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Journal Articles SIAM Journal on Applied Mathematics Year : 2020

Hypocoercivity properties of adaptive Langevin dynamics


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.

Dates and versions

hal-02273261 , version 1 (28-08-2019)



Benedict Leimkuhler, Matthias Sachs, Gabriel Stoltz. Hypocoercivity properties of adaptive Langevin dynamics. SIAM Journal on Applied Mathematics, 2020, 80 (3), pp.1197-1222. ⟨10.1137/19M1291649⟩. ⟨hal-02273261⟩
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