{
  "abstract": "Low-rank adaptation (LoRA) optimizes $J(B,A)=\\mathcal L(W_\\mathrm{base}+sBA)$ over two adapters $B \\in \\mathbb{R}^{m \\times r}$ and $A \\in \\mathbb{R}^{r \\times n}$ that form a low-rank update to a frozen pretrained weight matrix $W_\\mathrm{base} \\in \\mathbb{R}^{m \\times n}$. The prior analysis shows LoRA-GD takes $\\exp\\{\\mathcal{O}(ε^{-2})\\}$ oracle calls to find an $ε$-stationary point such that $\\|\\nabla J(B,A)\\|\\leq ε$ in the deterministic setting. We sharpen the analysis and show that $\\mathcal{O}(ε^{-4})$ full-gradient evaluations suffice for the same first-order criterion. We further study stochastic LoRA under unbiased gradient estimates and finite variance. We propose LoRA-NSGDM, which finds an $ε$-stationary point with $\\mathcal{O}(ε^{-8})$ stochastic oracle complexity. Under the additional mean-square smoothness condition, we use variance reduction strategy and propose LoRA-STORM, which improves the stochastic oracle complexity to $\\mathcal{O}(ε^{-6})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.21975",
  "author_total": 3,
  "authors": [
    {
      "name": "Ru Wang"
    },
    {
      "name": "Chengchang Liu"
    },
    {
      "name": "John C. S. Lui"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ckbsw99wmkzm8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.21975",
  "fulltext_url": null,
  "id": "ckbsw99wmkzm8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.21975",
  "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": "W7171416351",
  "pdf_url": "https://arxiv.org/pdf/2607.21975",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ckbsw99wmkzm8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2607.21975",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.21975"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "On the Convergence of Stochastic Low-Rank Adaptation",
  "title_text": "On the Convergence of Stochastic Low-Rank Adaptation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/ckbsw99wmkzm8",
  "venue": "arXiv",
  "versions": []
}