Approximation of BV functions by neural networks: A regularity theory approach

TL;DR

This paper develops a regularity theory approach to analyze how single hidden layer neural networks with ReLU activation functions can approximate BV functions, providing convergence guarantees and error bounds especially when weights are bounded.

Contribution

It introduces a novel localization theorem for BV function approximation by neural networks with bounded weights, inspired by elliptic PDE regularity techniques.

Findings

Proves a Poincaré inequality for the stochastic gradient flow.

Establishes an error bound of order R^{-1/9} for bounded weight networks.

Provides a quantitative universal approximation theorem.

Abstract

In this paper we are concerned with the approximation of functions by single hidden layer neural networks with ReLU activation functions on the unit circle. In particular, we are interested in the case when the number of data-points exceeds the number of nodes. We first study the convergence to equilibrium of the stochastic gradient flow associated with the cost function with a quadratic penalization. Specifically, we prove a Poincar\'e inequality for a penalized version of the cost function with explicit constants that are independent of the data and of the number of nodes. As our penalization biases the weights to be bounded, this leads us to study how well a network with bounded weights can approximate a given function of bounded variation (BV). Our main contribution concerning approximation of BV functions, is a result which we call the localization theorem. Specifically, it states that the expected error of the constrained problem, where the length of the weights are less than $R$, is of order $R^{-1/9}$ with respect to the unconstrained problem (the global optimum). The proof is novel in this topic and is inspired by techniques from regularity theory of elliptic partial differential equations. Finally we quantify the expected value of the global optimum by proving a quantitative version of the universal approximation theorem.