Zero-Temperature Limit of a Convergent Algorithm to Minimize the Bethe Free Energy

TL;DR

This paper introduces a convergent algorithm for zero-temperature inference in graphical models, effectively approximating ground states and max-marginals by extending existing methods to the max-product setting.

Contribution

It presents the zero-temperature limit of Heskes' double-loop algorithm, enabling convergence to max-product fixed points and combining advantages of belief propagation and diffusion.

Findings

Algorithm converges to max-product fixed points.

Provides good approximation of ground states.

Combines benefits of belief propagation and diffusion.

Abstract

After the discovery that fixed points of loopy belief propagation coincide with stationary points of the Bethe free energy, several researchers proposed provably convergent algorithms to directly minimize the Bethe free energy. These algorithms were formulated only for non-zero temperature (thus finding fixed points of the sum-product algorithm) and their possible extension to zero temperature is not obvious. We present the zero-temperature limit of the double-loop algorithm by Heskes, which converges a max-product fixed point. The inner loop of this algorithm is max-sum diffusion. Under certain conditions, the algorithm combines the complementary advantages of the max-product belief propagation and max-sum diffusion (LP relaxation): it yields good approximation of both ground states and max-marginals.