesis (LTH), according to which randomly initialized dense neural networks contain subnetworks that can be trained to achieve similar accuracy as the original network (Frankle & Carbin, 2019). That also led to work on the strong LTH, according to which one can improve the accuracy of randomly initialized models by pruning instead of training (Zhou et al., 2019; Ramanujan et al., 2020; Malach et al., 2020; Pensia et al., 2020; Orseau et al., 2020; Qian & Klabjan, 2021; Chijiwa et al., 2021).

Another common theme is the formulation of optimization models for network pruning, which include other references not mentioned above (He et al., 2017; Luo et al., 2017; Aghasi et al., 2017; ElAraby et al., 2020; Ye et al., 2020; Verma & Pesquet, 2021; Ebrahimi & Klabjan, 2021).

There are also more general approaches that would be better described as compression than as pruning methods, such as combining neurons and low-rank approximation, factorization, and random projection of weight matrices (Jaderberg et al., 2014; Denton et al., 2014; Lebedev et al., 2015; Srinivas & Babu, 2015; Mariet & Sra, 2016; Arora et al., 2018b; Wang et al., 2018; Su et al., 2018; Wang et al., 2019; Suzuki et al., 2020a;b; Suau et al., 2020; Li et al., 2020).

3. Pruning as an Optimization Problem: the Combinatorial Brain Surgeon

We formulate the Combinatorial Brain Surgeon (CBS) as an optimization problem based on the local quadratic model of the loss function previously described.

For a sparsity rate r (0, 1] corresponding to the fraction of the weights w RN of a neural network to be pruned, we formulate the problem of selecting the rN weights to remove and the remaining N - rN weights to update with minimum approximate loss. In order to model the loss, we use w? to denote the weights of the trained neural network before pruning, as well as H to denote the Hessian matrix 2L(w?) and Hi,j to denote the element at the i-th row and j-th column. That yields the following Mixed-Integer Quadratic Programming (MIQP) formulation:

min subject to

1N 2

N

(wi - w?i)Hi,j (wj - w?j )

(2)

i=1 j=1

N

yi = rN

(3)

i=1

yi wi = 0

i {1, . . . , N } (4)

yi {0, 1}

i {1, . . . , N } (5)

wi R

i {1, . . . , N } (6)

The decision variables of this formulation are, for each weight i, the updated value wi of that weight and a binary variable yi denoting whether the weight is pruned or not.

In order to obtain a tractable approach to CBS, we consider two special cases of this formulation in what follows.

CBS Selection First we consider the CBS Selection (CBSS) formulation, in which we aim to minimize the approximate loss from pruning a selection of weights without updating the unpruned weights. For the formulation above, that means wi = w?i if yi = 0 and wi = 0 if yi = 1. With each weight having a binary domain, wi {0, w?i}, only pairs of weights which are both removed affect the objective function. Hence, we can abstract the decision variables associated with weights and avoid implementing the indicator constraint in Equation (4), which could potentially lead to numerical difficulties. That yields the following Integer Quadratic Programming (IQP) formulation:

min subject to

1N 2

N

w?iyiHi,j w?j yj

(7)

i=1 j=1

N

yi = rN

(8)

i=1

yi {0, 1}

i {1, . . . , N } (9)

The formulation above is at the core of how we identify a

combination of pruned weights affecting the local approx-

imation of the loss function by the least amount. Namely,

the impact of pruning both weights i and j is captured by

1 2

(wiHi,j wj

+

wj Hj,iwi).

In

other

words,

the

impact

of

pruning a weight depends on the other pruned weights. If

weight updates are not considered, CBS-S is all you need.

We can linearize this formulation by replacing yiyi with yi and replacing yiyj, i = j, with a binary decision variable zi,j and the constraints zi,j yi, zi,j yj, and zi,j yi + yj - 1 (Padberg, 1989). However, that would make

the number of variables and constraints grow quadratically,

which would be impractical for large neural networks.

CBS Update Next we consider the CBS Update (CBS-U) formulation, in which we aim to minimize the approximate loss from updating the unpruned weights. Given a solution y~ of CBS-S, we state that wi = 0 if y~i = 0. That yields the following Quadratic Programming (QP) formulation:

min subject to

1N 2

N

(wi - w?i)Hi,j (wj - w?j )

(10)

i=1 j=1

wi = 0

i {1, . . . , N } : y~i = 1 (11)

wi R

i {1, . . . , N } : y~i = 0 (12)

For generality, we do not assume H to be symmetric.

The Combinatorial Brain Surgeon: Pruning Weights That Cancel One Another in Neural Networks

By abstracting the pruned weights altogether, we can reformulate CBS-U as an unconstrained quadratic optimization problem which can be efficiently solved (see Section 6).

Dissociating CBS into two subproblems has consequences, good and bad. On the one hand, solving each subproblem to optimality does not necessarily imply that we would obtain an optimal solution to CBS. However, it is very unlikely that we would obtain an optimal solution even to CBS-S for neural networks of reasonable size in the first place. If we were to nevertheless contemplate such a possibility, we could use a Benders-type decomposition (Benders, 1962; Hooker & Ottosson, 2003) through which we would alternate between solving a variant of CBS-S and CBS-U as formulated above by iteratively adding additional constraints to the initial formulation of CBS-S. These constraints would capture the change to the loss function due to weight updates for each selection of weights to prune obtained by solving the variant of CBS-S in prior steps.

On the other hand, this dissociation of the weight selection and weight update problems leaves most of the computational difficulty to the former. This is particularly beneficial because it allows us to explore tried-and-tested techniques commonly used to solve discrete optimization problems.

Although CBS-S -- and even CBS -- could potentially be fed into Mixed-Integer Programming (MIP) solvers capable of producing optimal solutions, that would not scale to neural networks of reasonable sizes. We refer the reader interested in formulations and algorithms for MIP problems to Conforti et al. (2014). Interestingly, the algorithmic improvements of MIP solvers have been comparable to their contemporary hardware improvements for decades (Bixby, 2012), but before that happened the tricks of the trade were different: they involved designing good heuristics. In the next sections, we will resort to heuristic techniques that allowed optimizers to obtain good solutions for seemingly intractable problems subject to the solvers and hardware of their time -- and even to those of today in cases like ours.

4. A Greedy Swapping Local Search Algorithm to Improve Pruning Selection

The local search is where we take full advantage of the interdependence between pruned weights in our approach. Local search methods are used to produce better solutions -- or a diversified pool of solutions -- based on adjusting an initial solution, which in our case would be a selection of weights P to be pruned. We refer the reader interested in local search to Aarts & Lenstra (1996).

In particular, our local search method iteratively swaps a weight in P with a weight not in P. Similar operations have been long used for the traveling salesperson problem (Applegate et al., 2006), in which the 2-opt (Flood, 1956; Croes,

1958) and the 3-opt (Lin, 1965) algorithms respectively swap two and three arcs from the tour with other arcs having a smaller sum of weights as long as an improvement is found. In our case, since minimizing the loss function on the training set may not necessarily lead to better test set accuracy, we found that avoiding swaps with negligible loss improvement led to better results.

Algorithm 1 describes our local search method. The outer loop repeats for stepsmax steps, unless the sample loss is not improved in noimpmax consecutive steps or a step concludes without changing the set of pruned weights P. At every repetition of the outer loop, we initialize the sets I P and J P? := {1, . . . , N } \ P that will respectively keep track of the weights in P that are no longer pruned and the weights in P? that are pruned in lieu of those in I. For each weight i P, we calculate i in Line 7 as the impact of pruning weight i if the other pruned weights are also those in P. The weights in P are then sorted in a sequence of nonincreasing values in Line 9, hence from largest to the smallest impact. For each weight j P?, we calculate j in Line 11 as the impact of having weight j removed in addition to all the weights in P. The weights in P? are then sorted in a sequence of nondecreasing impact if swapped with the first element i := 1 in Line 13. Hence, the first element j := 1 yields the greatest reduction if swapped with i. If swapping i and j does not yield a reduction of at least , the local search stops in Line 15. Otherwise, we loop with variable i over the currently pruned weighs in P and with variable j over the unpruned weights in P? between Line 18 and Line 29. The weights in P are visited according to the sequence , and for the element at the ii-th position of we consider only the elements in P? between positions ii - and ii + of sequence . There is no guarantee that the subsequent pairs of weights are sorted from largest to smallest reduction if swapped, but we loop on those to amortize the large cost of sorting the parameters in sequences and . We keep the number of steps approximately linear by limiting that each element in P can only be swapped with at most 2 + 1 elements in P?. In addition, the loop is interrupted once the number of parameters in P that could not be swapped reaches . Parameter is used in Line 14 and Line 21 to restrict changes to the pruning set P to cases in which the local estimate of the loss function improves by at least that much. Otherwise, we can observe that after a few swapping only minor changes on the loss occur and thus one weight may swap in and out of P again. To simplify notation, we do not divide all the calculated impacts by 2. For a concrete illustration, we visualize the basic swapping operation in Figure 1.

We refer to Appendix B for the matrix operations to calculate , , and with GPUs; and we refer to Appendix C for how to approximate and decompose the Hessian matrix H in order to reduce time and space complexity.

The Combinatorial Brain Surgeon: Pruning Weights That Cancel One Another in Neural Networks

Algorithm 1 Prune Selection Swapping Local Search

1: Input: P - initial set of pruned weights, - min impact variation for changing P, - max failed weight swap attempts in P, - range of candidates for each weight swap, stepsmax - max number of steps, noimpmax max number of non-improving steps

2: Output: updated set of pruned weights PF

3: Initialize PF P, sImprove 0,

4: for s 1 to stepsmax do

5: I ; J

6: for i P do

7:

i w?iHi,iw?i

+ jP:j=i (w?iHi,j w?j + w?j Hj,iw?i)

8: end for

9: Compute a sequence of i P by nonincreasing i 10: for j P? := {1, . . . , N } \ P do

11:

j w?j Hj,j w?j

+ iP (w?iHi,j w?j + w?j Hj,iw?i) 12: end for

13: Compute a sequence of j P? by nondecreasing

j - 1,j , where i,j := w?iHi,j w?j + w?j Hj,iw?i 14: if 1 - 1,1 - 1 > - then

15: Terminate

16: end if

17: c = 0

18: for i [1, . . . , |P|] do 19: ii index of i in ; c c + 1

20:

for j [max{1,ii-}, . . . , min{|P?|,ii+}] \ J do

21:

if j + j,j - i ,j - (i,j ) -

j J

i I

i + i,j - i,i - then

j J

i I

22:

I I + i; J J + j; c c - 1

23:

Goto Line 26

24:

end if

25: end for

26:

if c then

27:

Goto Line 30

28: end if

29: end for

30: if I = then

31: Terminate

32: end if

33: P P J \ I

34: if L(P) < L(PF ) then

35:

PF P; sImprov s

36: else if (s - sImprov) > noimpmax then

37: Terminate

38: end if

39: end for

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?

?

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1

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i

?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

??

AN 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

j

?AAAE6XicnVRNb9MwGM7WAqN8bXDkYjEV7VBVyTQBQpq0wWWnaSD2Ic0lchxnc+ckre10VFb6C7hwACGu/CNu/Bp47aRl3TgwnK/3eT+e57UtJxoIrrTv/1xYbDRv3Ly1dLt15+69+w+WVx4eqLyQlO3TXOTyKCKKCZ6xfc21YEcDyUgaCXYYnb228cMRk4rn2Ts9HrBeSk4ynnBKNLjClcVfOGYJZqNzY87LsmURAjg2ZgwQIdTGKc/QBAbCiSTUBKVZLxFWRRoavhmU73dr0K8BVIcc4YhIY4lLADuQ2umXVax/MQYA/5fMvMAUjC3jVG3WCiSF/UpFsw9apkZ1dbes5J7Oi9iCTYQFZVwgiUBVOtOVQ7IjlPzkVBMp8/OK3Jb4CA+HBYlnnySXRAgE6TAzbIIOsMa5Vh1LClMTJGLCsOFLnsV2Q3JZThdjquMK/Q4KsGv2ugozsrpJl5YSfRpF5m01/cnk+ryOVbrgmt8JegBRq222y9DsBqUl9GcvTG3hZBtCu66j9p9Y/a5TJv40XN3zbjyqkM3/N9OVzZry4ZrrpVWhcHnV7/puoKtGUBurXj32wuUfOM5pkbJMU0GUOg78ge4ZIjWnggFtodiA0DNywo7BzEjKVM+4k1qiNnhiBMsLT6aR816sMCRVapxGkGm3SV2OWeffYseFTl70DM8GhWYZrYSSQiCdI3vsUcwlo1qMwSBUcugV0VMC50zDz6EFixBcnvJV42C9GzzrbrzZWN16VS/HkvfYe+KteYH33Nvydrw9b9+jjaTxsfG58aV51vzU/Nr8VqUuLtQ1j7y50fz+G+hfm/c=

??

AN 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

N

Figure 1. Swapping weight i P and j P? in Algorithm 1. Assume that the first i weights are in P and others are in P? before pruning (left). Each Akm := w?kHk,mw?m. Note that only when both weights k and m are selected, namely yk = ym = 1, Akm counts in the objective as in equation 7. We highlight such elements

in orange in both left and right sub-figures for before and after

swapping respectively. We further highlight the elements used by i in dark orange, elements used by j in dark blue, elements used by ij with red circles before swapping (left). The objective change after pruning is -i + j - i,j.

5. A Greedy Randomized Constructive Algorithm for Pruning Selection

The success of local search typically depends on the quality of the initial solution. Since magnitude-based pruning can be rather effective as a stating point, even if not always perfect, we consider how to leverage its guidance while applying the diversification tricks of old-school optimizers.

In particular, we propose a constructive heuristic algorithm along the lines of semi-greedy methods (Hart & Shogan, 1987; Feo & Resende, 1989; 1995). These methods rely on a metric for identifying elements that are generally associated with good solutions. In our case, we know that selecting the weights with the smallest absolute value tends to be effective. In addition, some randomness is introduced in the actual selection of elements by semi-greedy methods, by which we may not necessarily always obtain a solution with the top-ranked elements for the chosen metric. By repeating the construction a sufficient number of times, we sample several solutions that closely follow the metric rather than obtain one solution that follows it rather strictly.

Algorithm 2 describes our constructive method. The outer loop from Line 4 to Line 7 produces S selections of weights to prune, among which the one with smallest sample loss is chosen. The inner loop from Line 12 to Line 16 iteratively partitions the weights of the neural network into B buckets of roughly the same size, and in which of those we select the same proportion of weights to be pruned. The partitioning step introduces randomness to the weight selection while still making the weights with smallest absolute value more likely to be pruned. This construction is highly paralleliz-

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