{
  "abstract": "Let $π(\\mathrm{d} x)\\propto e^{-U(x)}\\, \\mathrm{d} x$ on $\\mathbb{R}^d$, where $U$ is continuously differentiable and $m$-strongly convex with a globally $L$-Lipschitz gradient, $0<m\\leq L<\\infty$, and $κ=L/m$. Fixed-step Metropolis-adjusted Langevin algorithm (MALA) has known warm-start mixing-time upper bounds of order $κ\\sqrt d$, up to logarithmic factors. A spectral-gap lower bound at the corresponding scale $(κ\\sqrt{d})^{-1}$ would give an upper bound of order $κ\\sqrt{d}$ on the Monte Carlo asymptotic variance relative to independent sampling, uniformly over all square-integrable functions. However, when $κ$ is bounded away from~1, the best fixed-step spectral gap that can be guaranteed uniformly over this target class is at most of order $\\max\\{\\log(κd)/(κd), e^{-cd}\\}$ for some positive universal constant~$c$. We show that uniform randomization of the step size improves this worst-case guarantee. At each iteration, the algorithm draws $h\\sim\\operatorname{Unif}(0,H)$ and performs one ordinary MALA transition. Choosing $H$ of order $[L\\sqrt{d(1+\\log d+\\logκ)}]^{-1}$ yields a right spectral-gap lower bound of order \\[ \\frac{1}{κ\\sqrt{d(1+\\log d+\\logκ)}}, \\] uniformly over the target class. Thus, given $κ> 1$, for all square-integrable functions, the ratio of the Monte Carlo asymptotic variance relative to independence sampling has an upper bound of order $\\sqrt{d \\log d}$. This contrasts with fixed-MALA, where the ratio can be as bad as $d/\\log d$ in thew worst-case scenario. The spectral gap also gives geometric convergence of the lazy kernel from every initial density in $L^2(π)$, central limit theorems, and nonstationary mean-square error bounds. This work was developed with substantial assistance from ChatGPT.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05847",
  "author_total": 1,
  "authors": [
    {
      "name": "Qian Qin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/52rgf79zvp9dc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05847",
  "fulltext_url": null,
  "id": "52rgf79zvp9dc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05847",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7211992556",
  "pdf_url": "https://arxiv.org/pdf/2609.05847",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/52rgf79zvp9dc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.05847",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05847"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A spectral gap for Metropolis-adjusted Langevin algorithm with a uniformly randomized step size",
  "title_text": "A spectral gap for Metropolis-adjusted Langevin algorithm with a uniformly randomized step size",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:41Z",
  "url": "https://research.draconitic.com/paper/52rgf79zvp9dc",
  "venue": "arXiv",
  "versions": []
}