Approximating Continuous Functions by ReLU Nets of Minimal Width

This article concerns the expressive power of depth in deep feed-forward neural nets with ReLU activations. Specifically, we answer the following question: for a fixed din≥ 1, what is the minimal width w so that neural nets with ReLU activations, input dimension din, hidden layer widths at most w, and arbitrary depth can approximate any continuous, real-valued function of din variables arbitrarily well? It turns out that this minimal width is exactly equal to din+1. That is, if all the hidden layer widths are bounded by din, then even in the infinite depth limit, ReLU nets can only express a very limited class of functions, and, on the other hand, any continuous function on the din-dimensional unit cube can be approximated to arbitrary precision by ReLU nets in which all hidden layers have width exactly din+1. Our construction in fact shows that any continuous function f:[0,1]^din→mathbb R^dout can be approximated by a net of width din+dout. We obtain quantitative depth estimates for such an approximation in terms of the modulus of continuity of f.