Message-passing for graph-structured linear programs: Proximal methods and rounding schemes
The problem of computing a maximum a posteriori (MAP) configuration is a central computational challenge associated with Markov random fields. A line of work has focused on ``tree-based'' linear programming (LP) relaxations for the MAP problem. This paper develops a family of super-linearly convergent algorithms for solving these LPs, based on proximal minimization schemes using Bregman divergences. As with standard message-passing on graphs, the algorithms are distributed and exploit the underlying graphical structure, and so scale well to large problems. Our algorithms have a double-loop character, with the outer loop corresponding to the proximal sequence, and an inner loop of cyclic Bregman divergences used to compute each proximal update. Different choices of the Bregman divergence lead to conceptually related but distinct LP-solving algorithms. We establish convergence guarantees for our algorithms, and illustrate their performance via some simulations. We also develop two classes of graph-structured rounding schemes, randomized and deterministic, for obtaining integral configurations from the LP solutions. Our deterministic rounding schemes use a ``re-parameterization'' property of our algorithms so that when the LP solution is integral, the MAP solution can be obtained even before the LP-solver converges to the optimum. We also propose a graph-structured randomized rounding scheme that applies to iterative LP solving algorithms in general. We analyze the performance of our rounding schemes, giving bounds on the number of iterations required, when the LP is integral, for the rounding schemes to obtain the MAP solution. These bounds are expressed in terms of the strength of the potential functions, and the energy gap, which measures how well the integral MAP solution is separated from other integral configurations. We also report simulations comparing these rounding schemes.