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

The max-product algorithm, which attempts to compute the maximizing assignment of a given objective function, 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 optimal 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 that 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. Finally, we adopt an asynchronous message passing schedule and prove that, under mild assumptions, such a schedule guarantees the convergence of our algorithm.