Hypocoercivity Properties of Adaptive Langevin Dynamics

Leimkuhler, Benedict and Sachs, Matthias and Stoltz, Gabriel (2020) Hypocoercivity Properties of Adaptive Langevin Dynamics. SIAM Journal on Applied Mathematics, 80 (3). pp. 1197-1222. ISSN 0036-1399

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.