  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.
