Demystifying the Global Convergence Puzzle of Learning Over-parameterized ReLU Nets in Very High Dimensions

Abstract

This theoretical paper is devoted to developing a rigorous theory for demystifying the global convergence phenomenon in a challenging scenario: learning over-parameterized Rectified Linear Unit (ReLU) nets for very high dimensional dataset under very mild assumptions. A major ingredient of our analysis is a fine-grained analysis of random activation matrices. The essential virtue of dissecting activation matrices is that it bridges the dynamics of optimization and angular distribution in high-dimensional data space. This angle-based detailed analysis leads to asymptotic characterizations of gradient norm and directional curvature of objective function at each gradient descent iteration, revealing that the empirical loss function enjoys nice geometrical properties in the overparameterized setting. Along the way, we significantly improve existing theoretical bounds on both over-parameterization condition and learning rate with very mild assumptions for learning very high dimensional data. Moreover, we uncover the role of the geometrical and spectral properties of the input data in determining desired over-parameterization size and global convergence rate. All these clues allow us to discover a novel geometric picture of nonconvex optimization in deep learning: angular distribution in high-dimensional data space $\mapsto$ spectrums of overparameterized activation matrices $\mapsto$ favorable geometrical properties of empirical loss landscape $\mapsto$ global convergence phenomenon. Furthremore, our theoretical results imply that gradient-based nonconvex optimization algorithms have much stronger statistical guarantees with much milder over-parameterization condition than exisiting theory states for learning very high dimensional data, which is rarely explored so far.