An Analog of Nesterov Acceleration for MCMC

We formulate gradient-based Markov chain Monte Carlo sampling as an optimization on the space of probability measures, with Kullback-Leibler 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.