Langevin Monte-Carlo Provably Learns Depth Two Neural Nets at Any Size and Data

Abstract

In this work, we will establish that the Langevin Monte-Carlo algorithm can learn depth-2 neural nets of any size and for any data and we give non-asymptotic convergence rates for it. We achieve this via showing that in q-Renyi divergence, the iterates of Langevin Monte Carlo converge to the Gibbs distribution of Frobenius norm regularized losses for any of these nets, when using smooth activations and in both classification and regression settings. Most critically, the amount of regularization needed for our results is independent of the size of the net. This result achieves a synthesis of several recent observations about isoperimetry conditions under which LMC converges and that two-layer neural loss functions can always be regularized by a certain constant amount such that they satisfy the Villani conditions, and thus their Gibbs measures satisfy a Poincare inequality.