Complexity of Bethe Approximation

The sum-product Belief Propagation (BP) algorithm has been a popular heuristic method for estimating marginal probabilities of a joint distribution of 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 converges? and how fast it converges? In this paper, we fix this 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 arbitrary sparse graphical models (i.e. bounded degrees in the underlying graph). Our algorithm is the first fully polynomial-time approximation scheme (FPTAS) for the BP fixed point computation in such a large class of graphical models. Moreover, our result is of broader interest to understand the computational complexity of the so-called Bethe approximation.