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.
Cite
@article{arxiv.1908.09363,
title = {Hypocoercivity properties of adaptive Langevin dynamics},
author = {Benedict Leimkuhler and Matthias Sachs and Gabriel Stoltz},
journal= {arXiv preprint arXiv:1908.09363},
year = {2023}
}