Learning Sparse Multivariate Polynomials over a Field with Queries and Counterexamples

摘要

We consider the problem of learning a polynomial over an arbitrary fieldFdefined on a set of boolean variables. We present the first provably effective algorithm for exactly identifying such polynomials using membership and equivalence queries. Our algorithm runs in time polynomial inn, the number of variables, andt, the number of nonzero terms appearing in the polynomial. The algorithm makes at mostnt+2 equivalence queries, and at most (nt+1)(t2+3t)/2 membership queries. Our algorithm is equally effective for learning a generalized type of polynomial defined on certain kinds of semilattices. We also present an extension of our algorithm for learning multilinear polynomials when the domain of each variable is the entire fieldF.