Convergent and Correct Message Passing Schemes for Optimization Problems over Graphical Models

The max-product algorithm, which attempts to compute the most probable assignment (MAP) of a given probability distribution, has recently found applications in quadratic minimization and combinatorial optimization. Unfortunately, the max-product algorithm is not guaranteed to converge and, even if it does, is not guaranteed to produce the MAP assignment. In this work, we provide a simple derivation of a new family of message passing algorithms. We first show how to arrive at this general message passing scheme by "splitting" the factors of our graphical model and then we demonstrate that this construction can be extended beyond integral splitting. We prove that, for any objective function which attains its maximum value over its domain, this new family of message passing algorithms always contains a message passing scheme that guarantees correctness upon convergence to a unique estimate. We then adopt a serial message passing schedule and prove that, under mild assumptions, such a schedule guarantees the convergence of our algorithm. We demonstrate that this new algorithm generalizes the known message passing schemes for integer programming problems, and we illustrate the usefulness of our generalization by applying it to provably minimize any convex quadratic function of several variables.