The Vertex Sample Complexity of Free Energy is Polynomial

We study the following question: given a massive Markov random field on $n$ nodes, can a small sample from it provide a rough approximation to the free energy $\mathcal{F}_n = \log{Z_n}$? Results in graph limit literature by Borgs, Chayes, Lov\'asz, S\'os, and Vesztergombi show that for Ising models on $n$ nodes and interactions of strength $\Theta(1/n)$, an $\epsilon$ approximation to $\log Z_n / n$ can be achieved by sampling a randomly induced model on $2^{O(1/\epsilon^2)}$ nodes. We show that the sampling complexity of this problem is {\em polynomial in} $1/\epsilon$. We further show a polynomial dependence on $\epsilon$ cannot be avoided. Our results are very general as they apply to higher order Markov random fields. For Markov random fields of order $r$, we obtain an algorithm that achieves $\epsilon$ approximation using a number of samples polynomial in $r$ and $1/\epsilon$ and running time that is $2^{O(1/\epsilon^2)}$ up to polynomial factors in $r$ and $\epsilon$. For ferromagnetic Ising models, the running time is polynomial in $1/\epsilon$. Our results are intimately connected to recent research on the regularity lemma and property testing, where the interest is in finding which properties can tested within $\epsilon$ error in time polynomial in $1/\epsilon$. In particular, our proofs build on results from a recent work by Alon, de la Vega, Kannan and Karpinski, who also introduced the notion of polynomial vertex sample complexity. Another critical ingredient of the proof is an effective bound by the authors of the paper relating the variational free energy and the free energy.Comment: arXiv admin note: text overlap with arXiv:1802.06126 Updated bibliograph

Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective

The free energy is a key quantity of interest in Ising models, but unfortunately, computing it in general is computationally intractable. Two popular (variational) approximation schemes for estimating the free energy of general Ising models (in particular, even in regimes where correlation decay does not hold) are: (i) the mean-field approximation with roots in statistical physics, which estimates the free energy from below, and (ii) hierarchies of convex relaxations with roots in theoretical computer science, which estimate the free energy from above. We show, surprisingly, that the tight regime for both methods to compute the free energy to leading order is identical. More precisely, we show that the mean-field approximation is within $O((n\|J\|_{F})^{2/3})$ of the free energy, where $\|J\|_F$ denotes the Frobenius norm of the interaction matrix of the Ising model. This simultaneously subsumes both the breakthrough work of Basak and Mukherjee, who showed the tight result that the mean-field approximation is within $o(n)$ whenever $\|J\|_{F} = o(\sqrt{n})$, as well as the work of Jain, Koehler, and Mossel, who gave the previously best known non-asymptotic bound of $O((n\|J\|_{F})^{2/3}\log^{1/3}(n\|J\|_{F}))$. We give a simple, algorithmic proof of this result using a convex relaxation proposed by Risteski based on the Sherali-Adams hierarchy, automatically giving sub-exponential time approximation schemes for the free energy in this entire regime. Our algorithmic result is tight under Gap-ETH. We furthermore combine our techniques with spin glass theory to prove (in a strong sense) the optimality of correlation rounding, refuting a recent conjecture of Allen, O'Donnell, and Zhou. Finally, we give the tight generalization of all of these results to $k$-MRFs, capturing as a special case previous work on approximating MAX-$k$-CSP.Comment: This version: minor formatting changes, added grant acknowledgement

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

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.Comment: Updated bibliograph