Convergence of Generalized Belief Propagation Algorithm on Graphs with Motifs

TL;DR

This paper investigates the convergence properties of the generalized belief propagation algorithm on graphs containing motifs, demonstrating that under certain conditions it converges to the global optimum for specific models.

Contribution

It provides theoretical insights into the convergence of generalized belief propagation on loopy graphs with motifs, a topic previously not well-understood.

Findings

Convergence to the global optimum is guaranteed under specific initialization.

The results apply to ferromagnetic Ising models on graphs with motifs.

The study extends understanding of belief propagation in complex graph structures.

Abstract

Belief propagation is a fundamental message-passing algorithm for numerous applications in machine learning. It is known that belief propagation algorithm is exact on tree graphs. However, belief propagation is run on loopy graphs in most applications. So, understanding the behavior of belief propagation on loopy graphs has been a major topic for researchers in different areas. In this paper, we study the convergence behavior of generalized belief propagation algorithm on graphs with motifs (triangles, loops, etc.) We show under a certain initialization, generalized belief propagation converges to the global optimum of the Bethe free energy for ferromagnetic Ising models on graphs with motifs.