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

Abstract : 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 Unajusted 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.
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https://hal.archives-ouvertes.fr/hal-03082311
Contributor : Fabien Panloup Connect in order to contact the contributor
Submitted on : Friday, December 18, 2020 - 3:30:08 PM
Last modification on : Tuesday, November 16, 2021 - 5:20:19 AM

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  • HAL Id : hal-03082311, version 1
  • ARXIV : 2012.14310

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Gilles Pages, Fabien Panloup. Unajusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds. 2020. ⟨hal-03082311⟩

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