Fast Convergence of Belief Propagation to Global Optima: Beyond Correlation Decay

TL;DR

This paper proves that belief propagation converges rapidly to the global optimum in ferromagnetic Ising models on arbitrary graphs, even with long-range correlations, under natural initialization.

Contribution

It establishes dimension-free convergence guarantees for belief propagation and mean-field equations in ferromagnetic Ising models beyond correlation decay assumptions.

Findings

BP converges quickly to the global Bethe free energy optimum

Results hold for arbitrary graphs with ferromagnetic interactions

A constant number of iterations suffices for good estimates

Abstract

Belief propagation is a fundamental message-passing algorithm for probabilistic reasoning and inference in graphical models. While it is known to be exact on trees, in most applications belief propagation is run on graphs with cycles. Understanding the behavior of "loopy" belief propagation has been a major challenge for researchers in machine learning, and positive convergence results for BP are known under strong assumptions which imply the underlying graphical model exhibits decay of correlations. We show that under a natural initialization, BP converges quickly to the global optimum of the Bethe free energy for Ising models on arbitrary graphs, as long as the Ising model is \emph{ferromagnetic} (i.e. neighbors prefer to be aligned). This holds even though such models can exhibit long range correlations and may have multiple suboptimal BP fixed points. We also show an analogous result for iterating the (naive) mean-field equations; perhaps surprisingly, both results are dimension-free in the sense that a constant number of iterations already provides a good estimate to the Bethe/mean-field free energy.