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

05/18/2022

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by   Itay Safran, et al.

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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 𝒪(r) linear regions, implying a generalization bound. Our result may already hold for mild over-parameterization, where the width is 𝒪̃(r) and independent of the sample size.