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# How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods

*29th Annual Conference on Learning Theory*, PMLR 49:1402-1416, 2016.

#### Abstract

We consider the problem of approximating partition functions for Ising models. We make use of recent tools in combinatorial optimization: the Sherali-Adams and Lasserre convex programming hierarchies, in combination with variational methods to get algorithms for calculating partition functions in these families. These techniques give new, non-trivial approximation guarantees for the partition function beyond the regime of correlation decay. They also generalize some classical results from statistical physics about the Curie-Weiss ferromagnetic Ising model, as well as provide a partition function counterpart of classical results about max-cut on dense graphs (Arora, 1995). With this, we connect techniques from two apparently disparate research areas – optimization and counting/partition function approximations. (i.e. #-P type of problems). Furthermore, we design to the best of our knowledge the first provable, convex variational methods. Though in the literature there are a host of convex versions of variational methods, they come with no guarantees (apart from some extremely special cases, like e.g. the graph has a single cycle). We consider dense and low rank graphs, and interestingly, the reason our approach works on these types of graphs is because local correlations propagate to global correlations – completely the opposite of algorithms based on correlation decay. In the process we design novel entropy approximations based on the low-order moments of a distribution. Our proof techniques are very simple and generic, and likely to be applicable to many other settings other than Ising models.