{
  "abstract": "Many iterative algorithms rely on bootstrapping. A variable is updated using a second, frozen copy as a target, which is periodically replaced with the updated variable. Majorize-minimize and inexact proximal-point methods share this structure, as does temporal-difference (TD) learning. However, existing convergence guarantees for scenarios that combine sampled updates with targets refreshed only every $K$ steps rely on the specific structure of the update, such as linear approximation or gradient-based inner steps, and on uniformly bounded sampling error. We instead model the sampled update as a stochastic operator on the parameter space, which reduces the analysis to a contraction argument that needs no gradient structure and allows the sampling error to grow with the iterates. Within this framework, we derive a finite-time bound for i.i.d. samples and any target-update period $K$. We show that the iterates converge geometrically in root mean square to a ball around the fixed point, provided the sensitivity to the frozen target is smaller than the contraction slack of the inner map. Existing deterministic frozen-target contraction and stochastic-gradient-type bounds follow as special cases of our framework, and simulations of TD learning reproduce the predicted contraction rate and scaling of the error floor with the step size.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29961",
  "author_total": 3,
  "authors": [
    {
      "name": "Ids van der Werf"
    },
    {
      "name": "Sergio Rozada"
    },
    {
      "name": "Antonio G. Marques"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/58cs3ban63yf1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29961",
  "fulltext_url": null,
  "id": "58cs3ban63yf1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29961",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "eess.SP"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Reinforcement Learning in Robotics"
    ]
  ],
  "openalex": "W7214325216",
  "pdf_url": "https://arxiv.org/pdf/2609.29961",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/58cs3ban63yf1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29961",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29961"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Contraction Framework for Stochastic Operators with Bootstrapping: Application to TD Learning",
  "title_text": "A Contraction Framework for Stochastic Operators with Bootstrapping: Application to TD Learning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/58cs3ban63yf1",
  "venue": "arXiv",
  "versions": []
}