The Complexity of Approximating a Bethe Equilibrium
Abstract
This paper resolves a common complexity issue in the Bethe approximation of statistical physics and the Belief Propagation (BP) algorithm of artificial intelligence. The Bethe approximation and the BP algorithm are heuristic methods for estimating the partition function and marginal probabilities in graphical models, respectively. The computational complexity of the Bethe approximation is decided by the number of operations required to solve a set of nonlinear equations, the socalled Bethe equation. Although the BP algorithm was inspired and developed independently, Yedidia, Freeman and Weiss (2004) showed that the BP algorithm solves the Bethe equation if it converges (however, it often does not). This naturally motivates the following question to understand limitations and empirical successes of the Bethe and BP methods: is the Bethe equation computationally easy to solve? We present a messagepassing algorithm solving the Bethe equation in a polynomial number of operations for general binary graphical models of n variables where the maximum degree in the underlying graph is O(log n). Our algorithm can be used as an alternative to BP fixing its convergence issue and is the first fully polynomialtime approximation scheme for the BP fixedpoint computation in such a large class of graphical models, while the approximate fixedpoint computation is known to be (PPAD)hard in general. We believe that our technique is of broader interest to understand the computational complexity of the cavity method in statistical physics.
 Publication:

arXiv eprints
 Pub Date:
 September 2011
 arXiv:
 arXiv:1109.1724
 Bibcode:
 2011arXiv1109.1724S
 Keywords:

 Computer Science  Artificial Intelligence;
 Computer Science  Computational Complexity