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 d₈₍ 1, what is the minimal width w so that neural nets with ReLU activations, input dimension d₈₍, hidden layer widths at most w, and arbitrary depth can approximate any continuous, real-valued function of d₈₍ variables arbitrarily well? It turns out that this minimal width is exactly equal to d₈₍+1. That is, if all the hidden layer widths are bounded by d₈₍, 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 d₈₍-dimensional unit cube can be approximated to arbitrary precision by ReLU nets in which all hidden layers have width exactly d₈₍+1. Our construction in fact shows that any continuous function f: 0, 1^d₈₍ R^d₎ₔₓ can be approximated by a net of width d₈₍+d₎ₔₓ. We obtain quantitative depth estimates for such an approximation in terms of the modulus of continuity of f.