Speaker: Andrea Montanari (Stanford)

Title: A Mean Field View of the Landscape of Two-Layers Neural Networks

Abstract: Multi-layer neural networks are among the most powerful models in machine learning and yet, the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires to optimize a highly non-convex and high-dimensional objective (risk function), a problem which is usually attacked using stochastic gradient descent (SGD). Does SGD converge to a global optimum of the risk or only to a local optimum? In the first case, does this happen because local minima are absent, or because SGD somehow avoids them? In the second, why do local minima reached by SGD have good generalization properties? We consider a simple case, namely two-layers neural networks, and prove that –in a suitable scaling limit– the SGD dynamics is captured by a certain non-linear partial differential equation. We then consider several specific examples, and show how the asymptotic description can be used to prove convergence of SGD to network with nearly-ideal generalization error. This description allows to ‘average-out’ some of the complexities of the landscape of neural networks, and can be used to capture some important variants of SGD as well. [Based on joint work with Song Mei and Phan-Minh Nguyen]