Tree consistency and bounds on the performance of the max-product algorithm and its generalizations

Finding the maximum a posteriori (MAP) assignment of a discrete-state distribution specified by a graphical model requires solving an integer program. The max-product algorithm, also known as the max-plus or min-sum algorithm, is an iterative method for (approximately) solving such a problem on graphs with cycles. We provide a novel perspective on the algorithm, which is based on the idea of reparameterizing the distribution in terms of so-called pseudo-max-marginals on nodes and edges of the graph. This viewpoint provides conceptual insight into the max-product algorithm in application to graphs with cycles. First, we prove the existence of max-product fixed points for positive distributions on arbitrary graphs. Next, we show that the approximate max-marginals computed by max-product are guaranteed to be consistent, in a suitable sense to be defined, over every tree of the graph. We then turn to characterizing the nature of the approximation to the MAP assignment computed by max-product. We generalize previous work by showing that for any graph, the max-product assignment satisfies a particular optimality condition with respect to any subgraph containing at most one cycle per connected component. We use this optimality condition to derive upper bounds on the difference between the log probability of the true MAP assignment, and the log probability of a max-product assignment. Finally, we consider extensions of the max-product algorithm that operate over higher-order cliques, and show how our reparameterization analysis extends in a natural manner.