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