Multilevel Monte Carlo methods for the approximation of invariant measures of stochastic differential equations
This work addresses computational efficiency in statistical inference for large datasets, offering a novel method that is incremental but provides specific gains over current stochastic gradient MCMC techniques.
The paper tackles the problem of approximating invariant measures of stochastic differential equations using multilevel Monte Carlo methods, achieving a root mean square error of O(ε) with O(ε^{-2}) complexity, matching MCMC methods but with improved efficiency for large datasets, and introduces a stochastic gradient version with O(ε^{-2}|log ε|^{3}) complexity, outperforming existing O(ε^{-3}) methods.
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.