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The Langevin Monte Carlo algorithm in the non-smooth log-concave case
Joseph Lehec · 2021-01-26 · via math.ST updates on arXiv.org

We prove non asymptotic 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.

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