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We show that any target network of width d and depth l can be approximated by pruning a random network that is a factor O(log(dl)) wider and twice as deep

Optimal Lottery Tickets via Subset Sum: Logarithmic Over-Parameterization is Sufficient

NIPS 2020, (2020)

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Abstract

The strong lottery ticket hypothesis (LTH) postulates that one can approximate any target neural network by only pruning the weights of a sufficiently overparameterized random network. A recent work by Malach et al. [<a class="ref-link" id="c1" href="#r1">1</a>] establishes the first theoretical analysis for the strong LTH: one can provab...More

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Introduction
  • Many of the recent unprecedented successes of machine learning can be partially attributed to stateof-the-art neural network architectures that come with up to tens of billions of trainable parameters.
  • Test accuracy is one of the gold standards in choosing one of these architectures, in many applications having a “compressed” model is of practical interest, due to typically reduced energy, memory, and computational footprint [2,3,4,5,6,7,8,9,10,11,12,13,14,15]
  • Such a compressed form can be achieved by either modifying the architecture to be leaner in terms of number of weights, or by starting with a highaccuracy network and pruning it down to one that is sparse in some representation domain, while not sacrificing much of the original network’s accuracy.
  • Several of these pruning methods require many rounds of pruning and retraining, resulting in a time-consuming and hard to tune iterative meta-algorithm
Highlights
  • Many of the recent unprecedented successes of machine learning can be partially attributed to stateof-the-art neural network architectures that come with up to tens of billions of trainable parameters
  • We provide a lower bound for 2-layered networks that matches the upper bound proposed in Theorem 1, up to logarithmic terms with regards to the width: Theorem 2. There exists a 2-layer neural network with width d which cannot be approximated to error within by pruning a randomly initialized 2-layer network, unless the random network has width at least Ω(d log(1/ ))
  • We present our results for approximating a target network by pruning a sufficiently overparameterized neural network
  • We verify our results empirically by approximating a target network via SUBSETSUM in Experiment 1, and by pruning a sufficiently over-parameterized neural network that implements the structures in Figures 1b and 1c in Experiment 2
  • It would be interesting to extend the results to convolutional neural networks
  • As remarked in Malach et al [1], the strong lottery ticket hypothesis (LTH) implies that pruning an over-parameterized network to obtain good accuracy is NP-Hard in the worst case
Methods
  • The authors verify the results empirically by approximating a target network via SUBSETSUM in Experiment 1, and by pruning a sufficiently over-parameterized neural network that implements the structures in Figures 1b and 1c in Experiment 2.
  • The 397, 000 weights in the target network were approximated with 3, 725, 871 coefficients in 21.5 hours on 36 cores of a c5.18xlarge AWS EC2 instance
  • Such a running time is attributed to solving many instances of this nontrivial combinatorial problem
Results
  • The authors approximate a two-layer, 500 hidden node target network with a final test set accuracy of 97.19%.
Conclusion
  • In this paper the authors establish a tight version of the strong lottery ticket hypothesis: there always exist subnetworks of randomly initialized over-parameterized networks that can come close to the accuracy of a target network; further this can be achieved by random networks that are only a logarithmic factor wider than the original network.
  • Other interesting structures that come up in neural networks are sparsity and low-rank weight matrices.
  • This leads to the question of whether the authors can leverage the additional structure in the target network to improve the results.
  • An interesting question from a computational point of view is whether the analysis gives insights to improve the existing pruning algorithms [26].
  • It is an interesting future direction to find efficient algorithms for pruning which provably work under mild assumptions on the data
Summary
  • Introduction:

    Many of the recent unprecedented successes of machine learning can be partially attributed to stateof-the-art neural network architectures that come with up to tens of billions of trainable parameters.
  • Test accuracy is one of the gold standards in choosing one of these architectures, in many applications having a “compressed” model is of practical interest, due to typically reduced energy, memory, and computational footprint [2,3,4,5,6,7,8,9,10,11,12,13,14,15]
  • Such a compressed form can be achieved by either modifying the architecture to be leaner in terms of number of weights, or by starting with a highaccuracy network and pruning it down to one that is sparse in some representation domain, while not sacrificing much of the original network’s accuracy.
  • Several of these pruning methods require many rounds of pruning and retraining, resulting in a time-consuming and hard to tune iterative meta-algorithm
  • Objectives:

    It does not reflect the findings in Ramanujan et al [26] that only seem to require a constant factor over-parameterization, e.g., a randomly initialized Wide ResNet50, can be pruned to a model that has the accuracy of a fully trained ResNet34.
  • The authors' goal is to address the following question:.
  • The authors' goal is to approximate a target network f (x) by pruning a larger network g(x), where x ∈ Rd0.
  • The authors' goal is to obtain a pruned version of g by eliminating weights, i.e.,.
  • The authors' objective is to obtain a good approximation while controlling the size of g(·), i.e., the width of Mi’s
  • Methods:

    The authors verify the results empirically by approximating a target network via SUBSETSUM in Experiment 1, and by pruning a sufficiently over-parameterized neural network that implements the structures in Figures 1b and 1c in Experiment 2.
  • The 397, 000 weights in the target network were approximated with 3, 725, 871 coefficients in 21.5 hours on 36 cores of a c5.18xlarge AWS EC2 instance
  • Such a running time is attributed to solving many instances of this nontrivial combinatorial problem
  • Results:

    The authors approximate a two-layer, 500 hidden node target network with a final test set accuracy of 97.19%.
  • Conclusion:

    In this paper the authors establish a tight version of the strong lottery ticket hypothesis: there always exist subnetworks of randomly initialized over-parameterized networks that can come close to the accuracy of a target network; further this can be achieved by random networks that are only a logarithmic factor wider than the original network.
  • Other interesting structures that come up in neural networks are sparsity and low-rank weight matrices.
  • This leads to the question of whether the authors can leverage the additional structure in the target network to improve the results.
  • An interesting question from a computational point of view is whether the analysis gives insights to improve the existing pruning algorithms [26].
  • It is an interesting future direction to find efficient algorithms for pruning which provably work under mild assumptions on the data
Funding
  • Acknowledgments and Disclosure of Funding DP wants to thank Costis Daskalakis and Alex Dimakis for early discussions of the problem over a nice sushi lunch during NeurIPS2019 in Vancouver, BC. This research is supported by an NSF CAREER Award #1844951, a Sony Faculty Innovation Award, an AFOSR & AFRL Center of Excellence Award FA9550-18-1-0166, and an NSF TRIPODS Award #1740707
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Author
Ankit Pensia
Ankit Pensia
Shashank Rajput
Shashank Rajput
Alliot Nagle
Alliot Nagle
Harit Vishwakarma
Harit Vishwakarma
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