Part of Advances in Neural Information Processing Systems 32 (NeurIPS 2019)
Lili Su, Pengkun Yang
We consider training over-parameterized two-layer neural networks with Rectified Linear Unit (ReLU) using gradient descent (GD) method. Inspired by a recent line of work, we study the evolutions of network prediction errors across GD iterations, which can be neatly described in a matrix form. When the network is sufficiently over-parameterized, these matrices individually approximate {\em an} integral operator which is determined by the feature vector distribution ρ only. Consequently, GD method can be viewed as {\em approximately} applying the powers of this integral operator on the underlying/target function f∗ that generates the responses/labels. We show that if f∗ admits a low-rank approximation with respect to the eigenspaces of this integral operator, then the empirical risk decreases to this low rank approximation error at a linear rate which is determined by f∗ and ρ only, i.e., the rate is independent of the sample size n. Furthermore, if f∗ has zero low-rank approximation error, then, as long as the width of the neural network is Ω(nlogn), the empirical risk decreases to Θ(1/√n). To the best of our knowledge, this is the first result showing the sufficiency of nearly-linear network over-parameterization. We provide an application of our general results to the setting where ρ is the uniform distribution on the spheres and f∗ is a polynomial. Throughout this paper, we consider the scenario where the input dimension d is fixed.