Benign Overfitting for Regression with Trained Two-Layer ReLU Networks

TL;DR

This paper demonstrates that finite-width two-layer ReLU neural networks trained with gradient flow can perfectly fit data without sacrificing generalization, establishing the first benign overfitting results in this setting.

Contribution

It provides the first theoretical analysis of benign overfitting for finite-width ReLU networks trained by gradient flow, without assumptions on the target function or noise.

Findings

Neural tangent kernel regime analysis of generalization.

Gradient flow acts as an implicit regularizer.

Networks can overfit yet still generalize well.

Abstract

We study the least-square regression problem with a two-layer fully-connected neural network, with ReLU activation function, trained by gradient flow. Our first result is a generalization result, that requires no assumptions on the underlying regression function or the noise other than that they are bounded. We operate in the neural tangent kernel regime, and our generalization result is developed via a decomposition of the excess risk into estimation and approximation errors, viewing gradient flow as an implicit regularizer. This decomposition in the context of neural networks is a novel perspective of gradient descent, and helps us avoid uniform convergence traps. In this work, we also establish that under the same setting, the trained network overfits to the data. Together, these results, establishes the first result on benign overfitting for finite-width ReLU networks for arbitrary regression functions.