An application of Bernstein polynomials in PAC model

Abstract

We study a problem for learning the class of Lipschitz bounded, continuous and real valued functions in terms of Bernstein polynomials in the PAC model [2]. Let f be a Lipschitz bounded continuous function with constant L. We intend to approximate the function f with accuracy ε and confidence δ. By using Bernstein polynomials of degree n=[(3L/e)²], we will construct a polynomial time algorithm which will learn an ε-approximation to the function in probability 1−δ on the uniform distribution over [0, 1]. This algorithm requires a sample of size [(n+1) ln (n+1/δ)]. This approximate learning is assumed to ideal machine but in practice we have to do the task by using real machine with finite resources. We also consider the robustness of Bernstein polynomial for machine epsilons.