Hardness of reconstructing multivariate polynomials over finite fields

We study the polynomial reconstruction problem for low-degree multivariate polynomials over double-struck F sign[2]. In this problem, we are given a set of points x ∈{0, 1}ⁿ and target values f(x) ∈ {0, 1} for each of these points, with the promise that there is a polynomial over double-struck F sign[2] of degree at most d that agrees with f at 1 - ε fraction of the points. Our goal is to find a degree d polynomial that has good agreement with f. We show that it is NP-hard to find a polynomial that agrees with f on more than 1 - 2^-d + δ fraction of the points for any ε, δ > 0. This holds even with the stronger promise that the polynomial that fits the data is in fact linear, whereas the algorithm is allowed to find a polynomial of degree d. Previously the only known hardness of approximation (or even NP-completeness) was for the case when d = 1, which follows from a celebrated result of Håstad [16]. In the setting of Computational Learning, our result shows the hardness of (non-proper)agnostic learning of parities, where the learner is allowed a low-degree polynomial over double-struck F sign[2] as a hypothesis. This is the first nonproper hardness result for this central problem in computational learning. Our results extend to multivariate polynomial reconstruction over any finite field.