{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/log-concave-sampling-metropolis-hastings","title":"Log-concave sampling: Metropolis-Hastings algorithms are fast","arxiv_id":"1801.02309","date":"2018-01-08","proceeding":null,"authors":["Raaz Dwivedi","Yuansi Chen","Martin J. Wainwright","Bin Yu"],"abstract":"We consider the problem of sampling from a strongly log-concave density in $\\mathbb{R}^d$, and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by simulating a Markov chain obtained from the discretization of an appropriate Langevin diffusion, combined with an accept-reject step. Relative to known guarantees for the unadjusted Langevin algorithm (ULA), our bounds show that the use of an accept-reject step in MALA leads to an exponentially improved dependence on the error-tolerance. Concretely, in order to obtain samples with TV error at most $\\delta$ for a density with condition number $\\kappa$, we show that MALA requires $\\mathcal{O} \\big(\\kappa d \\log(1/\\delta) \\big)$ steps, as compared to the $\\mathcal{O} \\big(\\kappa^2 d/\\delta^2 \\big)$ steps established in past work on ULA. We also demonstrate the gains of MALA over ULA for weakly log-concave densities. Furthermore, we derive mixing time bounds for the Metropolized random walk (MRW) and obtain $\\mathcal{O}(\\kappa)$ mixing time slower than MALA. We provide numerical examples that support our theoretical findings, and demonstrate the benefits of Metropolis-Hastings adjustment for Langevin-type sampling algorithms.","url_abs":"https://arxiv.org/abs/1801.02309v4","url_pdf":"https://arxiv.org/pdf/1801.02309v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"log-concave-sampling-metropolis-hastings","repo_url":"https://github.com/yuachen/mala_public","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1801.02309","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1801.02309"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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