Multi-level Monte Carlo methods for the approximation of invariant measures of stochastic differential equations

Michael B. Giles, Mateusz B. Majka, Lukasz Szpruch, Sebastian J. Vollmer, Konstantinos C. Zygalakis

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Abstract

We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles in Acta Numer. 24:259–328, 2015. https://doi.org/10.1017/S096249291500001X) to calculate expectations with respect to the invariant measure of an ergodic SDE. In that context, we study the (over-damped) Langevin equations with a strongly concave potential. We show that when appropriate contracting couplings for the numerical integrators are available, one can obtain a uniform-in-time estimate of the MLMC variance in contrast to the majority of the results in the MLMC literature. As a consequence, a root mean square error of O(ε) is achieved with O(ε- 2) complexity on par with Markov Chain Monte Carlo (MCMC) methods, which, however, can be computationally intensive when applied to large datasets. Finally, we present a multi-level version of the recently introduced stochastic gradient Langevin dynamics method (Welling and Teh, in: Proceedings of the 28th ICML, 2011) built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity O(ε- 2| log ε| 3) , in contrast to the complexity O(ε- 3) of currently available methods. Numerical experiments confirm our theoretical findings.

Original languageEnglish
Pages (from-to)507-524
Number of pages18
JournalStatistics and Computing
Volume30
Issue number3
Early online date10 Sep 2019
DOIs
Publication statusPublished - May 2020

Keywords

  • Monte Carlo methods
  • Numerical analysis
  • Stochastic Gradient methods

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Computational Theory and Mathematics

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