{
  "abstract": "In this paper, we establish sharp lower bounds for Gaussian approximation of Polyak-Ruppert averaged stochastic gradient descent with polynomially decreasing step sizes. For every fixed exponent $γ\\in(1/2,1)$, a single one-dimensional smooth, strongly convex example has approximation error with the limiting variance of order $n^{1/2-γ}+n^{γ-1}$ in Kolmogorov and convex distances. This matches the existing upper bound throughout the range of $γ$. The nonlinear bias contributes $n^{1/2-γ}$ even when the Gaussian reference uses the finite linearized variance. We also derive a lower bound of order $n^{-(1-γ)/2}$ for the relative mean error of estimating the limiting variance from one SGD trajectory, uniformly over a scalar quadratic family. With the limiting variance known, achieving approximation error $\\varepsilon$ on this example takes order $\\varepsilon^{-4}$ iterations for the best choice $γ=3/4$. Achieving both Gaussian approximation and relative mean variance-estimation errors at most $\\varepsilon$ uniformly over the model class requires at least order $\\varepsilon^{-6}$ iterations for every fixed $γ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01155",
  "author_total": 3,
  "authors": [
    {
      "name": "Denis Belomestny"
    },
    {
      "name": "Alexey Naumov"
    },
    {
      "name": "Sergey Samsonov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8bwtg7saygssd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01155",
  "fulltext_url": null,
  "id": "8bwtg7saygssd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01155",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01155",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8bwtg7saygssd.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01155",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01155"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Lower Bounds on Inference Complexity for Polyak-Ruppert Averaged Stochastic Gradient Descent",
  "title_text": "Lower Bounds on Inference Complexity for Polyak-Ruppert Averaged Stochastic Gradient Descent",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/8bwtg7saygssd",
  "venue": "arXiv",
  "versions": []
}