2023/01/23 by Maxime Egéa, Egéa, Maxime
Mathematics · Medicine · #37M25 #65C05 (Primary) 65C40 #93E35 (Secondary) #Advanced Neuroimaging Techniques and Applications #FOS: Mathematics #G.1 #G.3 #Markov Chains and Monte Carlo Methods #Numerical Analysis (math.NA) #Probability (math.PR) #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.2301.09471
openalex publication_date 2023/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study and develop multilevel methods for the numerical approximation of a log-concave probability π on ℝd, based on (over-damped) Langevin diffusion. In the continuity of \citeart: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 \citeart:DalalyanRiouKaragulyan by adding a penalization term to the potential to recover the uniformly convex setting. Such approach leads to an ε-complexity of the order ε-5 π(|.|2)3 d (up to logarithmic terms). Then, in the spirit of \citeart: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 ∈ (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 C3 and U(x)-r control the largest eigenvalue of the Hessian, we obtain an ε-complexity of the order cρ,δε-2-ρ d^1+\fracρ2+(4-ρ+δ) r for any ρ>0 (but with a constant cρ,δ which increases when ρ and δ go to 0).