{"paper":{"title":"A Simple Proof of the Mixing of Metropolis-Adjusted Langevin Algorithm under Smoothness and Isoperimetry","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CC","cs.LG","stat.CO"],"primary_cat":"stat.ML","authors_text":"Khashayar Gatmiry, Yuansi Chen","submitted_at":"2023-04-08T20:17:29Z","abstract_excerpt":"We study the mixing time of Metropolis-Adjusted Langevin algorithm (MALA) for sampling a target density on $\\mathbb{R}^d$. We assume that the target density satisfies $\\psi_\\mu$-isoperimetry and that the operator norm and trace of its Hessian are bounded by $L$ and $\\Upsilon$ respectively. Our main result establishes that, from a warm start, to achieve $\\epsilon$-total variation distance to the target density, MALA mixes in $O\\left(\\frac{(L\\Upsilon)^{\\frac12}}{\\psi_\\mu^2} \\log\\left(\\frac{1}{\\epsilon}\\right)\\right)$ iterations. Notably, this result holds beyond the log-concave sampling setting "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.04095","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2304.04095/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}