On the Effective Number of Linear Regions in Shallow Univariate ReLU Networks: Convergence Guarantees and Implicit Bias

Itay Safran, Gal Vardi, Jason D. Lee

Keywords: Deep Learning Theory, Non-convex Optimization, Gradient Flow

TL;DR: End-to-end learning guarantee for gradient flow on shallow univariate neural networks in a binary classification setting.

Abstract: We study the dynamics and implicit bias of gradient flow (GF) on univariate ReLU neural networks with a single hidden layer in a binary classification setting. We show that when the labels are determined by the sign of a target network with $r$ neurons, with high probability over the initialization of the network and the sampling of the dataset, GF converges in direction (suitably defined) to a network achieving perfect training accuracy and having at most $\mathcal{O}(r)$ linear regions, implying a generalization bound. Unlike many other results in the literature, under an additional assumption on the distribution of the data, our result holds even for mild over-parameterization, where the width is $\tilde{\mathcal{O}}(r)$ and independent of the sample size.