Learning Sparse Polynomial Functions

Alexandr Andoni, Rina Panigrahy, Gregory Valiant, Li Zhang

Abstract: We study the question of learning a sparse multivariate polynomial over the real domain. In particular, for some unknown polynomial f(x) of degree-d and k monomials, we show how to reconstruct f, within error ∊, given only a set of examples xi drawn uniformly from the n-dimensional cube (or an n-dimensional Gaussian distribution), together with evaluations f( i) on them. The result holds even in the “noisy setting”, where we have only values f( i) + g where g is noise (say modeled as a Gaussian random variable). The runtime of our algorithm is polynomial in n, k, 1/∊ and Cd where Cd depends only on d. Note that, in contrast, in the “boolean version” of this problem, where is drawn from the hypercube, the problem is at least as hard as the “noisy parity problem,” where we do not know how to break the nΩ(d) time barrier, even for k = 1, and some believe it may be impossible to do so.