{
  "abstract": "We prove a joint stochastic zeroth-order lower bound for Goldstein stationarity on a Euclidean query ball, even when the ball is guaranteed to contain a stationary point. In dimension $d$, let $f=\\mathbb{E}[F(\\cdot;ξ)]$, assume $\\mathbb{E}[\\operatorname{Lip}(F(\\cdot;ξ))^2]\\le L_0^2$, and bound the initial objective gap over the ball by $Δ$. For neighborhood radius $δ>0$ and residual tolerance $\\varepsilon>0$, our smooth hard family requires $Ω(dL_0^2Δ/(δ\\varepsilon^3))$ scalar evaluations for success probability $1/2$, against randomized adaptive algorithms that may retain and repeatedly query each sampled function. The result holds for $\\varepsilon\\le cL_0$, $Δ\\ge Cδ\\varepsilon$, and $d\\ge C[1+\\log(2+ΔL_0^2/(δ\\varepsilon^3))]$, on a constructed ball of radius $Θ(Δ/\\varepsilon)$. A sequence of localized regions forces repeated direction estimation, and an adaptive Gaussian posterior argument controls sample reuse. The result establishes a joint dimension and accuracy obstruction on solvable bounded-domain instances. The gap is local to the query ball; a matching minimax characterization at a common radius and the corresponding unrestricted global-gap lower bound remain open here.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00275",
  "author_total": 8,
  "authors": [
    {
      "name": "Haihan Zhang"
    },
    {
      "name": "Wendao Wu"
    },
    {
      "name": "Chenheng Zhang"
    },
    {
      "name": "Yanyi Li"
    },
    {
      "name": "Chunyuan Zheng"
    },
    {
      "name": "Cong Fang"
    },
    {
      "name": "Haoxuan Li"
    },
    {
      "name": "Zhouchen Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3dp8dxt7khtx5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00275",
  "fulltext_url": null,
  "id": "3dp8dxt7khtx5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00275",
  "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",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7219155339",
  "pdf_url": "https://arxiv.org/pdf/2610.00275",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3dp8dxt7khtx5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.00275",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00275"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Joint Lower Bounds for Zeroth-Order Nonconvex Optimization on Euclidean Balls",
  "title_text": "Joint Lower Bounds for Zeroth-Order Nonconvex Optimization on Euclidean Balls",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/3dp8dxt7khtx5",
  "venue": "arXiv",
  "versions": []
}