Approximation with One-Bit Polynomials in Bernstein Form

Abstract

We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from \(\{ \pm 1\}\) only. A basic case of our results states that for any Lipschitz function \(f:[0,1] \rightarrow [-1,1]\) and for any positive integer n, there are signs \(\sigma _0,\dots ,\sigma _n \in \{\pm 1\}\) such that $$\begin{aligned} \left| f(x) - \sum _{k=0}^n \sigma _k \, \left( {\begin{array}{c}n\\ k\end{array}}\right) x^k (1-x)^{n-k} \right| \le \frac{C (1+|f|_{\textrm{Lip}})}{1+\sqrt{nx(1-x)}} ~ \text{ for } \text{ all } x \in [0,1]. \end{aligned}$$More generally, we show that higher accuracy is achievable for smoother functions: For any integer \(s\ge 1\), if f has a Lipschitz \((s{-}1)\)st derivative, then approximation accuracy of order \(O(n^{-s/2})\) is achievable with coefficients in \(\{\pm 1\}\) provided \(\Vert f \Vert _\infty < 1\), and of order \(O(n^{-s})\) with unrestricted integer coefficients, both uniformly on closed subintervals of (0, 1) as above. Hence these polynomial approximations are not constrained by the saturation of classical Bernstein polynomials. Our approximations are constructive and can be implemented using feedforward neural networks whose weights are chosen from \(\{\pm 1\}\) only.