Learning polynomials with queries: The highly noisy case

Given a function f mappping 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, δ, of the input space. We give a randomized algorithm for solving this task which accesses f as a black box and runs in time polynomial in 1/δ, n and exponential in d, provided δ is Ω(√(d/|F|)). For the special case when d=1, we solve this problem for all εder/=δ-1/|F|>0. In this case the running time of 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)