Probably Approximately Correct Maximum A Posteriori Inference

Computing the conditional mode of a distribution, better known as the maximum a posteriori (MAP) assignment, is a fundamental task in probabilistic inference. However, MAP is generally intractable, and remains hard even under many common structural constraints and approximation schemes. We take a novel approach inspired by multi-armed bandits, recasting MAP as a best arm identification task. We introduce probably approximately correct (PAC) algorithms for MAP that provide provably optimal solutions in both the fixed-confidence and fixed-budget regimes. We characterize tractability conditions using information theoretic measures that can be estimated from finite samples. Our PAC-MAP solvers are efficiently implemented using probabilistic circuits and graphical models with appropriate architectures. The algorithms we develop can be used either as standalone MAP estimators or to improve on standard heuristics, fortifying their solutions with rigorous guarantees. Experiments confirm the benefits of our method in a range of benchmarks.