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Depth separation -- why a deeper network is more powerful than a shallowerone -- has been a major problem in deep learning theory. Previous results oftenfocus on representation power. For example, arXiv:1904.06984 constructed afunction that is easy to approximate using a 3-layer network but notapproximable by any 2-layer network. In this paper, we show that thisseparation is in fact algorithmic: one can learn the function constructed byarXiv:1904.06984 using an overparameterized network with polynomially manyneurons efficiently. Our result relies on a new way of extending the mean-fieldlimit to multilayer networks, and a decomposition of loss that factors out theerror introduced by the discretization of infinite-width mean-field networks.

Depth Separation with Multilayer Mean-Field Networks