{
  "abstract": "We study the sample complexity of finding a point with expected hypergradient norm at most $ε>0$ in nonconvex--strongly-convex stochastic bilevel optimization using fresh, globally unbiased first-order samples with bounded variance. For lower-variable smoothness order $p\\ge1$, F$^2$SA-$p$ achieves $\\widetilde O(pε^{-4-2/p})$ (Chen et al., 2026), but necessity of the exponent $4+2/p$ was open. We prove matching upper and lower bounds $Θ_p(ε^{-4-2/p})$ under fixed nondegenerate regularity budgets, with constants allowed to depend on $p$. The lower bound holds for unrestricted randomized algorithms under a fixed sample budget, allows dimension to grow with accuracy, and preserves global unbiasedness and all prescribed lower-variable smoothness bounds, even with a scalar lower variable and exact upper gradients. We further characterize the minimax fixed-budget complexity under $L_j=MΛ^j(j!)^β$ for $j\\ge3$: $Q_{p,β}^*(ε)=Θ\\left(ε^{-4}\\left[\\inf_{1\\le r\\le p,,r\\in\\mathbb N} r^βε^{-1/r}\\right]^2\\right)$, for $p\\in\\mathbb N\\cup{\\infty}$, with constants independent of $p$. At infinite order, this gives $Θ(ε^{-4})$ for $β=0$ and $Θ(ε^{-4}\\log^2(1/ε))$ for $β=1$. A gradient-difference method with weighted independent batches attains these bounds uniformly in order. Thus quantitative smoothness yields precise gains, whereas qualitative analyticity alone still permits $Θ(ε^{-6})$ complexity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03638",
  "author_total": 8,
  "authors": [
    {
      "name": "Wendao Wu"
    },
    {
      "name": "Haihan Zhang"
    },
    {
      "name": "Chenheng Zhang"
    },
    {
      "name": "Yanyi Li"
    },
    {
      "name": "Chunyuan Zheng"
    },
    {
      "name": "Cong Fang"
    },
    {
      "name": "Haoxuan Li"
    },
    {
      "name": "Zhouchen Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6s8k4vapnjyt0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03638",
  "fulltext_url": null,
  "id": "6s8k4vapnjyt0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03638",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7220418326",
  "pdf_url": "https://arxiv.org/pdf/2610.03638",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6s8k4vapnjyt0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.03638",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03638"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The Minimax Cost of Smoothness in Fully First-Order Stochastic Bilevel Optimization",
  "title_text": "The Minimax Cost of Smoothness in Fully First-Order Stochastic Bilevel Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/6s8k4vapnjyt0",
  "venue": "arXiv",
  "versions": []
}