| Issue |
ESAIM: M2AN
Volume 58, Number 6, November-December 2024
Special issue - To commemorate Assyr Abdulle
|
|
|---|---|---|
| Page(s) | 2255 - 2286 | |
| DOI | https://doi.org/10.1051/m2an/2024038 | |
| Published online | 04 December 2024 | |
Contraction rate estimates of stochastic gradient kinetic Langevin integrators*
School of Mathematics, University of Edinburgh, Edinburgh EH9 2NX, Scotland
** Corresponding author: p.a.whalley@sms.ed.ac.uk
Received:
14
June
2023
Accepted:
16
May
2024
In previous work, we introduced a method for determining convergence rates for integration methods for the kinetic Langevin equation for M-▽Lipschitz m-log-concave densities [Leimkuhler et al., SIAM J. Numer. Anal. 62 (2024) 1226–1258]. In this article, we exploit this method to treat several additional schemes including the method of Brunger, Brooks and Karplus (BBK) and stochastic position/velocity Verlet. We introduce a randomized midpoint scheme for kinetic Langevin dynamics, inspired by the recent scheme of Bou-Rabee and Marsden [arXiv:2211.11003, 2022]. We also extend our approach to stochastic gradient variants of these schemes under minimal extra assumptions. We provide convergence rates of O(m/M), with explicit stepsize restriction, which are of the same order as the stability thresholds for Gaussian targets and are valid for a large interval of the friction parameter. We compare the contraction rate estimates of many kinetic Langevin integrators from molecular dynamics and machine learning. Finally, we present numerical experiments for a Bayesian logistic regression example.
Mathematics Subject Classification: 65C05 / 65C30 / 65C40
Key words: Stochastic gradient / contractive numerical method / Wasserstein convergence / kinetic Langevin dynamics / underdamped Langevin dynamics / MCMC sampling / Brunger–Brooks–Karplus / stochastic Verlet / Bayesian logistic regression / MNIST classification
© The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.