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Arising from many applications at the intersection of decision making andmachine learning, Marginal Maximum A Posteriori (Marginal MAP) Problems unifythe two main classes of inference, namely maximization (optimization) andmarginal inference (counting), and are believed to have higher complexity thanboth of them. We propose XOR_MMAP, a novel approach to solve the Marginal MAPProblem, which represents the intractable counting subproblem with queries toNP oracles, subject to additional parity constraints. XOR_MMAP provides aconstant factor approximation to the Marginal MAP Problem, by encoding it as asingle optimization in polynomial size of the original problem. We evaluate ourapproach in several machine learning and decision making applications, and showthat our approach outperforms several state-of-the-art Marginal MAP solvers.

Solving Marginal MAP Problems with NP Oracles and Parity Constraints