  We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as
optimization on the space of probability measures, with Kullback-Leibler (KL)
divergence as the objective functional. We show that an underdamped form of the
Langevin algorithm performs accelerated gradient descent in this metric. To
characterize the convergence of the algorithm, we construct a Lyapunov
functional and exploit hypocoercivity of the underdamped Langevin algorithm. As
an application, we show that accelerated rates can be obtained for a class of
nonconvex functions with the Langevin algorithm.
