The Mean-Field Approximation: Information Inequalities, Algorithms, and Complexity

TL;DR

This paper establishes optimal bounds for the mean field approximation error in Ising models, introduces algorithms for estimating free energy efficiently, and proves NP-hardness results for certain approximation levels.

Contribution

It provides the first tight bounds on KL error for mean field approximation in general graphs and develops algorithms with provable guarantees for free energy estimation.

Findings

Bound for KL error is tight up to lower order terms.

Algorithms approximate free energy within specified additive errors.

NP-hardness of approximation within certain bounds is established.

Abstract

The mean field approximation to the Ising model is a canonical variational tool that is used for analysis and inference in Ising models. We provide a simple and optimal bound for the KL error of the mean field approximation for Ising models on general graphs, and extend it to higher order Markov random fields. Our bound improves on previous bounds obtained in work in the graph limit literature by Borgs, Chayes, Lov\'asz, S\'os, and Vesztergombi and another recent work by Basak and Mukherjee. Our bound is tight up to lower order terms. Building on the methods used to prove the bound, along with techniques from combinatorics and optimization, we study the algorithmic problem of estimating the (variational) free energy for Ising models and general Markov random fields. For a graph $G$ on $n$ vertices and interaction matrix $J$ with Frobenius norm $\| J \|_F$, we provide algorithms that approximate the free energy within an additive error of $\epsilon n \|J\|_F$ in time $\exp(poly(1/\epsilon))$. We also show that approximation within $(n \|J\|_F)^{1-\delta}$ is NP-hard for every $\delta > 0$. Finally, we provide more efficient approximation algorithms, which find the optimal mean field approximation, for ferromagnetic Ising models and for Ising models satisfying Dobrushin's condition.