The Complexity of Approximating a Bethe Equilibrium

The sum-product Belief Propagation (BP) algorithm has been a popular heuristic method for estimating marginal probabilities of a joint distribution of n random variables represented by a graphical model. This is primarily due to the ease of implementation derived from its iterative, message-passing nature. One of fundamental questions on BP (and message-passing algorithms in general) is about its convergence: when it does converge? and how fast it converges? In this paper, we fix the convergence issue via developing a deterministic, iterative algorithm for forcing fast-convergence of BP to its fixed point in a polynomial number of iterations for graphical models where the maximum degree in the underlying graph is O(log n). Our algorithm is the first fully polynomial-time approximation scheme for the BP fixed point computation in such a large class of sparse graphical models. Moreover, our result is of broader interest to understand the computational complexity of the so-called Bethe approximation.