(Non)-penalized Multilevel methods for non-uniformly log-concave distributions

Abstract

We study and develop multilevel methods for the numerical approximation of a log-concave probability $\pi$ on $\mathbb{R}^d$, based on (over-damped) Langevin diffusion. In the continuity of \cite{art:egeapanloup2021multilevel} concentrated on the uniformly log-concave setting, we here study the procedure in the absence of the uniformity assumption. More precisely, we first adapt an idea of \cite{art:DalalyanRiouKaragulyan} by adding a penalization term to the potential to recover the uniformly convex setting. Such approach leads to an \textit{$\varepsilon$-complexity} of the order $\varepsilon^{-5} \pi(|.|^2)^{3} d$ (up to logarithmic terms). Then, in the spirit of \cite{art:gadat2020cost}, we propose to explore the robustness of the method in a weakly convex parametric setting where the lowest eigenvalue of the Hessian of the potential $U$ is controlled by the function $U(x)^{-r}$ for $r \in (0,1)$. In this intermediary framework between the strongly convex setting ($r=0$) and the ``Laplace case'' ($r=1$), we show that with the help of the control of exponential moments of the Euler scheme, we can adapt some fundamental properties for the efficiency of the method. In the ``best'' setting where $U$ is ${\mathcal{C}}^3$ and $U(x)^{-r}$ control the largest eigenvalue of the Hessian, we obtain an $\varepsilon$-complexity of the order $c_{\rho,\delta}\varepsilon^{-2-\rho} d^{1+\frac{\rho}{2}+(4-\rho+\delta) r}$ for any $\rho>0$ (but with a constant $c_{\rho,\delta}$ which increases when $\rho$ and $\delta$ go to $0$).