Learning Sparse Multivariate Polynomials over a Field with Queries and Counterexamples

Abstract

We consider the problem of learning a polynomial over an arbitrary field F defined 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 in n, the number of variables, and t, the number of nonzero terms appearing in the polynomial. The algorithm makes at most nt+2 equivalence queries, and at most (nt+1)(t 2 +3t)Â2 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 field F. ] 1996 Academic Press, Inc.

A deterministic multivariate interpolation algorithm for small finite fields

“…We believe that this special case was known before, although we found no direct reference to such an algorithm. An algorithm in [5] uses the poset structure in a similar way for GF(2) function learning, rather than for interpolation.…”

“…In addition to the traditional applications over real numbers [2], interpolations over finite fields have been used for decoding errorcorrecting codes [3], testing [4], and in learning algorithms [5]. The application considered in our previous work [1] was that of obtaining an RM transform for functions with possibly many "don't care" points.…”

Applying learning algorithms to preference elicitation

“…Schapire and Sellie [17] give a learning algorithm for sparse multivariate polynomials that can be used as the basis for a combinatorial auction protocol. The equivalence queries made by this algorithm are all proper.…”

“…A polynomial "over the real numbers" has coefficients drawn from the real numbers. Polynomials are expressive: every valuation function v : 2 M → R+ can be uniquely written as a polynomial [17].…”

“…The learning algorithm for polynomials makes at most mti + 2 equivalence queries and at most (mti + 1)(t 2 i + 3ti)/2 membership queries to an agent i, where ti is the sparcity of the polynomial representing vi [17]. We therefore obtain an algorithm that elicits general valuations with a polynomial number of queries and polynomial communication.…”