Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds

Gilles Pages (1) , Fabien Panloup (2)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

In this paper, we focus on non-asymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (non-constant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step ({usually called Unadjusted Langevin Algorithm in the Monte Carlo literature}) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under {uniform} ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for Total Variation and $L^1$-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using {Stochastic Analysis} adapted to decreasing step setting.

Domaines: Statistiques [math.ST]; Probabilités [math.PR]

Gilles Pages, Fabien Panloup. Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds. 2022. ⟨hal-03082311v3⟩