Sharp Lower Bounds on the Approximation Rate of Shallow Neural Networks

TL;DR

This paper establishes sharp lower bounds on the approximation capabilities of shallow neural networks, demonstrating fundamental limits and differences in approximation efficiency based on activation functions.

Contribution

It provides the first sharp lower bounds on approximation rates for shallow neural networks, resolving an open problem and analyzing the complexity of neural network approximation spaces.

Findings

Lower bounds on $L^2$-metric entropy of neural network basis functions.

Lower bounds on Kolmogorov $n$-widths of the convex hull of basis functions.

Quantification of the difference between Barron spectral norm and variation norm.

Abstract

We consider the approximation rates of shallow neural networks with respect to the variation norm. Upper bounds on these rates have been established for sigmoidal and ReLU activation functions, but it has remained an important open problem whether these rates are sharp. In this article, we provide a solution to this problem by proving sharp lower bounds on the approximation rates for shallow neural networks, which are obtained by lower bounding the $L^2$-metric entropy of the convex hull of the neural network basis functions. In addition, our methods also give sharp lower bounds on the Kolmogorov $n$-widths of this convex hull, which show that the variation spaces corresponding to shallow neural networks cannot be efficiently approximated by linear methods. These lower bounds apply to both sigmoidal activation functions with bounded variation and to activation functions which are a power of the ReLU. Our results also quantify how much stronger the Barron spectral norm is than the variation norm and, combined with previous results, give the asymptotics of the $L^\infty$-metric entropy up to logarithmic factors in the case of the ReLU activation function.