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

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 citearora1995polynomial. 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 citewainwright2003tree, wainwright2005new, heskes2006convexity, meshi2009convexifying, they come with no guarantees (apart from some extremely special cases, like e.g. the graph has a single cycle citeweiss2000correctness). We consider dense and low threshold 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.