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 that agree with f on a tiny but nonnegligible fraction, $\delta$, of the input space. We give a randomized algorithm for solving this task. The algorithm accesses f as a black box and runs in time polynomial in ${\frac{n}\d}$ and exponential in d, provided $\delta$ is $\Omega(\sqrt{d/|F|})$. For the special case when d = 1, we solve this problem for all $\epsilon\eqdef\delta - \frac1{|F|} >0$. In this case the running time of our algorithm is bounded by a polynomial in $\frac1\e$ and $n$. Our algorithm generalizes a previously known algorithm, due to Goldreich and Levin [in Proceedings of the 21st Annual ACM Symposium on Theory of Computing, Seattle, WA, ACM Press, New York, 1989, pp. 25--32.], that solves this task for the case when F = GF(2) (and d = 1). In the process we provide new bounds on the number of degree $d$ polynomials that may agree with any given function on $\d \geq \sqrt{d/|F|}$ fraction of the inputs. This result is derived by generalizing a well-known bound from coding theory on the number of codewords from an error-correcting code that can be "close" to an arbitrary word; our generalization works for codes over arbitrary alphabets, while the previous result held only for binary alphabets.