{
  "abstract": "We study the oracle complexity of finding $ε$-stationary points of smooth bilevel optimization problems with a nonconvex upper-level objective and a strongly convex lower-level problem. We consider a stochastic first-order oracle that returns unbiased stochastic gradients of both the upper- and lower-level objectives, with variance bounded by $σ^2$. We prove that, for any initial optimality gap $Δ>0$ and all sufficiently small $ε>0$, every adaptive randomized first-order algorithm requires $Ω\\!\\left(Δκ_y^2 ε^{-2}\\max\\{1,σ^2κ_y^6ε^{-4}\\}\\right)$ oracle queries to find an $ε$-stationary point of its hyper-objective function, where $κ_y$ denotes the condition number of the lower-level problem. In particular, in the noise-dominated regime, the lower bound is $Ω\\!\\left(Δσ^2κ_y^8ε^{-6}\\right)$. This establishes the optimality of the $ε^{-6}$ dependence achieved by the best-known first-order stochastic methods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21905",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhihao Gu"
    },
    {
      "name": "Qilong Wu"
    },
    {
      "name": "Junchi Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dhygwj5jvh2sk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21905",
  "fulltext_url": null,
  "id": "dhygwj5jvh2sk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21905",
  "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": "W7213919814",
  "pdf_url": "https://arxiv.org/pdf/2609.21905",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dhygwj5jvh2sk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21905",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21905"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "An $Ω(κ_y^8ε^{-6})$ Lower Bound for Stochastic NC-SC Bilevel Optimization with First-order Oracles",
  "title_text": "An Ω(κ_y^8ε^−6) Lower Bound for Stochastic NC-SC Bilevel Optimization with First-order Oracles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/dhygwj5jvh2sk",
  "venue": "arXiv",
  "versions": []
}