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

• URL: http://arxiv.org/abs/2205.09072v1
• Date: Wed, 18 May 2022 16:57:10 GMT
• Title: On the Effective Number of Linear Regions in Shallow Univariate ReLU Networks: Convergence Guarantees and Implicit Bias
• Authors: Itay Safran, Gal Vardi, Jason D. Lee
• Abstract summary: We show that gradient flow converges in direction when labels are determined by the sign of a target network with $r$ neurons. Our result may already hold for mild over- parameterization, where the width is $tildemathcalO(r)$ and independent of the sample size.
• Score: 50.84569563188485
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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. Our result may already hold for mild over-parameterization, where the width is $\tilde{\mathcal{O}}(r)$ and independent of the sample size.

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