Learning Polynomials with Queries: The Highly Noisy Case

Given a function f mapping n-variate inputs from a finite field F into F , we consider the task of reconstructing a list of all n-variate degree d polynomials which agree with f on a tiny but non-negligible fraction, , of the input space. We give a randomized algorithm for solving this task which accessesf as a black box and runs in time polynomial in 1 ; n and exponential in d, provided is (pd=jF j). For the special case when d = 1, we solve this problem for all def = 1 jF j > 0. In this case the running time of our algorithm is bounded by a polynomial in 1 ; n and exponential in d. Our algorithm generalizes a previously known algorithm, due to Goldreich and Levin, that solves this task for the case when F = GF(2) (and d = 1).