Depth Separation with Multilayer Mean-Field Networks

Yunwei Ren, Mo Zhou, Rong Ge

Keywords: depth separation, mean-field, nonconvex optimization

Abstract: Depth separation—why a deeper network is more powerful than a shallow one—has been a major problem in deep learning theory. Previous results often focus on representation power, for example, Safran et al. (2019) constructed a function that is easy to approximate using a 3-layer network but not approximable by any 2-layer network. In this paper, we show that this separation is in fact algorithmic: one can learn the function constructed by Safran et al. (2019) using an overparametrized network with polynomially many neurons efficiently. Our result relies on a new way of extending the mean-field limit to multilayer networks, and a decomposition of loss that factors out the error introduced by the discretization of infinite-width mean-field networks.

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TL;DR: We show that, using gradient flow, 3-layer networks can efficiently learn a function that no 2-layer networks can efficiently approximate.