{
  "abstract": "Recent advances in penalty-based methods for stochastic bilevel optimization (SBO) have eliminated the need for second-order derivative oracles. However, for stochastic nonconvex-strongly convex bilevel problems, existing first-order methods typically rely on nested loops and/or large batch sizes for attaining $O(ε^{-6})$ or $O(ε^{-4})$ sample complexity under standard bounded-variance assumption or mean-square smoothness assumption. Achieving these rates with a single-loop penalty method and a constant batch size remains challenging due to a large penalty value needed for an accurate approximation. To address this challenge, we develop a stochastic SIngle-loop COnstant-Batch first-order penalty method (SICO) that combines two complementary ingredients. First, it performs one stochastic-gradient update per-iteration for both the original lower-level and penalized problems, with a projection that controls the separation between their iterates. Second, it applies an exponential moving average to stabilize the upper-level gradient estimator. We show that this combination achieves $ O(ε^{-6}) $ sample complexity using only $O(1)$ stochastic-gradient samples per iteration under unbiased, bounded-variance stochastic gradients. Under the additional mean-square smoothness assumption on the lower-level stochastic gradients, the same algorithm improves the complexity to $O(ε^{-4})$ also with $O(1)$ batch size. To the best of our knowledge, this is the first work to match the best-known convergence rate for fully first-order SBO methods using a single loop and a constant batch size. This result addresses an open problem posed in the literature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07290",
  "author_total": 4,
  "authors": [
    {
      "name": "Xingyu Chen"
    },
    {
      "name": "Ming Yang"
    },
    {
      "name": "Quanqi Hu"
    },
    {
      "name": "Tianbao Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a226r237qwk91.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07290",
  "fulltext_url": null,
  "id": "a226r237qwk91",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07290",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7220909258",
  "pdf_url": "https://arxiv.org/pdf/2610.07290",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a226r237qwk91.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07290",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07290"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Single-Loop, Constant-Batch First-Order Penalty Method for Stochastic Bilevel Optimization",
  "title_text": "A Single-Loop, Constant-Batch First-Order Penalty Method for Stochastic Bilevel Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/a226r237qwk91",
  "venue": "arXiv",
  "versions": []
}