Neural network compression techniques have become increasingly popular as they can drastically reduce the storage and computation requirements for very large networks. Recent empirical studies have illustrated that even simple pruning strategies can be surprisingly effective, and several theoretical studies have shown that compressible networks (in specific senses) should achieve a low generalization error. Yet, a theoretical characterization of the underlying cause that makes the networks amenable to such simple compression schemes is still missing. In this study, we address this fundamental question and reveal that the dynamics of the training algorithm has a key role in obtaining such compressible networks. Focusing our attention on stochastic gradient descent (SGD), our main contribution is to link compressibility to two recently established properties of SGD: (i) as the network size goes to infinity, the system can converge to a mean-field limit, where the network weights behave independently, (ii) for a large step-size/batch-size ratio, the SGD iterates can converge to a heavy-tailed stationary distribution. In the case where these two phenomena occur simultaneously, we prove that the networks are guaranteed to be '$\ell_p$-compressible', and the compression errors of different pruning techniques (magnitude, singular value, or node pruning) become arbitrarily small as the network size increases. We further prove generalization bounds adapted to our theoretical framework, which indeed confirm that the generalization error will be lower for more compressible networks. Our theory and numerical study on various neural networks show that large step-size/batch-size ratios introduce heavy-tails, which, in combination with overparametrization, result in compressibility.

本研究揭示了训练算法的动态特性在实现神经网络压缩方案上起着关键作用，通过将压缩性与随机梯度下降的近期性质联系起来，本文证明了当神经网络遵循一定条件时，网络参数具有稀疏性，并且压缩误差可以任意减小，同时可证明良好的压缩性可降低泛化误差，该理论得到了各种神经网络的数值实验验证。

随机梯度下降中的重尾以及过度参数化神经网络的可压缩性