{
  "abstract": "We prove that first-order algorithms require $Ω(δ^{-1}ε^{-3})$ gradient queries (in the worst case) to find a $(δ,ε)$-Goldstein stationary point of a Lipschitz function, at which there is a convex combination of gradients within distance $δ$ whose norm is at most $ε$. This lower bound is tight, matching known algorithms up to absolute constants, therefore resolving the complexity of convergence to stationarity in nonsmooth nonconvex optimization. We further prove a tight lower bound of $Ω(λ^{1/2}ε^{-7/2})$ for finding points satisfying the recently proposed relaxed notion of $(λ,ε)$-stationarity, which allows combining further-away gradients. Our results reveal that convergence rates to nonsmooth stationarity are not affected by gradient stochasticity, in sharp contrast to smooth optimization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17780",
  "author_total": 1,
  "authors": [
    {
      "name": "Guy Kornowski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/btrmzmq83fqz9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17780",
  "fulltext_url": null,
  "id": "btrmzmq83fqz9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17780",
  "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": "W7213523973",
  "pdf_url": "https://arxiv.org/pdf/2609.17780",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/btrmzmq83fqz9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17780",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17780"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The Complexity of Finding Stationary Points in Nonsmooth Nonconvex Optimization",
  "title_text": "The Complexity of Finding Stationary Points in Nonsmooth Nonconvex Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/btrmzmq83fqz9",
  "venue": "arXiv",
  "versions": []
}