{
  "abstract": "Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise is a fundamental theoretical challenge. We make progress towards this challenge for a class of stochastic approximation algorithms whose expected updates are contractive, a setting that arises in many reinforcement learning algorithms such as $Q$-learning and linear temporal difference learning. Specifically, for a power-law learning rate $O(n^{-η})$ with $η\\in (1/2, 1)$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{1 - 2η})$. For the harmonic learning rate $α_n=α/(n+1)$, the almost sure squared-error rate is $o(n^{-ζ})$ for every $ζ<\\min\\{1,2(1-κ)α\\}$, where $κ$ is the contraction factor. Thus exponents arbitrarily close to one are available when $2(1-κ)α\\geq1$. This approaches the $n^{-1}$ polynomial exponent of classical law-of-the-iterated-logarithm benchmarks. Key to our analysis is a novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the well-established Moreau-envelope smoothing for the contractive mapping.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.07104",
  "author_total": 3,
  "authors": [
    {
      "name": "Xinyu Liu"
    },
    {
      "name": "Zixuan Xie"
    },
    {
      "name": "Shangtong Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dvvds85p8hv1v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.07104",
  "fulltext_url": null,
  "id": "dvvds85p8hv1v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.07104",
  "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|Reinforcement Learning in Robotics"
    ]
  ],
  "openalex": "W7160852816",
  "pdf_url": "https://arxiv.org/pdf/2605.07104",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dvvds85p8hv1v.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:28Z",
      "native_id": "2605.07104",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.07104"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift",
  "title_text": "Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:50Z",
  "url": "https://research.draconitic.com/paper/dvvds85p8hv1v",
  "venue": "arXiv",
  "versions": []
}