• DOI:10.1145/3313276.3316299
• Corpus ID: 52067488

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

@article{Jain2018MeanfieldAC,
  title={Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective},
  author={Vishesh Jain and Frederic Koehler and Andrej Risteski},
  journal={Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing},
  year={2018}
}

• Vishesh Jain, Frederic Koehler, Andrej Risteski
• Published 22 August 2018
• Computer Science
• Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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 to the free energy is within O((n||J||F)2/3) of the true 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(√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)2/3log1/3(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.