{
  "abstract": "We develop single-loop stochastic projected damped extragradient methods for nonconvex--(strongly) concave minimax optimization, with complexity guarantees for both game stationarity (GS) and optimization stationarity (OS). Our approach combines a stochastic projected damped extragradient (SPDE) method with a recursive variance-reduced variant, VR-SPDE, both of which retain a single-loop structure. Under an unbiased stochastic gradient oracle with uniformly bounded variance, SPDE finds an $\\varepsilon$-game-stationary point with stochastic first-order oracle (SFO) complexities of $O(κ\\varepsilon^{-4})$ and $O(\\varepsilon^{-5})$ in the nonconvex--strongly concave and nonconvex--concave settings, respectively, where $κ=L/μ$. Under an additional mean-square Lipschitz condition on the stochastic gradients, VR-SPDE improves these GS complexities to $O(κ^{3/2}\\varepsilon^{-3})$ and $O(\\varepsilon^{-9/2})$, respectively. For an $\\varepsilon$-optimization-stationary point, SPDE achieves SFO complexities of $O(κ\\varepsilon^{-4})$ and $O(\\varepsilon^{-6})$, while VR-SPDE achieves $O(κ^{3/2}\\varepsilon^{-3})$ and $O(\\varepsilon^{-6})$, in the two settings, respectively. These OS guarantees match the best-known bounds achieved by multi-loop methods while preserving a single-loop implementation. To the best of our knowledge, our results provide the best-known SFO complexity guarantees among single-loop stochastic first-order methods for the respective stationarity criteria and problem classes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21747",
  "author_total": 4,
  "authors": [
    {
      "name": "Huiling Zhang"
    },
    {
      "name": "Minhao Zhang"
    },
    {
      "name": "Zi Xu"
    },
    {
      "name": "Yuhong Dai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/49s0fh4steebg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21747",
  "fulltext_url": null,
  "id": "49s0fh4steebg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21747",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7213862717",
  "pdf_url": "https://arxiv.org/pdf/2609.21747",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/49s0fh4steebg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.21747",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21747"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Single-Loop Stochastic Projected Damped Extragradient Methods for Nonconvex--(Strongly) Concave Minimax Optimization",
  "title_text": "Single-Loop Stochastic Projected Damped Extragradient Methods for Nonconvex--(Strongly) Concave Minimax Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/49s0fh4steebg",
  "venue": "arXiv",
  "versions": []
}