The Sum-Product Theorem: A Foundation for Learning Tractable Models

Inference in expressive probabilistic models is generally intractable, which makes them difficult to learn and limits their applicability. Sum-product networks are a class of deep models where, surprisingly, inference remains tractable even when an arbitrary number of hidden layers are present. In this paper, we generalize this result to a much broader set of learning problems: all those where inference consists of summing a function over a semiring. This includes satisfiability, constraint satisfaction, optimization, integration, and others. In any semiring, for summation to be tractable it suffices that the factors of every product have disjoint scopes. This unifies and extends many previous results in the literature. Enforcing this condition at learning time thus ensures that the learned models are tractable. We illustrate the power and generality of this approach by applying it to a new type of structured prediction problem: learning a nonconvex function that can be globally optimized in polynomial time. We show empirically that this greatly outperforms the standard approach of learning without regard to the cost of optimization.