{
  "abstract": "We characterize the optimal accuracy dependence for smooth nonconvex--strongly-convex bilevel optimization with stochastic first-order oracles. For globally unbiased fresh-gradient observations with bounded variance, we prove an $Ω(ε^{-6})$ lower bound matching the accuracy exponent of existing upper bounds. We consider globally Lipschitz gradients, a bounded upper gradient in the lower variable, and Lipschitz lower Hessian blocks, with target $\\mathbb{E}|\\nabla F(\\widehat{x})|\\leε$. Let $L$ be the common first-order scale, $μ$ the lower strong-convexity modulus, $ρ$ the lower Hessian-variation budget, $κ=L/μ$, and $χ=1+ρ/μ$. For $ρ\\gtrsimμ$, $χ\\lesssimκ$, sufficiently high accuracy, and sufficiently large dimension, we establish $Ω\\left(Lκ^2Δε^{-2}+L^3χ^2κ^8σ_g^2Δε^{-6}\\right)$, where $Δ$ is the initial gap budget and $σ_g^2$ is the lower-gradient variance budget. The bound holds over full Euclidean spaces against arbitrary randomized adaptive algorithms, even when every sample is the gradient of a scalar function. In the common-scale regime $ρ=Θ(L)$, the leading stochastic term has condition-number dependence $κ^{10}$, compared with the achievable $κ^{11}$ dependence. The proof embeds a sequential hard objective into an exact lower response while preserving global regularity and finite gap. A multiscale decomposition limits the gradient signal carrying each new direction, yielding the required sample complexity. We complement this lower bound with a frozen-linear-tilt estimator whose bias is proportional to lower Hessian variation. Its analysis makes this structural dependence explicit and yields matching leading rates in the specified curvature-controlled and small-gap regimes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03633",
  "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/2rf3q8461a3te.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03633",
  "fulltext_url": null,
  "id": "2rf3q8461a3te",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03633",
  "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": "W7220748465",
  "pdf_url": "https://arxiv.org/pdf/2610.03633",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2rf3q8461a3te.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.03633",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03633"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Closing the Accuracy Gap in Stochastic First-Order Bilevel Optimization",
  "title_text": "Closing the Accuracy Gap in Stochastic First-Order Bilevel Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/2rf3q8461a3te",
  "venue": "arXiv",
  "versions": []
}