User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient

Arnak Dalalyan and Avetik Karagulyan (avetik.karagulyan@ensae-paristech.fr)
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Avetik Karagulyan: CREST;ENSAE

No 2017-20, Working Papers from Center for Research in Economics and Statistics

Abstract: In this paper, we revisit the recently established theoretical guarantees for the convergence of the Langevin Monte Carlo algorithm of sampling from a smooth and (strongly) log-concave density. We improve, in terms of constants, the existing results when the accuracy of sampling is measured in the Wasserstein distance and provide further insights on relations between, on the one hand, the Langevin Monte Carlo for sampling and, on the other hand, the gradient descent for optimization. More importantly, we establish non-asymptotic guarantees for the accuracy of a version of the Langevin Monte Carlo algorithm that is based on inaccurate evaluations of the gradient. Finally, we propose a variable-step version of the Langevin Monte Carlo algorithm that has two advantages. First, its step-sizes are independent of the target accuracy and, second, its rate provides a logarithmic improvement over the constant-step Langevin Monte Carlo algorithm ;Classification-JEL: Primary 62J05; secondary 62H12

Keywords: Markov Chain Monte Carlo; Approximate sampling; Rates of convergence; Langevin algorithm; Gradient descent (search for similar items in EconPapers)
Pages: 30 pages
Date: 2017-11-06
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Citations: View citations in EconPapers (2)

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http://crest.science/RePEc/wpstorage/2017-20.pdf CREST working paper version (application/pdf)

Related works:
Journal Article: User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient (2019)
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