2019/05/23 by Frederic Koehler, Koehler, Frederic · 1 citation
Computer Science · Mathematics · #Bayesian Modeling and Causal Inference #Data Structures and Algorithms (cs.DS) #Error Correcting Code Techniques #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1905.09992
openalex publication_date 2019/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.