Hypocoercivity properties of adaptive Langevin dynamics

Benedict Leimkuhler, Matthias Sachs, Gabriel Stoltz

Research output: Contribution to journalArticlepeer-review

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 Nose-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 languageEnglish
Pages (from-to)1197–1222
Number of pages26
JournalSiam Journal on Applied Mathematics
Volume80
Issue number3
Early online date19 May 2020
DOIs
Publication statusE-pub ahead of print - 19 May 2020

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