Topology and Geometry of Half-Rectified Network Optimization

Abstract

The loss surface of deep neural networks has recently attracted interest in the optimization and machine learning communities as a prime example of high-dimensional non-convex problem. Some insights were recently gained using spin glass models and mean-field approximations, but at the expense of strongly simplifying the nonlinear nature of the model. In this work, we do not make any such assumption and study conditions on the data distribution and model architecture that prevent the existence of bad local minima. Our theoretical work quantifies and formalizes two important \emph{folklore} facts: (i) the landscape of deep linear networks has a radically different topology from that of deep half-rectified ones, and (ii) that the energy landscape in the non-linear case is fundamentally controlled by the interplay between the smoothness of the data distribution and model over-parametrization. Our main theoretical contribution is to prove that half-rectified single layer networks are asymptotically connected, and we provide explicit bounds that reveal the aforementioned interplay. The conditioning of gradient descent is the next challenge we address. We study this question through the geometry of the level sets, and we introduce an algorithm to efficiently estimate the regularity of such sets on large-scale networks. Our empirical results show that these level sets remain connected throughout all the learning phase, suggesting a near convex behavior, but they become exponentially more curvy as the energy level decays, in accordance to what is observed in practice with very low curvature attractors.