Complexity of Bethe Approximation

This paper resolves a common complexity issue in the Bethe approximation of statistical physics and the sum-product Belief Propagation (BP) algorithm of artificial intelligence. The Bethe approximation reduces the problem of computing the partition function in a graphical model to that of solving a set of non-linear equations, so-called the Bethe equation. On the other hand, the BP algorithm is a popular heuristic method for estimating marginal distribution in a graphical model. Although they are inspired and developed from different directions, Yedidia, Freeman and Weiss (2004) established a somewhat surprising connection: the BP algorithm solves the Bethe equation if it converges (however, it often does not). This naturally motivates the following important question to understand their limitations and empirical successes: the Bethe equation is computationally easy to solve? We present a message passing algorithm solving the Bethe equation in polynomial number of bitwise operations for arbitrary binary graphical models of n nodes where the maximum degree in the underlying graph is O(log n). Our algorithm, an alternative to BP fixing its convergence issue, is the first fully polynomial-time approximation scheme for the BP fixed point computation in such a large class of graphical models. Moreover, we believe that our technique is of broader interest to understand the computational complexity of the cavity method in statistical physics. Appearing in Proceedings of the 15 International Conference on Artificial Intelligence and Statistics (AISTATS) 2012, La Palma, Canary Islands. Volume XX of JMLR: W&CP XX. Copyright 2012 by the authors.