{
  "abstract": "We study the problem of sampling from a target distribution $π(x)\\propto e^{-f(x)}$ supported on a convex set $ X\\subseteq\\mathbb R^d$, when the potential $f$ is accessible only through a stochastic first-order oracle. Mirror Langevin algorithms provide a natural approach to constrained sampling by transporting the problem to an unconstrained dual space and discretizing the resulting Mirror Langevin diffusion. Existing implementations, however, typically use a fixed discretization step size and consequently retain a nonvanishing discretization bias at any fixed step size. Moreover, their direct extension to settings with noisy gradient information entails the challenge of controlling both discretization and stochastic-oracle error. We study a stochastic first-order version of the Mirror Langevin Algorithm (sFO-MLA) and, as our main contribution, develop a warm-started two-loop implementation in which an outer loop progressively decreases the step size while an inner loop runs sFO-MLA (with a fixed step size) for an appropriately chosen epoch length. The construction provides a principled schedule linking step sizes and epoch lengths, so that successive epochs warm-start from increasingly accurate distributions rather than repeatedly paying the cost of mixing from a cold start. We establish finite-time Wasserstein guarantees for sFO-MLA that explicitly separate mixing, Euler--Maruyama discretization, and stochastic-gradient errors. These bounds yield a fixed horizon rate of $\\widetilde O(T^{-1/2})$ and show that the two-loop scheme removes the associated logarithmic penalty, attaining the canonical $O(T^{-1/2})$ rate under a geometric step-size schedule and corresponding epoch lengths. We illustrate the methodology in two statistically distinct settings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09402",
  "author_total": 3,
  "authors": [
    {
      "name": "Ruiting Tong"
    },
    {
      "name": "Antik Chakraborty"
    },
    {
      "name": "Raghu Pasupathy"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/atgsgj8yhymsv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09402",
  "fulltext_url": null,
  "id": "atgsgj8yhymsv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09402",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "stat.CO"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7220927328",
  "pdf_url": "https://arxiv.org/pdf/2610.09402",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/atgsgj8yhymsv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:48Z",
      "native_id": "2610.09402",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09402"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.CO",
      "subfield": "statistics"
    }
  ],
  "title": "Two-Loop Stochastic Mirror Langevin Algorithms for Constrained Sampling",
  "title_text": "Two-Loop Stochastic Mirror Langevin Algorithms for Constrained Sampling",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:48Z",
  "url": "https://research.draconitic.com/paper/atgsgj8yhymsv",
  "venue": "arXiv",
  "versions": []
}