Replicated Bethe Free Energy: A Variational Principle behind Survey Propagation

TL;DR

This paper introduces a unified variational framework based on replicated Bethe free energy to derive and understand survey propagation and related algorithms for analyzing disordered systems.

Contribution

It presents a novel scheme employing replicated Bethe free energy with analytical continuation to unify mean-field approximation algorithms, including survey propagation, belief propagation, and their variants.

Findings

Survey propagation is derived from the simplest replica symmetry assumption.

Belief propagation and generalized survey propagation are obtained under different replica symmetry assumptions.

The framework offers a unified perspective on various mean-field algorithms for disordered systems.

Abstract

A scheme to provide various mean-field-type approximation algorithms is presented by employing the Bethe free energy formalism to a family of replicated systems in conjunction with analytical continuation with respect to the number of replicas. In the scheme, survey propagation (SP), which is an efficient algorithm developed recently for analyzing the microscopic properties of glassy states for a fixed sample of disordered systems, can be reproduced by assuming the simplest replica symmetry on stationary points of the replicated Bethe free energy. Belief propagation and generalized SP can also be offered in the identical framework under assumptions of the highest and broken replica symmetries, respectively.