{
  "abstract": "We study whether the linear condition-number dependence in the stochastic complexity of SAPD+ is necessary for nonconvex-strongly-concave minimax optimization. For jointly $L$-smooth objectives with dual strong-concavity parameter $μ$, we prove a lower bound that matches the SAPD+ upper bound under the same Moreau-envelope stationarity criterion and the same primal-dual initialization gap. Specifically, when $σ\\ge\\varepsilon$, the worst-case complexity of zero-respecting algorithms is $Θ(κLGσ^2\\varepsilon^{-4})$ in the stated accuracy regime, where $κ=L/μ$, $G$ bounds the initial primal-dual gap, and $σ^2$ bounds the variance of a general unbiased first-order oracle. The lower bound is realized on a smooth problem class with a bounded dual box. Our construction routes each link of a nonconvex zero-chain through a dual gradient of magnitude proportional to $\\varepsilon/\\sqrtκ$, while an undiscovered primal coordinate prevents stationarity. It also yields the primal-gradient lower bound $Ω(LΔ(\\sqrtκ\\varepsilon^{-2}+κσ^2\\varepsilon^{-4}))$ after combination with the known deterministic bound, where $Δ$ bounds the initial primal function gap.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30877",
  "author_total": 1,
  "authors": [
    {
      "name": "Qihao Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2rknkrnydmm43.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30877",
  "fulltext_url": null,
  "id": "2rknkrnydmm43",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30877",
  "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": "W7214629919",
  "pdf_url": "https://arxiv.org/pdf/2609.30877",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2rknkrnydmm43.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30877",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30877"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Tight Stochastic Condition-Number Dependence in Nonconvex-Strongly-Concave Minimax Optimization",
  "title_text": "Tight Stochastic Condition-Number Dependence in Nonconvex-Strongly-Concave Minimax Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/2rknkrnydmm43",
  "venue": "arXiv",
  "versions": []
}