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Volume 2 (2006) Article 10 pp. 185-206
Learning Restricted Models of Arithmetic Circuits
Received: August 31, 2005
Published: September 28, 2006
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Keywords: learning, exact learning, arithmetic circuit, partial derivative, multiplicity automata
ACM Classification: I.2.6, F.2.2
AMS Classification: 68Q32

Abstract: [Plain Text Version]

We present a polynomial time algorithm for learning a large class of algebraic models of computation. We show that any arithmetic circuit whose partial derivatives induce a low-dimensional vector space is exactly learnable from membership and equivalence queries. As a consequence, we obtain polynomial-time algorithms for learning restricted algebraic branching programs as well as noncommutative set-multilinear arithmetic formulae. In addition, we observe that the algorithms of Bergadano et al. (1996) and Beimel et al. (2000) can be used to learn depth-3 set-multilinear arithmetic circuits. Previously only versions of depth-2 arithmetic circuits were known to be learnable in polynomial time. Our learning algorithms can be viewed as solving a generalization of the well known polynomial interpolation problem where the unknown polynomial has a succinct representation. We can learn representations of polynomials encoding exponentially many monomials. Our techniques combine a careful algebraic analysis of the partial derivatives of arithmetic circuits with “multiplicity automata” learning algorithms due to Bergadano et al. (1997) and Beimel et al. (2000).