A general approximation lower bound in $L^p$ norm, with applications to feed-forward neural networks

TL;DR

This paper establishes a general lower bound on the approximation capabilities of neural networks in $L^p$ norm, revealing fundamental limits and comparing approximation in $L^p$ versus sup norm.

Contribution

It introduces a novel lower bound framework for neural network approximation in $L^p$ norm, applicable to various function classes and measures, and clarifies differences between $L^p$ and sup norm approximations.

Findings

Lower bounds depend on packing number, range, and fat-shattering dimension.

Application to piecewise-polynomial neural networks and specific function classes.

Matching upper bounds up to log factors and insights into $L^p$ vs sup norm approximation.

Abstract

We study the fundamental limits to the expressive power of neural networks. Given two sets $F$, $G$ of real-valued functions, we first prove a general lower bound on how well functions in $F$ can be approximated in $L^p(\mu)$ norm by functions in $G$, for any $p \geq 1$ and any probability measure $\mu$. The lower bound depends on the packing number of $F$, the range of $F$, and the fat-shattering dimension of $G$. We then instantiate this bound to the case where $G$ corresponds to a piecewise-polynomial feed-forward neural network, and describe in details the application to two sets $F$: H{\"o}lder balls and multivariate monotonic functions. Beside matching (known or new) upper bounds up to log factors, our lower bounds shed some light on the similarities or differences between approximation in $L^p$ norm or in sup norm, solving an open question by DeVore et al. (2021). Our proof strategy differs from the sup norm case and uses a key probability result of Mendelson (2002).