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) = Πi=1n Ai(xi), where, for each 1 ≤ i ≤ n , Ai is a mapping Ai : X → Rmi× mi+1 and m1 = mn+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 Zn 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+].