{
  "abstract": "We establish the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in every fixed dimension $d\\geq1$. For $μ$-strongly convex, $L$-smooth potentials with unknown minimizers $x_f^\\star$ in the ball $\\mathbb{B}(0,μ^{-1/2})$, and unbiased gradient oracles with variance at most $σ^2$, the minimax worst-case expected query complexity is $Θ\\left(\\log(1+κ)+\\frac{σ^2}{μ\\varepsilon}\\right)$, jointly optimal for the condition number $κ:=L/μ$, variance $σ^2\\ge 0$, and TV accuracy $0<\\varepsilon\\leq 1/10$. In the noiseless setting $σ=0$, this tight complexity $Θ\\left(\\log(1+κ)\\right)$ is independent of $\\varepsilon$. Moreover, a gradient-only sampler generates an exact sample with ${O}(\\log(1+κ))$ worst-case expected queries. Without previous ball $\\mathbb{B}(0,μ^{-1/2})$, exact sampling from any initial point $x_0$ can be implemented with expected cost ${O}\\left(\\log(1+κ)+\\log(1+\\sqrtμ\\|x_0-x_f^\\star\\|)\\right)$, without knowing the initial distance. We also show that dependence on initial distance is generally unavoidable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12590",
  "author_total": 2,
  "authors": [
    {
      "name": "Weiming Ou"
    },
    {
      "name": "Xiao Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1dss4pb2j9wc1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12590",
  "fulltext_url": null,
  "id": "1dss4pb2j9wc1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12590",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7212933924",
  "pdf_url": "https://arxiv.org/pdf/2609.12590",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1dss4pb2j9wc1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.12590",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12590"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Tight Sampling Complexity with Stochastic Gradient Oracles in Fixed Dimensions",
  "title_text": "Tight Sampling Complexity with Stochastic Gradient Oracles in Fixed Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/1dss4pb2j9wc1",
  "venue": "arXiv",
  "versions": []
}