{
  "abstract": "We introduce a new single-loop algorithmic framework for smooth nonconvex--concave minimax optimization. The resulting projected damped extragradient method combines projected extragradient updates, dual momentum, and a moving proximal center. Under both the optimization-stationarity and game-stationarity criteria, our method achieves the best-known complexity among single-loop first-order methods. For optimization stationarity, our method achieves a gradient complexity of $O(L^2D_Y\\barΔ_0\\varepsilon^{-3})$, where $L$ is the gradient Lipschitz constant, $D_Y$ bounds the diameter of the dual feasible set, and $\\barΔ_0$ is an initialization quantity involving the value-function gap and the initial gradients. Moreover, by incorporating a fixed-center warm-up phase, the complexity can be improved to $O(L^2D_YΔ_ϕ\\varepsilon^{-3})$, up to an additive lower-order cost, where $Δ_ϕ:=ϕ(x_0)-\\inf_xϕ(x)$. We further establish a lower bound of $Ω(L^2D_YΔ_ϕ\\varepsilon^{-3})$ for optimization stationarity over projected zero-respecting first-order methods. This lower bound proves that the warm-started version of our algorithm is optimal up to a constant factor for optimization stationarity within this oracle class. For game stationarity, our method achieves $\\mathcal{O}\\!(L^{3/2}D_Y^{1/2}Δ_ϕ\\varepsilon^{-5/2})$ gradient complexity. This matches the best-known complexity of multi-loop first-order methods, thereby establishing the same complexity with a single-loop algorithmic structure. Under dual strong concavity, the proposed framework achieves $O\\!(\\sqrtκ\\,LΔ_ϕ\\varepsilon^{-2})$ leading complexity for both stationarity criteria, where $κ=L/μ$ is the dual condition number, up to an additive initialization cost. The $\\varepsilon^{-2}$ accuracy dependence is optimal under fixed regularity and initialization bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17973",
  "author_total": 2,
  "authors": [
    {
      "name": "Minghao Zhang"
    },
    {
      "name": "Zi Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/23z8gy4hbffxz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17973",
  "fulltext_url": null,
  "id": "23z8gy4hbffxz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17973",
  "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": "W7213530129",
  "pdf_url": "https://arxiv.org/pdf/2609.17973",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/23z8gy4hbffxz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17973",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17973"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Matching Multi-Loop Complexities with a Single Loop: Optimal Optimization Stationarity and Best-Known Game Stationarity in Nonconvex--Concave Minimax Optimization",
  "title_text": "Matching Multi-Loop Complexities with a Single Loop: Optimal Optimization Stationarity and Best-Known Game Stationarity in Nonconvex--Concave Minimax Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:10Z",
  "url": "https://research.draconitic.com/paper/23z8gy4hbffxz",
  "venue": "arXiv",
  "versions": []
}