NeurIPS 2020

Greedy Optimization Provably Wins the Lottery: Logarithmic Number of Winning Tickets is Enough

Review 1

Summary and Contributions: The paper proposed a greedy optimization algorithm for neural network pruning. The method has guaranteed discrepancy between pruned network and original network. The method is validated on several network structure and different tasks.

Strengths: 1. The paper proposed a new greedy optimization based prunning method with smaller error rate. 2. The paper is written clearly, and experiment is validated sufficiently.

Weaknesses: 1. More comparision and analysis on pruning compared with knowledge distill, network structure design, NAS is helpful for better understand the task. 2. Provide more details on practical efficiency of the prunning algorithm to reach a good small network

Correctness: Yes

Clarity: Yes

Relation to Prior Work: Yes. More analysis on knowledge distill, network structure design, and NAS based approach is helpful.

Reproducibility: Yes

Additional Feedback:

Review 2

Summary and Contributions: This paper proposes and analyzes greedy methods for neural network pruning. The methods are based on local or global reconstruction losses: greedily selecting neurons to include in the current layer to match the unpruned output of either the current layer or the final layer. Analysis demonstrates a better theoretical error rate than prior work, and shows competitive experimental results.

Strengths: The paper provides a simple greedy pruning algorithm (along with more complex variations for efficiency and better performance). The analysis is deeper than is typical in related papers and may provide some insight to future work. Experiments demonstrate reasonable performance on three large scale tasks.

Weaknesses: The paper is unclear in some critical areas and the overall structure of the argument takes some work to tease out (as detailed below). Further, the theoretical argument, while intriguing, seems to have little consequence in the experiments in which the loosely bound global method generally outperforms the more tightly bound local method. This raises the question of whether the long theoretical analysis was worth the effort and allocated space.

Correctness: I have not checked all proofs and derivations in the appendices. Much of what I have checked seems reasonable, however the crux of the argument lies with Theorem 1 and, as discussed below, I haven’t verified the proof. Theorem 1 claims an exponential convergence rate for global reconstruction error w.r.t. to local greedy reconstruction (while exhaustively selecting optimal neuron k at each step and computing the optimal \gamma via a convex optimization). However, each subsequent modification weakens this convergence rate from exponential to polynomial: using a fixed \gamma; selecting w.r.t. global reconstruction loss instead of local; and applying a Taylor approximation for speedup. (a) The argument that the full algorithm yields an exponential rate consists of selecting the better performing method—global or local—at each step, and then applying the local bound from Theorem 1 either directly—because the local solution was selected—or transitively—since picking the global solution means the global bound must have been too weak in that case. This broad outline of the proof structure between Theorem 1 and the convergence rate of Algorithm 1 seems to be only addressed in the supplementary section, but should be stated explicitly in the main text. (b) Given its importance, I tried to follow the proof of Theorem 1, but was repeatedly stuck on the notation. Some initial terms are undefined and just doing setup work using variables from Lemmas 1-3. However the references to \h and \bar\h confused me and I couldn’t find the right definitions to make sense of it. Perhaps this is my fault, but given the heavy lifting that this Theorem performs, it deserves a clearer proof, along with a brief intuition of the proof supplied in the main body. (c) Then there is the question of why the loosely-bounded global method would so consistently outperform the local method? Perhaps the slope is better, but the constant is worse?

Clarity: As noted above, there are things that should be clarified in the arguments for the convergence rate. In addition, there are numerous errors in grammar (e.g. verb tenses, missing conjunctions, capitalization). In addition, these items could use additional clarification: (d) Many methods apply pruning to outputs, e.g. pruning rows from FC layer or filter blocks from conv. layers. In Section 2, this method appears to prune inputs to a layer, e.g. pruning columns from an FC layer or a single filter from each filter block. These two should be equivalent in practice, but the distinction would be helpful to make explicit to avoid confusion. (Related, identifying which variables in line 53 are scalars or vectors would be clarifying.) (e) The definition and usage for U_i in line 97 is also unclear and may be incorrect. Plugging in 0.3 for a_i(k) yields [0.4, 1]. Does this make sense? (f) What are the memory requirements and algorithmic complexities for the greedy local optimization in (4), as well as subsequent alternative methods (e.g. Taylor)? (g) Line 159 mentions ancillary parameters—from the supplementary, there appears to be one additional parameter per “neuron”, denoted `b`. Is this correct? And is parameter `a`, seemingly a scale on the activation, standard in these networks? (h) As noted above, give reasoning for bound in line 164. (i) How is the method affected by batch normalization?

Relation to Prior Work: A brief summary of related work is made in the final section that broadly sets this work in related context. (j) Could the bound in Theorem 1 be adapted to other greedy methods, e.g. could the bound in Ye et al. (2020) be too loose? Specifically, how does your greedy method differ? (k) How does your Taylor method relate to the method proposed in Molchanov (2017b)? This method also uses auxiliary variables.

Reproducibility: Yes

Additional Feedback:

Review 3

Summary and Contributions: This paper proposed a greedy optimization based pruning and provided theoretical analysis to show that under some mild assumptions, sub-networks of size n with a significantly smaller O(exp(-cn)) error rate can be achieved. This work is highly related to Ye et al. (2020) and can be viewed as an extension which improves the error rates and relax the constraints in Ye et al. (2020). The proposed method is evaluated with ResNet, MobilenetV2/V3 on ImageNet and achieve better performance than Ye et al. (2020) and some other methods.

Strengths: -The contribution and novelty of this paper are clear。 -The theoretical analysis seems correct and reasonably good. -The extension of the conclusion from Ye et al. (2020) from two-layer conv nets and over-parameterized assumption to entire net with mild assumptions are meaningful and provides insights on that the pruning method is provably better than direct training with gradient descent. -Evaluation especially the ablation studies and analysis on local and global Imitation is solid and provides useful insights to better understand the method.

Weaknesses: -The discussion between the proposed method and existing ones can be improved. First, since this paper is highly related with Ye et al. (2020), it is better to specify with a little bit more details about the limitation of Ye et al. (2020) and how this paper addresses them. Meanwhile, despite the difference, the proposed local and global Imitation methods are related to some of the existing methods such as the ones that uses local or global activations to conduct network pruning. It would be nice to mention the similarity and difference among existing methods and the proposed one more explicit so the readers can better link them together. -Although the empirical experiments show better overall performance than Ye et al. (2020) and some other methods, but the margins are limited. It would be better to provide some discussions on why the theoretically significant improvements are not fully observed empirically.

Correctness: Yes, but can be improved.

Clarity: Yes, but can be improved.

Relation to Prior Work: Yes, but can be improved.

Reproducibility: No

Additional Feedback: ----After rebuttal: The authors' efforts on rebuttal are appreciated. My concerns regarding the comparison with Ye et al. (2020) are partially addressed. So I will remain my original rating and vote for acceptance.

Review 4

Summary and Contributions: This paper proposes to extend the previous GFS method as an optimization problem instead of searching. It gives a better decay rate and better accuracy of pruned models.

Strengths: The proposed method achieved much better results than previous GFS.

Weaknesses: The method of combining local and global imitation lacks theoretical analysis. Since it improves the performance, how does it affect convergence rate during pruning?

Correctness: Technically correct. However, I'm questioning about using Lipschitz continuous to prove convergence rate is tight enough analysis to explain the benefits of proposed methods.

Clarity: The paper is clearly written.

Relation to Prior Work: Previous work is well addressed. Since the proposed method is highly related to GFS, I suggested adding more ablation study to compare them in detail.

Reproducibility: Yes

Additional Feedback: