Swendsen-wang dynamics for general graphs in the tree uniqueness region

Antonio Blanca Pimentel, Zongchen Chen, Eric Vigoda

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G = (V,E). The dynamics is a "global" Markov chain which is conjectured to converge to equilibrium in O(|V |1/4) steps for any graph G at any (inverse) temperature β. It was recently proved by Guo and Jerrum (2017) that the Swendsen-Wang dynamics has polynomial mixing time on any graph at all temperatures, yet there are few results providing o(|V |) upper bounds on its convergence time. We prove fast convergence of the Swendsen-Wang dynamics on general graphs in the tree uniqueness region of the ferromagnetic Ising model. In particular, when β < βc(d) where βc(d) denotes the uniqueness/non-uniqueness threshold on infinite d-regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the Swendsen-Wang dynamics is α(1) on any graph of maximum degree d ≥ 3. Our proof utilizes a version of the Swendsen-Wang dynamics which only updates isolated vertices. We establish that this variant of the Swendsen-Wang dynamics has mixing time O(log |V |) and relaxation time α(1) on any graph of maximum degree d for all β < βc(d). We believe that this Markov chain may be of independent interest, as it is a monotone Swendsen-Wang type chain. As part of our proofs, we provide modest extensions of the technology of Mossel and Sly (2013) for analyzing mixing times and of the censoring result of Peres and Winkler (2013). Both of these results are for the Glauber dynamics, and we extend them here to general monotone Markov chains. This class of dynamics includes for example the heat-bath block dynamics, for which we obtain new tight mixing time bounds.

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018
EditorsEric Blais, Jose D. P. Rolim, David Steurer, Klaus Jansen
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print)9783959770859
DOIs
StatePublished - Aug 1 2018
Event21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 - Princeton, United States
Duration: Aug 20 2018Aug 22 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume116
ISSN (Print)1868-8969

Conference

Conference21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018
CountryUnited States
CityPrinceton
Period8/20/188/22/18

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Markov processes
Ising model
Relaxation time
Polynomials
Sampling
Temperature
Hot Temperature

All Science Journal Classification (ASJC) codes

  • Software

Cite this

Pimentel, A. B., Chen, Z., & Vigoda, E. (2018). Swendsen-wang dynamics for general graphs in the tree uniqueness region. In E. Blais, J. D. P. Rolim, D. Steurer, & K. Jansen (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018 [32] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 116). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.32
Pimentel, Antonio Blanca ; Chen, Zongchen ; Vigoda, Eric. / Swendsen-wang dynamics for general graphs in the tree uniqueness region. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018. editor / Eric Blais ; Jose D. P. Rolim ; David Steurer ; Klaus Jansen. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018. (Leibniz International Proceedings in Informatics, LIPIcs).
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title = "Swendsen-wang dynamics for general graphs in the tree uniqueness region",
abstract = "The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G = (V,E). The dynamics is a {"}global{"} Markov chain which is conjectured to converge to equilibrium in O(|V |1/4) steps for any graph G at any (inverse) temperature β. It was recently proved by Guo and Jerrum (2017) that the Swendsen-Wang dynamics has polynomial mixing time on any graph at all temperatures, yet there are few results providing o(|V |) upper bounds on its convergence time. We prove fast convergence of the Swendsen-Wang dynamics on general graphs in the tree uniqueness region of the ferromagnetic Ising model. In particular, when β < βc(d) where βc(d) denotes the uniqueness/non-uniqueness threshold on infinite d-regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the Swendsen-Wang dynamics is α(1) on any graph of maximum degree d ≥ 3. Our proof utilizes a version of the Swendsen-Wang dynamics which only updates isolated vertices. We establish that this variant of the Swendsen-Wang dynamics has mixing time O(log |V |) and relaxation time α(1) on any graph of maximum degree d for all β < βc(d). We believe that this Markov chain may be of independent interest, as it is a monotone Swendsen-Wang type chain. As part of our proofs, we provide modest extensions of the technology of Mossel and Sly (2013) for analyzing mixing times and of the censoring result of Peres and Winkler (2013). Both of these results are for the Glauber dynamics, and we extend them here to general monotone Markov chains. This class of dynamics includes for example the heat-bath block dynamics, for which we obtain new tight mixing time bounds.",
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Pimentel, AB, Chen, Z & Vigoda, E 2018, Swendsen-wang dynamics for general graphs in the tree uniqueness region. in E Blais, JDP Rolim, D Steurer & K Jansen (eds), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018., 32, Leibniz International Proceedings in Informatics, LIPIcs, vol. 116, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018, Princeton, United States, 8/20/18. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.32

Swendsen-wang dynamics for general graphs in the tree uniqueness region. / Pimentel, Antonio Blanca; Chen, Zongchen; Vigoda, Eric.

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018. ed. / Eric Blais; Jose D. P. Rolim; David Steurer; Klaus Jansen. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018. 32 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 116).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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AU - Pimentel, Antonio Blanca

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N2 - The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G = (V,E). The dynamics is a "global" Markov chain which is conjectured to converge to equilibrium in O(|V |1/4) steps for any graph G at any (inverse) temperature β. It was recently proved by Guo and Jerrum (2017) that the Swendsen-Wang dynamics has polynomial mixing time on any graph at all temperatures, yet there are few results providing o(|V |) upper bounds on its convergence time. We prove fast convergence of the Swendsen-Wang dynamics on general graphs in the tree uniqueness region of the ferromagnetic Ising model. In particular, when β < βc(d) where βc(d) denotes the uniqueness/non-uniqueness threshold on infinite d-regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the Swendsen-Wang dynamics is α(1) on any graph of maximum degree d ≥ 3. Our proof utilizes a version of the Swendsen-Wang dynamics which only updates isolated vertices. We establish that this variant of the Swendsen-Wang dynamics has mixing time O(log |V |) and relaxation time α(1) on any graph of maximum degree d for all β < βc(d). We believe that this Markov chain may be of independent interest, as it is a monotone Swendsen-Wang type chain. As part of our proofs, we provide modest extensions of the technology of Mossel and Sly (2013) for analyzing mixing times and of the censoring result of Peres and Winkler (2013). Both of these results are for the Glauber dynamics, and we extend them here to general monotone Markov chains. This class of dynamics includes for example the heat-bath block dynamics, for which we obtain new tight mixing time bounds.

AB - The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G = (V,E). The dynamics is a "global" Markov chain which is conjectured to converge to equilibrium in O(|V |1/4) steps for any graph G at any (inverse) temperature β. It was recently proved by Guo and Jerrum (2017) that the Swendsen-Wang dynamics has polynomial mixing time on any graph at all temperatures, yet there are few results providing o(|V |) upper bounds on its convergence time. We prove fast convergence of the Swendsen-Wang dynamics on general graphs in the tree uniqueness region of the ferromagnetic Ising model. In particular, when β < βc(d) where βc(d) denotes the uniqueness/non-uniqueness threshold on infinite d-regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the Swendsen-Wang dynamics is α(1) on any graph of maximum degree d ≥ 3. Our proof utilizes a version of the Swendsen-Wang dynamics which only updates isolated vertices. We establish that this variant of the Swendsen-Wang dynamics has mixing time O(log |V |) and relaxation time α(1) on any graph of maximum degree d for all β < βc(d). We believe that this Markov chain may be of independent interest, as it is a monotone Swendsen-Wang type chain. As part of our proofs, we provide modest extensions of the technology of Mossel and Sly (2013) for analyzing mixing times and of the censoring result of Peres and Winkler (2013). Both of these results are for the Glauber dynamics, and we extend them here to general monotone Markov chains. This class of dynamics includes for example the heat-bath block dynamics, for which we obtain new tight mixing time bounds.

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Pimentel AB, Chen Z, Vigoda E. Swendsen-wang dynamics for general graphs in the tree uniqueness region. In Blais E, Rolim JDP, Steurer D, Jansen K, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2018. 32. (Leibniz International Proceedings in Informatics, LIPIcs). https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.32