On Learning Over-parameterized Neural Networks: A Functional Approximation Prospective
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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 the network prediction errors across GD iterations, which can be neatly described in a matrix form. It turns out that 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 apply 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, even with constant stepsize, the empirical risk decreases to this low-rank approximation error at a linear rate in iteration t. In addition, this linear rate is determined by f^* and ρ only. Furthermore, if f^* has zero low-rank approximation error, then Ω(n^2) network over-parameterization is enough, and the empirical risk decreases to Θ(1/√(n)). 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.
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