(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 π on ${\mathbb{R}}^d$, based on (over-damped) Langevin diffusion. In the continuity of [12] 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 [8] by adding a penalization term to the potential to recover the uniformly convex setting. Such approach leads to an ε-complexity of the order ${\mathit{\epsilon }}^{-5}\mathrm{\pi }{(|.{|}^2)}^{9∕2}d$ (up to logarithmic terms). Then, in the spirit of [14], 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 ε-complexity of the order $c_{\mathit{\rho },\mathit{\delta }}{\mathit{\epsilon }}^{-2-\mathit{\rho }}{d}^{1+\frac{\mathit{\rho }}{2}+(4-\mathit{\rho }+\mathit{\delta })r}$ for any $\mathit{\rho }>0$ (but with a constant $c_{\mathit{\rho },\mathit{\delta }}$ which increases when ρ and δ go to 0).