High-dimensional classification problems with Barron regular boundaries under margin conditions

TL;DR

This paper demonstrates that neural networks with ReLU activations can efficiently approximate high-dimensional classifiers with Barron-regular boundaries under margin conditions, leading to fast learning rates.

Contribution

It establishes approximation rates for neural networks with Barron-regular boundaries under margin assumptions, extending understanding of high-dimensional classification.

Findings

Neural networks can approximate high-dimensional classifiers with Barron-regular boundaries at high polynomial rates.

Fast learning rates close to $n^{-1}$ are achievable under strong margin conditions.

Numerical experiments on binary classification, including MNIST, validate theoretical results.

Abstract

We prove that a classifier with a Barron-regular decision boundary can be approximated with a rate of high polynomial degree by ReLU neural networks with three hidden layers when a margin condition is assumed. In particular, for strong margin conditions, high-dimensional discontinuous classifiers can be approximated with a rate that is typically only achievable when approximating a low-dimensional smooth function. We demonstrate how these expression rate bounds imply fast-rate learning bounds that are close to $n^{-1}$ where $n$ is the number of samples. In addition, we carry out comprehensive numerical experimentation on binary classification problems with various margins. We study three different dimensions, with the highest dimensional problem corresponding to images from the MNIST data set.