Belief Optimization for Binary Networks: A Stable Alternative to Loopy Belief Propagation

TL;DR

This paper introduces a new stable inference algorithm for binary undirected graphs that directly minimizes the Bethe free energy, ensuring convergence even when loopy belief propagation fails, and is useful for learning graphical models.

Contribution

It proposes a novel belief optimization algorithm that guarantees convergence by descending on the Bethe free energy, unlike traditional loopy belief propagation.

Findings

The algorithm converges to local minima of the Bethe free energy.

It matches belief propagation solutions when BP converges.

It remains stable and converges where BP fails.

Abstract

We present a novel inference algorithm for arbitrary, binary, undirected graphs. Unlike loopy belief propagation, which iterates fixed point equations, we directly descend on the Bethe free energy. The algorithm consists of two phases, first we update the pairwise probabilities, given the marginal probabilities at each unit,using an analytic expression. Next, we update the marginal probabilities, given the pairwise probabilities by following the negative gradient of the Bethe free energy. Both steps are guaranteed to decrease the Bethe free energy, and since it is lower bounded, the algorithm is guaranteed to converge to a local minimum. We also show that the Bethe free energy is equal to the TAP free energy up to second order in the weights. In experiments we confirm that when belief propagation converges it usually finds identical solutions as our belief optimization method. However, in cases where belief propagation fails to converge, belief optimization continues to converge to reasonable beliefs. The stable nature of belief optimization makes it ideally suited for learning graphical models from data.