Hypocoercivity Properties of Adaptive Langevin Dynamics

Benedict Leimkuhler; Matthias Sachs; Gabriel Stoltz

doi:10.1137/19M1291649

Hypocoercivity Properties of Adaptive Langevin Dynamics

Benedict Leimkuhler, Matthias Sachs, Gabriel Stoltz

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

Adaptive Langevin dynamics is a method for sampling the Boltzmann--Gibbs distribution at prescribed temperature in cases where the potential gradient is subject to stochastic perturbation of unknown magnitude. The method replaces the friction in underdamped Langevin dynamics with a dynamical variable, updated according to a negative feedback loop control law as in the Nosé--Hoover thermostat. Using a hypocoercivity analysis we show that the law of Adaptive Langevin dynamics converges exponentially rapidly to the stationary distribution, with a rate that can be quantified in terms of the key parameters of the dynamics. This allows us in particular to obtain a central limit theorem with respect to the time averages computed along a stochastic path. Our theoretical findings are illustrated by numerical simulations involving classification of the MNIST data set of handwritten digits using Bayesian logistic regression.

Original language	English
Pages (from-to)	1197-1222
Number of pages	26
Journal	SIAM Journal on Applied Mathematics
Volume	80
Issue number	3
DOIs	https://doi.org/10.1137/19M1291649
Publication status	Published - 19 May 2020

Bibliographical note

Funding Information:
The work of the first author was supported by the ERC project RULE through grant 320823 and by EPSRC grant EP/P006175/1. The work of the second author was supported by the National Science Foundation under grant DMS-1638521 to the Statistical and Applied Mathematical Sciences Institute (SAMSI), North Carolina. The work of the third author was supported by the Agence Nationale de la Recherche, under grant ANR-14-CE23-0012 (COSMOS), and by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC grant agreement 614492. This work was initiated during the authors' stay at the Institut Henri Poincaré-Centre Emile Borei during the trimester "Stochastic Dynamics Out of Equilibrium" (April-July 2017). The authors warmly thank this institution for its hospitality. G.S. also benefited from the scientific environment of the Laboratoire International Associé between the Centre National de la Recherche Scientifique and the University of Illinois at Urbana-Champaign.

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

Keywords

Bayesian inference
Hypocoercivity
Langevin dynamics
Nose-Hoover
Sampling
Stochastic gradients

ASJC Scopus subject areas

Applied Mathematics

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10.1137/19M1291649

Cite this

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title = "Hypocoercivity Properties of Adaptive Langevin Dynamics",

abstract = "Adaptive Langevin dynamics is a method for sampling the Boltzmann--Gibbs distribution at prescribed temperature in cases where the potential gradient is subject to stochastic perturbation of unknown magnitude. The method replaces the friction in underdamped Langevin dynamics with a dynamical variable, updated according to a negative feedback loop control law as in the Nos{\'e}--Hoover thermostat. Using a hypocoercivity analysis we show that the law of Adaptive Langevin dynamics converges exponentially rapidly to the stationary distribution, with a rate that can be quantified in terms of the key parameters of the dynamics. This allows us in particular to obtain a central limit theorem with respect to the time averages computed along a stochastic path. Our theoretical findings are illustrated by numerical simulations involving classification of the MNIST data set of handwritten digits using Bayesian logistic regression.",

keywords = "Bayesian inference, Hypocoercivity, Langevin dynamics, Nose-Hoover, Sampling, Stochastic gradients",

author = "Benedict Leimkuhler and Matthias Sachs and Gabriel Stoltz",

note = "Funding Information: The work of the first author was supported by the ERC project RULE through grant 320823 and by EPSRC grant EP/P006175/1. The work of the second author was supported by the National Science Foundation under grant DMS-1638521 to the Statistical and Applied Mathematical Sciences Institute (SAMSI), North Carolina. The work of the third author was supported by the Agence Nationale de la Recherche, under grant ANR-14-CE23-0012 (COSMOS), and by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC grant agreement 614492. This work was initiated during the authors' stay at the Institut Henri Poincar{\'e}-Centre Emile Borei during the trimester {"}Stochastic Dynamics Out of Equilibrium{"} (April-July 2017). The authors warmly thank this institution for its hospitality. G.S. also benefited from the scientific environment of the Laboratoire International Associ{\'e} between the Centre National de la Recherche Scientifique and the University of Illinois at Urbana-Champaign. Publisher Copyright: {\textcopyright} 2020 Society for Industrial and Applied Mathematics.",

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PY - 2020/5/19

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N2 - Adaptive Langevin dynamics is a method for sampling the Boltzmann--Gibbs distribution at prescribed temperature in cases where the potential gradient is subject to stochastic perturbation of unknown magnitude. The method replaces the friction in underdamped Langevin dynamics with a dynamical variable, updated according to a negative feedback loop control law as in the Nosé--Hoover thermostat. Using a hypocoercivity analysis we show that the law of Adaptive Langevin dynamics converges exponentially rapidly to the stationary distribution, with a rate that can be quantified in terms of the key parameters of the dynamics. This allows us in particular to obtain a central limit theorem with respect to the time averages computed along a stochastic path. Our theoretical findings are illustrated by numerical simulations involving classification of the MNIST data set of handwritten digits using Bayesian logistic regression.

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Hypocoercivity Properties of Adaptive Langevin Dynamics

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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