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 ≥ 1, what is the minimal width w so that neural nets with ReLU activations, input dimension d, hidden layer widths at most w, and arbitrary depth can approximate any continuous function of d variables arbitrarily well. It turns out that this minimal width is exactly equal to d+1. That is, if all the hidden layer widths are bounded by d, then even in the infinite depth limit, ReLU nets can only express a very limited class of functions. On the other hand, we show that any continuous function on the d-dimensional unit cube can be approximated to arbitrary precision by ReLU nets in which all hidden layers have width exactly d+ 1. Our construction gives quantitative depth estimates for such an approximation.