MIXING OF HAMILTONIAN MONTE CARLO ON STRONGLY LOG-CONCAVE DISTRIBUTIONS: CONTINUOUS DYNAMICS

• 成果类型: Article
• 署名作者: Mangoubi, Oren; Smith, Aaron
• 署名单位: Worcester Polytechnic Institute; University of Ottawa
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1640
• 发表日期: 2021
• 页码: 2019-2045
• 关键词: algorithms

摘要

We obtain several quantitative bounds on the mixing properties of an ideal Hamiltonian Monte Carlo (HMC) Markov chain for a strongly log-concave target distribution pi on R-d. Our main result says that the HMC Markov chain generates a sample with Wasserstein error epsilon in roughly O(kappa(2) log(1/epsilon)) steps, where the condition number kappa = M-2/m(2) is the ratio of the maximum M-2 and minimum m(2) eigenvalues of the Hessian of - log(pi). In particular, this mixing bound does not depend explicitly on the dimension d. These results significantly extend and improve previous quantitative bounds on the mixing of ideal HMC, and can be used to analyze more realistic HMC algorithms. The main ingredient of our argument is a proof that initially parallel Hamiltonian trajectories contract over much longer steps than would be predicted by previous heuristics based on the Jacobi manifold.