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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.
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Bubeck, S.; Eldan, R.; Lehec, J. Sampling from a log-concave distribution with projected Langevin Monte Carlo. Discrete Comput. Geom. 59 (2018), no. 4, 757-783
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Chatterji, N.S., Diakonikolas, J., Jordan, M.I., Bartlett, P.L. Langevin Monte Carlo without smoothness. Preprint, arXiv: 1905.13285, 2019
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Durmus, A.; Moulines, E. High-dimensional Bayesian inference via the unadjusted Langevin algorithm. Bernoulli 25 (2019), no. 4A, 2854-2882
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Bakry, D.; Gentil, I.; Ledoux, M. Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 348. Springer, Cham, 2014
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Ding, Y. A note on quadratic transportation and divergence inequality, Statist. Probab. Lett. 100 (2015), 115-123
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Dalalyan, A.S. Theoretical guarantees for approximate sampling from smooth and log-concave densities. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 (2017), no. 3, 651-676
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Liu, Y. The Poincaré inequality and quadratic transportation-variance inequalities. Electron. J. Probab. 25 (2020), Paper no. 1, 16 p
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Chen, Y. An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture. Geom. Funct. Anal. 31 (2021),no. 1, 34-61
2021
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