{
  "abstract": "AdaGrad and AdaGrad-Norm are widely used adaptive methods, but their precise behavior in stochastic convex optimization remains less understood. We first show that AdaGrad-Norm and AdaGrad do not admit any universal \\textbf{last-iterate} rate, even under sub-Gaussian noise and bounded iterates. We then prove that bounded variance alone is too weak: even with bounded iterates, it cannot yield \\textbf{high-probability average-iterate} rates, and without bounded iterates it may not even guarantee convergence in expectation. Besides, we construct tight $Ω(\\log T/\\sqrt T)$ lower bounds for both AdaGrad-Norm and AdaGrad under sub-Gaussian noise, showing that $\\log T$ in existing average-iterate upper bounds is unavoidable. Finally, we show that this $\\log T$ loss disappears once bounded-iterate condition is imposed: under a general ABC condition, both methods achieve rates ${O}(1/\\sqrt{T})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32729",
  "author_total": 2,
  "authors": [
    {
      "name": "Weiming Ou"
    },
    {
      "name": "Xiao Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0gt29kc220jtr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32729",
  "fulltext_url": null,
  "id": "0gt29kc220jtr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32729",
  "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.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214748454",
  "pdf_url": "https://arxiv.org/pdf/2609.32729",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0gt29kc220jtr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32729",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32729"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Revisiting AdaGrad in Stochastic Convex Optimization: Last Iterates, High Probability, and Lower Bounds",
  "title_text": "Revisiting AdaGrad in Stochastic Convex Optimization: Last Iterates, High Probability, and Lower Bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/0gt29kc220jtr",
  "venue": "arXiv",
  "versions": []
}