Multilevel-Langevin pathwise average for Gibbs approximation

We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution $\pi$ on $\mathbb{R}^d$, based on (overdamped) Langevin diffusions. This method inspired by \cite{mainPPlangevin} and \cite{gilesszpruchinvariant} relies on a multilevel occupation measure, $i.e.$ on an appropriate combination of $R$ occupation measures of (constant-step) Euler schemes with respective steps $\gammar = \gamma0 2^{-r}$, $r=0,\ldots,R$. We first state a quantitative result under general assumptions which guarantees an \textit{$\varepsilon$-approximation} (in a $L^2$-sense) with a cost of the order $\varepsilon^{-2}$ or $\varepsilon^{-2}|\log \varepsilon|^3$ under less contractive assumptions. We then apply it to overdamped Langevin diffusions with strongly convex potential $U:\mathbb{R}^{d\rightarrow\mathbb{R}$} and obtain an \textit{$\varepsilon$-complexity} of the order ${\cal O}(d\varepsilon^{{-2}\log3(d\varepsilon{-2}))$} or ${\cal O}(d\varepsilon^{-2})$ under additional assumptions on $U$. More precisely, up to universal constants, an appropriate choice of the parameters leads to a cost controlled by ${(\bar{\lambda}U\vee 1)^{2}{\underline{\lambda}}U^{-3}} d\varepsilon^{-2}$ (where $\bar{\lambda}U$ and $\underline{\lambda}U$ respectively denote the supremum and the infimum of the largest and lowest eigenvalue of $D^2U$). We finally complete these theoretical results with some numerical illustrations including comparisons to other algorithms in Bayesian learning and opening to non strongly convex setting.