On Learning Over-parameterized Neural Networks: A Functional Approximation Perspective

This is the authors' abstract. We don't add key points for this paper.

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 { an} integral operator which is determined by the feature vector distribution $ρ$ only. Consequently, GD method can be viewed as { 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(nlog n), 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.