Learning Matrix Functions over Rings

Abstract

Let R be a commutative Artinian ring with identity and let X be a finite subset of R . We present an exact learning algorithm with a polynomial query complexity for the class of functions representable as f(x) = Π i=1 n A i (x i ), where, for each 1 ≤ i ≤ n , A i is a mapping A i : X → R mi× mi+1 and m 1 = m n+1 = 1 . We show that the above algorithm implies the following results: 1. Multivariate polynomials over a finite commutative ring with identity are learnable using equivalence and substitution queries. 2. Bounded degree multivariate polynomials over Z n can be interpolated using substitution queries. 3. The class of constant depth circuits that consist of bounded fan-in MOD gates, where the modulus are prime powers of some fixed prime, is learnable using equivalence and substitution queries. Our approach uses a decision tree representation for the hypothesis class which takes advantage of linear dependencies. This paper generalizes the learning algorithm for automata over fields given in [BBB+].