Optimization and Generalization of Gradient Descent for Shallow ReLU Networks with Minimal Width

Understanding the generalization and optimization of neural networks is a longstanding problem in modern learning theory. The prior analysis often leads to risk bounds of order $1/\sqrt{n}$ for ReLU networks, where $n$ is the sample size. In this paper, we present a general optimization and generalization analysis for gradient descent applied to shallow ReLU networks. We develop convergence rates of the order $1/T$ for gradient descent with $T$ iterations, and show that the gradient descent iterates fall inside local balls around either an initialization point or a reference point. Then we develop improved Rademacher complexity estimates by using the activation pattern of the ReLU function in these local balls. We apply our general result to NTK-separable data with a margin $\gamma$, and develop an almost optimal risk bound of the order $1/(n\gamma^2)$ for the ReLU network with a polylogarithmic width.