Abstract
We prove nonasymptotic polynomial bounds on the convergence of the Langevin Monte Carlo algorithm in the case where the potential is a convex function which is globally Lipschitz on its domain, typically the maximum of a finite number of affine functions on an arbitrary convex set. In particular the potential is not assumed to be gradient Lipschitz, in contrast with most existing works on the topic.
Acknowledgments
The author is grateful to Sébastien Bubeck and Ronen Eldan for a number of useful discussions related to this work. We are also grateful to Andre Wibisono who brought to our attention the /chi-square inequality used in the proof of Theorem 3. In the first version of the paper the formulation of that theorem was slightly weaker, with in place of . Last, we would like to thank the anonymous referee for his or her careful reading of the manuscript and accurate comments.
Citation
Joseph Lehec. "The Langevin Monte Carlo algorithm in the non-smooth log-concave case." Ann. Appl. Probab. 33 (6A) 4858 - 4874, December 2023. https://doi.org/10.1214/23-AAP1935
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