The Barron Space and the Flow-induced Function Spaces for Neural Network Models

TL;DR

This paper introduces the Barron space and flow-induced function spaces as the appropriate mathematical frameworks for analyzing two-layer and residual neural networks, providing approximation theorems and complexity bounds.

Contribution

It defines the Barron space and flow-induced spaces, establishing their suitability for neural network analysis with proven approximation theorems and complexity bounds.

Findings

Optimal approximation theorems hold for functions in these spaces.

Rademacher complexity bounds are optimal for bounded sets.

The spaces effectively characterize neural network function classes.

Abstract

One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation errors by a particular machine learning model. In this paper, we address this issue for two representative neural network models: the two-layer networks and the residual neural networks. We define the Barron space and show that it is the right space for two-layer neural network models in the sense that optimal direct and inverse approximation theorems hold for functions in the Barron space. For residual neural network models, we construct the so-called flow-induced function space, and prove direct and inverse approximation theorems for this space. In addition, we show that the Rademacher complexity for bounded sets under these norms has the optimal upper bounds.