Is There an Analog of Nesterov Acceleration for MCMC?

While optimization methodology has provided much of the underlying algorithmic machinery that has driven the theory and practice of machine learning in recent years, sampling-based methodology, in particular Markov chain Monte Carlo (MCMC), remains of critical importance, given its role in linking algorithms to statistical inference and, in particular, its ability to provide notions of confidence that are lacking in optimization-based methodology. However, the classical theory of MCMC is largely asymptotic and the theory has not developed as rapidly in recent years as the theory of optimization. Recently, however, a literature has emerged that derives nonasymptotic rates for MCMC algorithms [see, e.g., 9, 12, 10, 8, 6, 14, 21, 22, 2, 5]. This work has explicitly aimed at making use of ideas from optimization; in particular, whereas the classical literature on MCMC focused on reversible Markov chains, the recent literature has focused on nonreversible stochastic processes that are built on gradients [see, e.g., 18, 20, 3, 1]. In particular, the gradient-based Langevin algorithm [33, 32, 13] has been shown to be a form of gradient descent on the space of probabilities [see, e.g., 36]. What has not yet emerged is an analog of acceleration. Recall that the notion of acceleration has played a key role in gradient-based optimization methods [26]. In particular, the Nesterov accelerated gradient descent (AGD) method, an instance of the general family of "momentum methods," provably achieves faster convergence rate than gradient descent (GD) in a variety of settings [25]. Moreover, it achieves the optimal convergence rate under an oracle model of optimization complexity in the convex setting [24].