An entropic approach for Hamiltonian Monte Carlo: The idealized case

• 成果类型: Article
• 署名作者: Monmarche, Pierre
• 署名单位: Sorbonne Universite; Sorbonne Universite; Universite Paris Cite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP2021
• 发表日期: 2024
• 页码: 2243-2293
• 关键词: logarithmic sobolev inequalities; relaxation boltzmann-equation; fokker-planck equation; long-time behavior; spectral gap; MARKOV-PROCESSES; hypocoercivity; CONVERGENCE; equilibrium; contraction

摘要

Quantitative long-time entropic convergence and short-time regularization are established for an idealized Hamiltonian Monte Carlo chain which alternatively follows an Hamiltonian dynamics for a fixed time and then partially or totally refreshes its velocity with an auto-regressive Gaussian step. These results, in discrete time, are the analogues of similar results for the continuous-time kinetic Langevin diffusion, and the latter can be obtained from our bounds in a suitable limit regime. The dependency in the log-Sobolev constant of the target measure is sharp and is illustrated on a mean-field case and on a low-temperature regime, with an application to the simulated annealing algorithm. The practical unadjusted algorithm is briefly discussed.