{
  "abstract": "We study lower bounds on the first-order oracle complexity of smooth nonconvex-concave minimax optimization. We consider objectives $f$ that are jointly $L$-smooth in the primal and dual variables $(x,y)$, concave in $y$, and whose primal value function $Φ(x) := \\max_{y\\in\\mathcal Y} f(x,y)$ satisfies the initial-gap condition $Φ(0)-\\inf_{x\\in\\mathcal X}Φ(x)\\le Δ_Φ$, with a bounded dual domain satisfying $\\operatorname{diam}(\\mathcal Y)\\le D_{\\mathcal Y}$. We measure stationarity by the norm of the gradient of the Moreau envelope of $Φ+ι_{\\mathcal X}$ with parameter $1/(2L)$. We prove that any deterministic zero-respecting first-order algorithm requires $Ω\\left(L^2D_{\\mathcal Y}Δ_Φε^{-3}\\right)$ oracle evaluations to find an $ε$-stationary point. Under an unbiased stochastic first-order oracle with bounded variance, any stochastic zero-respecting algorithm requires $Ω\\left(L^3D_{\\mathcal Y}^2Δ_Φε^{-6}\\right)$ oracle evaluations. The same lower bounds hold when $Δ_Φ$ is replaced by the initial primal-dual gap $\\mathcal G_0$. These deterministic and stochastic lower bounds match the corresponding upper bounds of [14] and [29], respectively, up to a logarithmic factor in the deterministic setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14233",
  "author_total": 3,
  "authors": [
    {
      "name": "Qilong Wu"
    },
    {
      "name": "Zhihao Gu"
    },
    {
      "name": "Junchi Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dcc161fqpv0hg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14233",
  "fulltext_url": null,
  "id": "dcc161fqpv0hg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14233",
  "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",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.14233",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dcc161fqpv0hg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14233",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14233"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Lower Bounds for Nonconvex-Concave Minimax Optimization",
  "title_text": "Lower Bounds for Nonconvex-Concave Minimax Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/dcc161fqpv0hg",
  "venue": "arXiv",
  "versions": []
}