Lihan Wang
Carnegie Mellon University
Title
Quantitative estimates on convergence rates of kinetic dynamics for sampling
Abstract
Sampling has important applications in a wide range of areas, including molecular dynamics, Bayesian statistics and machine learning. Many of the most widely used dynamics for sampling have a kinetic structure, the two most prominent examples of which are underdamped Langevin dynamics and Hamiltonian Monte Carlo. We discuss the quantitative long-time convergence behavior of these kinetic dynamics for sampling, under different assumptions of the potential. For convex potentials, we show that kinetic sampling dynamics accelerate the convergence rate by a square root factor of the Poincaré constant, compared to the overdamped Langevin dynamics, which is the sampling analog of Nesterov acceleration in convex optimization. We also show in the weakly confining setting, how the growth rate of the potential impacts the convergence rates via weak variants of the Poincaré inequality. The analysis is inspired by the Armstrong-Mourrat variational framework for hypocoercivity, which combines a Poincaré-Lions inequality in time-augmented state space and an L^2 energy estimate.
