Approximation of functions with one-bit neural networks

This paper examines the approximation capabilities of coarsely quantized neural networks -- those whose parameters are selected from a small set of allowable values. We show that any smooth multivariate function can be arbitrarily well approximated by an appropriate coarsely quantized neural network and provide a quantitative approximation rate. For the quadratic activation, this can be done with only a one-bit alphabet; for the ReLU activation, we use a three-bit alphabet. The main theorems rely on important properties of Bernstein polynomials. We prove new results on approximation of functions with Bernstein polynomials, noise-shaping quantization on the Bernstein basis, and implementation of the Bernstein polynomials by coarsely quantized neural networks.