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

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, Lovász, Sós, 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 $\Vert J{\Vert }_F$, we provide algorithms that approximate the free energy within an additive error of $\epsilon n\Vert J{\Vert }_F$ in time $\exp (poly(1/\epsilon ))$. We also show that approximation within $(n\Vert J{\Vert }_F{)}^{1-\delta }$ 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.

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• Ising models on locally tree-like graphs
• Sufficient conditions for convergence of the Sum-Product Algorithm
• The computational hardness of counting in two-spin models on d-regular graphs
• How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods
• Approximating Partition Functions in Constant Time
• The Vertex Sample Complexity of Free Energy is Polynomial

Cited by in corpus (2)

• Nonlinear large deviation bounds with applications to traces of Wigner matrices and cycles counts in Erdös-Renyi graphs
• Variational Inference in high-dimensional linear regression