A Large Deviations Analysis of Certain Qualitative Properties of Parallel Tempering and Infinite Swapping Algorithms

Doll, J. D.; Dupuis, Paul; Nyquist, Pierre

doi:10.1007/s00245-017-9401-9

Cited by 14 publications

(13 citation statements)

References 17 publications

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“…On the other hand, Bhatnagar and Randall (2004) and Bhatnagar and Randall (2016) prove that both, the swapping algorithm and simulated tempering, are slowly mixing for the 3-state Potts model and conjecture that this is caused by the first order phase transition in the Potts model (also see our discussion in the remark following Lemma 4.3). Qualitative properties of the swapping algorithm and parallel tempering were studied in Doll et al (2018). A first rapid convergence result for the Swapping Algorithm in an disordered situation was proved in Löwe and Vermet (2009).…”

Section: Equi-energy Samplingmentioning

confidence: 99%

Equi-Energy sampling does not converge rapidly on the mean-field Potts model with three colors close to the critical temperature

Ebbers,

Löwe

2019

Preprint

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Equi-Energy Sampling (EES, for short) is a method to speed up the convergence of the Metropolis chain, when the latter is slow. We show that there are still models like the mean-field Potts model, where EES does not converge rapidly in certain temperature regimes. Indeed we will show that EES is slowly mixing on the mean-field Potts model, in a regime below the critical temperature. Though we will concentrate on the Potts model with three colors, our arguments remain valid for any number of colors q ≥ 3, if we adapt the temperature regime. For the situation of the mean-field Potts model this answers a question posed in Hua and Kou (2011).

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Section: Equi-energy Samplingmentioning

confidence: 99%

Equi-Energy sampling does not converge rapidly on the mean-field Potts model with three colors close to the critical temperature

Ebbers,

Löwe

2019

Preprint

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“…However, analogous results for discrete state systems are expected. See [9] for the formulation of infinite swapping for discrete state models.…”

Section: Introductionmentioning

confidence: 99%

Analysis and optimization of certain parallel Monte Carlo methods in the low temperature limit

Dupuis¹,

Wu²

2020

Preprint

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Metastability is a formidable challenge to Markov chain Monte Carlo methods. In this paper we present methods for algorithm design to meet this challenge. The design problem we consider is temperature selection for the infinite swapping scheme, which is the limit of the widely used parallel tempering scheme obtained when the swap rate tends to infinity. We use a recently developed tool for the analysis of the empirical measure of a small noise diffusion to transform the variance reduction problem into an explicit optimization problem. Our first analysis of the optimization problem is in the setting of a double well model, and it shows that the optimal selection of temperature ratios is a geometric sequence except possibly the highest temperature. In the same setting we identify two different sources of variance reduction, and show how their competition determines the optimal highest temperature. In the general multi-well setting we prove that a pure geometric sequence of temperature ratios is always nearly optimal, with a performance gap that decays geometrically in the number of temperatures.

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“…Therein empirical measure large deviations, specifically the associated rate function, was proposed as a tool for analysing parallel tempering, one of the computational workhorses of the physical sciences, leading to a new type of simulation method (infinite swapping). In the subsequent work [DDN18] empirical measure large deviations were again used, combined with associated stochastic control problems, to analyse the convergence properties of these algorithms. Similarly, in [RBS15] Rey-Bellet and Spiliopoulos use empirical measure large deviations to analyse the performance of certain non-reversible MCMC samplers.…”

Section: Introductionmentioning

confidence: 99%

Large deviations for the empirical measure of the zig-zag process

Bierkens¹,

Nyquist²,

Schlottke³

2019

Preprint

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The zig-zag process is a piecewise deterministic Markov process in position and velocity space. The process can be designed to have an arbitrary Gibbs type marginal probability density for its position coordinate, which makes it suitable for Monte Carlo simulation of continuous probability distributions. An important question in assessing the efficiency of this method is how fast the empirical measure converges to the stationary distribution of the process. In this paper we provide a partial answer to this question by characterizing the large deviations of the empirical measure from the stationary distribution. Based on the Feng-Kurtz approach, we develop an abstract framework aimed at encompassing piecewise deterministic Markov processes in position-velocity space. We derive explicit conditions for the zig-zag process to allow the Donsker-Varadhan variational formulation of the rate function, both for a compact setting (the torus) and one-dimensional Euclidean space. Finally we derive an explicit expression for the Donsker-Varadhan functional for the case of a compact state space and use this form of the rate function to address a key question concerning the optimal choice of the switching rate of the zig-zag process.

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A Large Deviations Analysis of Certain Qualitative Properties of Parallel Tempering and Infinite Swapping Algorithms

Cited by 14 publications

References 17 publications

Equi-Energy sampling does not converge rapidly on the mean-field Potts model with three colors close to the critical temperature

Equi-Energy sampling does not converge rapidly on the mean-field Potts model with three colors close to the critical temperature

Analysis and optimization of certain parallel Monte Carlo methods in the low temperature limit

Large deviations for the empirical measure of the zig-zag process

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