{
  "abstract": "Two-timescale stochastic approximation (TTSA) is a fundamental tool for analyzing coupled iterative algorithms in reinforcement learning, optimization, and stochastic control. However, finite-time guarantees for nonlinear two-timescale schemes remain difficult to obtain, especially under constant step-sizes. In this paper, we study nonlinear TTSA with step-sizes $α\\ggβ$. Under standard stability, regularity, and Markovian noise assumptions, we upper bound the mean-squared error and the bias of both iterates around their limiting equilibria. Our bounds scale as $O(α+β^2/α^2)$, which we prove to be tight when $β\\leα^{3/2}$. The analysis separates the contributions of initial conditions, fast-timescale tracking error, Markovian dependence, and timescale coupling, thereby clarifying the origin of the $β^2/α^2$ term. Our results reveal qualitative differences from the linear TTSA setting previously studied, showing that nonlinear dynamics introduce additional finite-time effects that are absent in the linear case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20409",
  "author_total": 3,
  "authors": [
    {
      "name": "Djamel Rassem Lamouri"
    },
    {
      "name": "Dorian Baudry"
    },
    {
      "name": "Nicolas Gast"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4gaz5dpgzk5hz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20409",
  "fulltext_url": null,
  "id": "4gaz5dpgzk5hz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20409",
  "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"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7213609383",
  "pdf_url": "https://arxiv.org/pdf/2609.20409",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4gaz5dpgzk5hz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20409",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20409"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "The Bias of Nonlinear Two-Time-scale Stochastic Approximation under Constant Step-Sizes",
  "title_text": "The Bias of Nonlinear Two-Time-scale Stochastic Approximation under Constant Step-Sizes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:29Z",
  "url": "https://research.draconitic.com/paper/4gaz5dpgzk5hz",
  "venue": "arXiv",
  "versions": []
}