A Mixed-Integer Linear Programming reformulation approach to Maximum A Posteriori inference in Sum-Product Networks - Katague, Gustavo Perez (2021)
Sum-Product Network (SPN) is a relatively new class of probabilistic graphical models. They differ from other probabilistic graphical models by allowing explicit representation of context-sensitive independence and marginal inference computation in linear time. Bayesian Networks and Markov Networks, for example, require #P-hard effort for performing marginal inference. However, it is still NP-hard to find the most probable configuration for a set of variables in an SPN, and there is currently a shortage of efficient techniques to solve the problem. A widely employed technique for solving NP-hard optimization problems consists in translating them into Mixed-Integer Linear Programming (MILP) programs, which hence can be solved by highly efficient commercial solvers. Besides harvesting the power of current solvers, formulating the problem as a MILP program immediately allows us to obtain an anytime algorithm that continuously improves its solution as more resources are given (time and memory), and can be stopped at any time with a feasible solution with error bounds. In this work, we developed a new algorithm that finds the most probable configuration for a set of variables in SPNs (Maximum A Posteriori inference) by reformulating it as a MILP program. This translation is rather intricate and relies on several results scattered throughout this field of study, such as the reformulation of SPNs as Bayesian Networks with latent variables, the compact representation of conditional probability tables through Algebraic Decision Diagrams and the symbolic manipulation of multilinear expressions by Parameterized Algebraic Decision Diagrams.