{
  "abstract": "We study the complexity of finding $(δ,ε)$-Goldstein stationary points of nonsmooth nonconvex Lipschitz functions. By now, it is known that randomized first-order algorithms can solve this task with a dimension-free oracle complexity [Zhang et al., 2020], whereas deterministic algorithms cannot, as their complexity must scale at least linearly with the dimension $d$ [Jordan et al., 2023, Tian and So, 2024]. This leaves open whether deterministic algorithms can nevertheless solve the problem with oracle complexity polynomial in $d$. We answer this question negatively by proving a lower bound of order $(1/ε)^{Ω(d)}$ for deterministic algorithm, closing the exponential gap between the previously known lower and upper bounds and resolving an open problem posed by Jordan et al. [2023]. We further discuss several extensions and implications of this result to weaker stationarity notions, finding a descent direction and deterministic smoothing. Overall, our results establish an exponential computational advantage in nonsmooth nonconvex optimization offered by randomization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23837",
  "author_total": 1,
  "authors": [
    {
      "name": "Guy Kornowski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1hhzxfew1q582.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23837",
  "fulltext_url": null,
  "id": "1hhzxfew1q582",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23837",
  "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"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214033371",
  "pdf_url": "https://arxiv.org/pdf/2609.23837",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1hhzxfew1q582.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23837",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23837"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The Exponential Price of Determinism in Nonsmooth Nonconvex Optimization",
  "title_text": "The Exponential Price of Determinism in Nonsmooth Nonconvex Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/1hhzxfew1q582",
  "venue": "arXiv",
  "versions": []
}