The Complexity of Approximating a Bethe Equilibrium

TL;DR

This paper introduces a polynomial-time message-passing algorithm for solving the Bethe equation in certain graphical models, addressing a key computational challenge in statistical physics and AI.

Contribution

It presents the first fully polynomial-time approximation scheme for BP fixed-point computation in large classes of graphical models with bounded degree.

Findings

Algorithm solves Bethe equation in polynomial time for models with max degree O(log n)

Provides an alternative to BP that avoids convergence issues

Shows the problem is PPAD-hard in general, but tractable in specific cases

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 non-linear equations, the so-called 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 message-passing 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 polynomial-time approximation scheme for the BP fixed-point computation in such a large class of graphical models, while the approximate fixed-point 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.