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

Andrej Risteski
; 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.

Cite this Paper


BibTeX
@InProceedings{pmlr-v49-risteski16, title = {How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods}, author = {Andrej Risteski}, booktitle = {29th Annual Conference on Learning Theory}, pages = {1402--1416}, year = {2016}, editor = {Vitaly Feldman and Alexander Rakhlin and Ohad Shamir}, volume = {49}, series = {Proceedings of Machine Learning Research}, address = {Columbia University, New York, New York, USA}, month = {23--26 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v49/risteski16.pdf}, url = {http://proceedings.mlr.press/v49/risteski16.html}, 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. } }
Endnote
%0 Conference Paper %T How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods %A Andrej Risteski %B 29th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2016 %E Vitaly Feldman %E Alexander Rakhlin %E Ohad Shamir %F pmlr-v49-risteski16 %I PMLR %J Proceedings of Machine Learning Research %P 1402--1416 %U http://proceedings.mlr.press %V 49 %W PMLR %X 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.
RIS
TY - CPAPER TI - How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods AU - Andrej Risteski BT - 29th Annual Conference on Learning Theory PY - 2016/06/06 DA - 2016/06/06 ED - Vitaly Feldman ED - Alexander Rakhlin ED - Ohad Shamir ID - pmlr-v49-risteski16 PB - PMLR SP - 1402 DP - PMLR EP - 1416 L1 - http://proceedings.mlr.press/v49/risteski16.pdf UR - http://proceedings.mlr.press/v49/risteski16.html AB - 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. ER -
APA
Risteski, A.. (2016). How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods. 29th Annual Conference on Learning Theory, in PMLR 49:1402-1416

Related Material