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Nonasymptotic bounds for sampling algorithms without log-concavity
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Majka, Mateusz B., Mijatović, Aleksandar and Szpruch, Lukasz (2020) Nonasymptotic bounds for sampling algorithms without log-concavity. The Annals of Applied Probability, 30 (4). pp. 1534-1581. doi:10.1214/19-AAP1535 ISSN 1050-5164.
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Official URL: https://doi.org/10.1214/19-AAP1535
Abstract
Discrete time analogues of ergodic stochastic differential equations (SDEs) are one of the most popular and flexible tools for sampling high-dimensional probability measures. Non-asymptotic analysis in the L2 Wasserstein distance of sampling algorithms based on Euler discretisations of SDEs has been recently developed by several authors for log-concave probability distributions. In this work we replace the log-concavity assumption with a log-concavity at infinity condition. We provide novel L2 convergence rates for Euler schemes, expressed explicitly in terms of problem parameters. From there we derive nonasymptotic bounds on the distance between the laws induced by Euler schemes and the invariant laws of SDEs, both for schemes with standard and with randomised (inaccurate) drifts. We also obtain bounds for the hierarchy of discretisation, which enables us to deploy a multi-level Monte Carlo estimator. Our proof relies on a novel construction of a coupling for the Markov chains that can be used to control both the L1 and L2 Wasserstein distances simultaneously. Finally, we provide a weak convergence analysis that covers both the standard and the randomised (inaccurate) drift case. In particular, we reveal that the variance of the randomised drift does not influence the rate of weak convergence of the Euler scheme to the SDE.
Item Type: | Journal Article | ||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||
Journal or Publication Title: | The Annals of Applied Probability | ||||||
Publisher: | Institute of Mathematical Statistics | ||||||
ISSN: | 1050-5164 | ||||||
Official Date: | 4 August 2020 | ||||||
Dates: |
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Volume: | 30 | ||||||
Number: | 4 | ||||||
Page Range: | pp. 1534-1581 | ||||||
DOI: | 10.1214/19-AAP1535 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 7 November 2019 | ||||||
Date of first compliant Open Access: | 23 February 2021 | ||||||
RIOXX Funder/Project Grant: |
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