Is There an Analog of Nesterov Acceleration for MCMC?
This provides a theoretical foundation for accelerating MCMC methods, which is incremental but addresses a known bottleneck in sampling for machine learning and statistics.
The paper tackles the problem of accelerating gradient-based Markov chain Monte Carlo (MCMC) sampling by formulating it as optimization on probability measures with KL divergence, showing that an underdamped Langevin algorithm achieves accelerated gradient descent and obtains accelerated rates for a class of nonconvex functions.
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.