Translator Disclaimer
October 2021 Mixing of Hamiltonian Monte Carlo on strongly log-concave distributions: Continuous dynamics
Oren Mangoubi, Aaron Smith
Author Affiliations +
Ann. Appl. Probab. 31(5): 2019-2045 (October 2021). DOI: 10.1214/20-AAP1640

Abstract

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 π on Rd. Our main result says that the HMC Markov chain generates a sample with Wasserstein error ϵ in roughly O(κ2log(1ϵ)) steps, where the condition number κ=M2 m2 is the ratio of the maximum M2 and minimum m2 eigenvalues of the Hessian of log(π). 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.

Funding Statement

Oren Mangoubi was supported by a Canadian Statistical Sciences Institute (CANSSI) Postdoctoral Fellowship, and by an NSERC Discovery grant. Aaron Smith was supported by an NSERC Discovery grant.

Acknowledgements

We are grateful to Natesh Pillai and Alain Durmus for helpful discussions. Oren Mangoubi was supported by a Canadian Statistical Sciences Institute (CANSSI) Postdoctoral Fellowship, and by an NSERC Discovery grant. Aaron Smith was supported by an NSERC Discovery grant. We are grateful to the anonymous reviewers of an earlier version of this paper for their helpful comments and suggestions.

Citation

Download Citation

Oren Mangoubi. Aaron Smith. "Mixing of Hamiltonian Monte Carlo on strongly log-concave distributions: Continuous dynamics." Ann. Appl. Probab. 31 (5) 2019 - 2045, October 2021. https://doi.org/10.1214/20-AAP1640

Information

Received: 1 November 2019; Revised: 1 July 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

Digital Object Identifier: 10.1214/20-AAP1640

Subjects:
Primary: 60J05
Secondary: 60J20, 65C40, 68W20

Rights: Copyright © 2021 Institute of Mathematical Statistics

JOURNAL ARTICLE
27 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.31 • No. 5 • October 2021
Back to Top