Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling

Abstract

Sampling from a high-dimensional distribution is a fundamental task in statistics, engineering, and the sciences. A canonical approach is the Langevin Algorithm, i.e., the Markov chain for the discretized Langevin Diffusion. This is the sampling analog of Gradient Descent. Despite being studied for several decades in multiple communities, tight mixing bounds for this algorithm remain unresolved even in the seemingly simple setting of log-concave distributions over a bounded domain. This paper completely characterizes the mixing time of the Langevin Algorithm to its stationary distribution in this setting (and others). This mixing result can be combined with any bound on the discretization bias in order to sample from the stationary distribution of the continuous Langevin Diffusion. In this way, we disentangle the study of the mixing and bias of the Langevin Algorithm.Our key insight is to introduce a technique from the differential privacy literature to the sampling literature. This technique, called Privacy Amplification by Iteration, uses as a potential a variant of Rényi divergence that is made geometrically aware via Optimal Transport smoothing. This gives a short, simple proof of optimal mixing bounds and has several additional appealing properties. First, our approach removes all unnecessary assumptions required by other sampling analyses. Second, our approach unifies many settings: it extends unchanged if the Langevin Algorithm uses projections, stochastic mini-batch gradients, or strongly convex potentials (whereby our mixing time improves exponentially). Third, our approach exploits convexity only through the contractivity of a gradient step-reminiscent of how convexity is used in textbook proofs of Gradient Descent. In this way, we offer a new approach towards further unifying the analyses of optimization and sampling algorithms.

Convergence Rates for Non-Log-Concave Sampling and Log-Partition Estimation

“…In the case where p f is (strongly) log-concave, that is, if f is (strongly) concave, convergence rates of Markov chain Monte Carlo (MCMC) sampling algorithms have been studied extensively. For example, good convergence rates in terms of the dimension d have been established for versions of the Langevin algorithm (Chewi et al, 2021;Altschuler and Talwar, 2022) and Hamiltonian Monte Carlo (Mangoubi and Vishnoi, 2018). Chewi et al (2022b) establish an algorithm with optimal convergence rate for the case d = 1, while not much is known about algorithm-independent lower bounds in other cases.…”

“…To study this, we need to make some assumptions on f . While efficient sampling algorithms for suitable classes of concave f are known, at least with access to gradients of f (Dwivedi et al, 2018;Mangoubi and Vishnoi, 2018;Chewi et al, 2021;Altschuler and Talwar, 2022), we are interested in larger classes of non-concave functions, which are defined in the following:…”