In this paper, we propose a new approach for sampling from probability measures in, possibly, high-dimensional spaces. By perturbing the standard overdamped Langevin dynamics by a suitable Stratonovich perturbation that preserves the invariant measure of the original system, we show that accelerated convergence to equilibrium and reduced asymptotic variance can be achieved, leading, thus, to a computationally advantageous sampling algorithm. The new perturbed Langevin dynamics is reversible with respect to the target probability measure and, consequently, does not suffer from the drawbacks of the nonreversible Langevin samplers that were introduced in C.-R. Hwang et al. (1993) [1] and studied in, e.g., T. Lelièvre et al. (2013) [2] and A.B. Duncan et al. (2016) [3], while retaining all of their advantages in terms of accelerated convergence and reduced asymptotic variance. In particular, the reversibility of the dynamics ensures that there is no oscillatory transient behaviour. The improved performance of the proposed methodology, in comparison to the standard overdamped Langevin dynamics and its nonreversible perturbation, is illustrated on an example of sampling from a two-dimensional warped Gaussian target distribution.
Dans cet article, nous proposons une nouvelle approche pour l'échantillonnage de mesures invariantes dans des espaces de grandes dimensions à l'aide d'une dynamique de Langevin perturbée. En modifiant la dynamique standard de l'équation de Langevin suramortie en introduisant une perturbation de Stratonovich convenable préservant la mesure invariante du système initial, nous montrons qu'il est possible d'obtenir une convergence accélérée vers l'équilibre et une variance asymptotique réduite, conduisant ainsi à un algorithme d'échantillonnage avantageux du point de vue du calcul. La nouvelle dynamique de Langevin perturbée est réversible par rapport à la mesure de probabilité cherchée et ne souffre donc pas des inconvénients des échantillonneurs de Langevin non réversibles introduits dans C.-R. Hwang et al. (1993) [1] et étudiés, par exemple, dans T. Lelièvre et al. (2013) [2] et A.B. Duncan et al. (2016) [3], tout en conservant tous leurs avantages en termes de convergence accélérée et de réduction de la variance asymptotique. En particulier, la réversibilité de la dynamique garantit l'absence de comportement transitoire oscillant. Les performances améliorées de la méthodologie proposée par rapport à la dynamique de Langevin suramortie standard et à sa perturbation irréversible sont illustrées par un exemple d'échantillonnage à partir d'une distribution gaussienne déformée à deux dimensions.
Accepted:
Published online:
@article{CRMATH_2019__357_4_349_0,
author = {Abdulle, Assyr and Pavliotis, Grigorios A. and Vilmart, Gilles},
title = {Accelerated convergence to equilibrium and reduced asymptotic variance for {Langevin} dynamics using {Stratonovich} perturbations},
journal = {Comptes Rendus. Math\'ematique},
pages = {349--354},
publisher = {Elsevier},
volume = {357},
number = {4},
year = {2019},
doi = {10.1016/j.crma.2019.04.008},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2019.04.008/}
}
TY - JOUR AU - Abdulle, Assyr AU - Pavliotis, Grigorios A. AU - Vilmart, Gilles TI - Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations JO - Comptes Rendus. Mathématique PY - 2019 SP - 349 EP - 354 VL - 357 IS - 4 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2019.04.008/ DO - 10.1016/j.crma.2019.04.008 LA - en ID - CRMATH_2019__357_4_349_0 ER -
%0 Journal Article %A Abdulle, Assyr %A Pavliotis, Grigorios A. %A Vilmart, Gilles %T Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations %J Comptes Rendus. Mathématique %D 2019 %P 349-354 %V 357 %N 4 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2019.04.008/ %R 10.1016/j.crma.2019.04.008 %G en %F CRMATH_2019__357_4_349_0
Abdulle, Assyr; Pavliotis, Grigorios A.; Vilmart, Gilles. Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations. Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 349-354. doi : 10.1016/j.crma.2019.04.008. http://archive.numdam.org/articles/10.1016/j.crma.2019.04.008/
[1] Accelerating Gaussian diffusions, Ann. Appl. Probab., Volume 3 (1993) no. 3, pp. 897-913
[2] Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion, J. Stat. Phys., Volume 152 (2013) no. 2, pp. 237-274
[3] Variance reduction using nonreversible Langevin samplers, J. Stat. Phys., Volume 163 (2016) no. 3, pp. 457-491
[4] Free Energy Computations: A Mathematical Perspective, Imperial College Press, London, 2010
[5] Inverse problems: a Bayesian perspective, Acta Numer., Volume 19 (2010), pp. 451-559
[6] An introduction to MCMC for machine learning, Machine Learning, Volume 50 (2003) no. 1, pp. 5-43
[7] Convergence of numerical time-averaging and stationary measures via Poisson equations, SIAM J. Numer. Anal., Volume 48 (2010) no. 2, pp. 552-577
[8] Using perturbed underdamped Langevin dynamics to efficiently sample from probability distributions, J. Stat. Phys., Volume 169 (2017) no. 6, pp. 1098-1131
[9] Accelerating diffusions, Ann. Appl. Probab., Volume 15 (2005) no. 2, pp. 1433-1444
[10] Irreversible Langevin samplers and variance reduction: a large deviations approach, Nonlinearity, Volume 28 (2015) no. 7, p. 2081
[11] Improving the convergence of reversible samplers, J. Stat. Phys., Volume 164 (2016) no. 3, pp. 472-494
[12] A.B. Duncan, G.A. Pavliotis, K.C. Zygalakis, Nonreversible Langevin samplers: splitting schemes, analysis and implementation, ArXiv e-prints, 2017.
[13] Stochastic Processes and Applications – Diffusion Processes, the Fokker–Planck and Langevin Equations, Texts in Applied Mathematics, vol. 60, Springer, New York, 2014
Cited by Sources:
