Multilevel Monte Carlo methods for the approximation of invariant measures of stochastic differential equations

TL;DR

This paper introduces a multilevel Monte Carlo framework for efficiently approximating invariant measures of ergodic SDEs, achieving lower complexity than traditional methods, and extends it to stochastic gradient Langevin dynamics for large datasets.

Contribution

It develops a novel MLMC approach for invariant measure estimation of ergodic SDEs and introduces a multi-level SGLD method with improved complexity for large data applications.

Findings

Achieves $ ext{O}( ext{ε}^{-2})$ complexity for invariant measure approximation.

First stochastic gradient MCMC method with $ ext{O}( ext{ε}^{-2}| ext{log} ext{ε}|^{3})$ complexity.

Numerical experiments confirm theoretical complexity improvements.

Abstract

We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles2015) 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 $\mathcal{O}(\varepsilon)$ is achieved with $\mathcal{O}(\varepsilon^{-2})$ complexity on par with Markov Chain Monte Carlo (MCMC) methods, which however can be computationally intensive when applied to large data sets. Finally, we present a multi-level version of the recently introduced Stochastic Gradient Langevin Dynamics (SGLD) method (Welling and Teh, 2011) built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity $\mathcal{O}(\varepsilon^{-2}|\log {\varepsilon}|^{3})$, in contrast to the complexity $\mathcal{O}(\varepsilon^{-3})$ of currently available methods. Numerical experiments confirm our theoretical findings.