Multilevel Langevin Pathwise Average for Gibbs Approximation

Maxime Egéa () and Fabien Panloup ()
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Maxime Egéa: Université d’Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France
Fabien Panloup: Université d’Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France

Mathematics of Operations Research, 2025, vol. 50, issue 1, 573-605

Abstract: We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution π on R d , based on (overdamped) Langevin diffusions. This method relies on a multilevel occupation measure, that is, on an appropriate combination of R occupation measures of (constant-step) Euler schemes with respective steps γ r = γ 0 2 − r , r = 0 , … , R . We first state a quantitative result under general assumptions that guarantees an ε-approximation (in an L 2 -sense) with a cost of the order ε − 2 or ε − 2 | log ε | 3 under less contractive assumptions. We then apply it to overdamped Langevin diffusions with strongly convex potential U : R d → R and obtain an ε-complexity of the order O ( d ε − 2 log 3 ( d ε − 2 ) ) or O ( d ε − 2 ) under additional assumptions on U . More precisely, up to universal constants, an appropriate choice of the parameters leads to a cost controlled by ( λ ¯ U ∨ 1 ) 2 λ ¯ U − 3 d ε − 2 (where λ ¯ U and λ ¯ U respectively denote the supremum and the infimum of the largest and lowest eigenvalue of D 2 U ). We finally complete these theoretical results with some numerical illustrations, including comparisons to other algorithms in Bayesian learning and opening to the non–strongly convex setting.

Keywords: Primary: 65C05-37M25; secondary: 65C40-93E35; multilevel Monte Carlo; ergodic diffusion; Langevin algorithm (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:50:y:2025:i:1:p:573-605

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