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\nunfortunately, computing it in general is computationally intractable. Two\npopular (variational) approximation schemes for estimating the free energy of\ngeneral Ising models (in particular, even in regimes where correlation decay\ndoes not hold) are: (i) the mean-field approximation with roots in statistical\nphysics, which estimates the free energy from below, and (ii) hierarchies of\nconvex relaxations with roots in theoretical computer science, which estimate\nthe free energy from above. We show, surprisingly, that the tight regime for\nboth methods to compute the free energy to leading order is identical.\n More precisely, we show that the mean-field approximation is within\n$O((n\\|J\\|_{F})^{2/3})$ of the free energy, where $\\|J\\|_F$ denotes the\nFrobenius norm of the interaction matrix of the Ising model. This\nsimultaneously subsumes both the breakthrough work of Basak and Mukherjee, who\nshowed the tight result that the mean-field approximation is within $o(n)$\nwhenever $\\|J\\|_{F} = o(\\sqrt{n})$, as well as the work of Jain, Koehler, and\nMossel, who gave the previously best known non-asymptotic bound of\n$O((n\\|J\\|_{F})^{2/3}\\log^{1/3}(n\\|J\\|_{F}))$. We give a simple, algorithmic\nproof of this result using a convex relaxation proposed by Risteski based on\nthe Sherali-Adams hierarchy, automatically giving sub-exponential time\napproximation schemes for the free energy in this entire regime. Our\nalgorithmic result is tight under Gap-ETH.\n We furthermore combine our techniques with spin glass theory to prove (in a\nstrong sense) the optimality of correlation rounding, refuting a recent\nconjecture of Allen, O'Donnell, and Zhou. Finally, we give the tight\ngeneralization of all of these results to $k$-MRFs, capturing as a special case\nprevious work on approximating MAX-$k$-CSP.\n