arXiv subjects

Maxime Egéa

Publications and source records attributed to Maxime Egéa.

2 recordsLinked to original sources

arXiv2024

Multilevel-Langevin pathwise average for Gibbs approximation

We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution $π$ on $\mathbb{R}^d$, based on (overdamped) Langevin diffusions. This method inspired by \cite{mainPPlangevin} and \cite{giles_szpruch_invariant} relies on a multilevel occupation measure, $i.e.$ on an appropriate combination of $R$ occupation measures of (constant-step) Euler schemes with respective steps $γ_r = γ_0 2^{-r}$, $r=0,\ldots,R$. 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 ${\cal O}(d\varepsilon^{-2}\log^3(d\varepsilon^{-2}))$ 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 ${(\barλ_U\vee 1)^2}{\underlineλ_U^{-3}} d\varepsilon^{-2}$ (where $\barλ_U$ and $\underlineλ_U$ 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.

math.NA↗

arXiv2023

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

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 \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 $\varepsilon^{-5} π(|.|^2)^{3} d$ (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 $c_{ρ,δ}\varepsilon^{-2-ρ} d^{1+\fracρ{2}+(4-ρ+δ) r}$ for any $ρ>0$ (but with a constant $c_{ρ,δ}$ which increases when $ρ$ and $δ$ go to $0$).

math.NA↗