Approximate Sherali-Adams Relaxations for MAP Inference via Entropy Regularization

Maximum a posteriori (MAP) inference is a fundamental computational paradigm for statistical inference. In the setting of graphical models, MAP inference entails solving a combinatorial optimization problem to find the most likely configuration of the discrete-valued model. Linear programming (LP) relaxations in the Sherali-Adams hierarchy are widely used to attempt to solve this problem. We leverage recent work in entropy-regularized linear programming to propose an iterative projection algorithm (SMPLP) for large scale MAP inference that is guaranteed to converge to a near-optimal solution to the relaxation. With an appropriately chosen regularization constant, we show the resulting rounded solution solves the exact MAP problem whenever the LP is tight. We further provide theoretical guarantees on the number of iterations sufficient to achieve $\epsilon$-close solutions. Finally, we show in practice that SMPLP is competitive for solving Sherali-Adams relaxations.