How Many Neurons Does it Take to Approximate the Maximum?

TL;DR

This paper investigates the minimal size of ReLU neural networks needed to approximate the maximum function over multiple inputs, establishing new bounds and depth separations, with implications for understanding neural network capacity.

Contribution

It provides new lower and upper bounds on network width for maximum function approximation, including depth separation results and a construction with logarithmic depth.

Findings

Depth 2 networks require exponential size to approximate the maximum.

Depth 3 networks can approximate the maximum with a polynomial number of neurons.

New lower bounds for depth 2 networks under exponential weight assumptions.

Abstract

We study the size of a neural network needed to approximate the maximum function over $d$ inputs, in the most basic setting of approximating with respect to the $L_2$ norm, for continuous distributions, for a network that uses ReLU activations. We provide new lower and upper bounds on the width required for approximation across various depths. Our results establish new depth separations between depth 2 and 3, and depth 3 and 5 networks, as well as providing a depth $\mathcal{O}(\log(\log(d)))$ and width $\mathcal{O}(d)$ construction which approximates the maximum function. Our depth separation results are facilitated by a new lower bound for depth 2 networks approximating the maximum function over the uniform distribution, assuming an exponential upper bound on the size of the weights. Furthermore, we are able to use this depth 2 lower bound to provide tight bounds on the number of neurons needed to approximate the maximum by a depth 3 network. Our lower bounds are of potentially broad interest as they apply to the widely studied and used \emph{max} function, in contrast to many previous results that base their bounds on specially constructed or pathological functions and distributions.