(Non)penalized Multilevel methods for nonuniformly logconcave distributions
Abstract
We study and develop multilevel methods for the numerical approximation of a logconcave probability $\pi$ on $\mathbb{R}^d$, based on (overdamped) Langevin diffusion. In the continuity of \cite{art:egeapanloup2021multilevel} concentrated on the uniformly logconcave 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 $0$).
 Publication:

arXiv eprints
 Pub Date:
 January 2023
 DOI:
 10.48550/arXiv.2301.09471
 arXiv:
 arXiv:2301.09471
 Bibcode:
 2023arXiv230109471E
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Probability;
 65C05 (Primary) 65C40;
 37M25;
 93E35 (Secondary);
 G.1;
 G.3