{
  "abstract": "Quantile temporal-difference learning (QTD) is an effective method for learning return distributions through quantile approximation, yet its finite-time behavior remains poorly understood. Its update is nonlinear and nonsmooth, and the stability needed for a sharp convergence rate holds only near the target. We establish a global high-probability last-iterate guarantee for synchronous tabular QTD under general positive, nonincreasing step-size sequences and arbitrary initialization in the natural parameter range. For polynomially decaying step sizes with exponent $a\\in(0,1)$, the last iterate converges to the target at rate $T^{-a/2}$ in the infinity norm, up to logarithmic and lower-order terms. A suitably tuned harmonic schedule recovers the $T^{-1/2}$ statistical rate up to logarithmic factors. For the $m$-quantile representation, its $\\infty$-Wasserstein error scales as $\\sqrt{m/T}$ up to logarithmic factors, matching the leading polynomial dependence on the quantile resolution and sample size of the corresponding model-based estimator. The proof uses a two-stage global-to-local argument. From arbitrary initialization, Bellman contraction and CDF monotonicity first bring the iterate close to the target, after which, a novel variance--drift matching argument sharpens the control of accumulated noise and local contraction reduces the remaining errors, yielding the sharp rate. Simulations verify the predicted polynomial decay and assess the finite-time entrance bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27313",
  "author_total": 4,
  "authors": [
    {
      "name": "Zijie Cheng"
    },
    {
      "name": "Xiang Li"
    },
    {
      "name": "Yang Peng"
    },
    {
      "name": "Zhihua Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/318tdpt9nk357.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27313",
  "fulltext_url": null,
  "id": "318tdpt9nk357",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27313",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Reinforcement Learning in Robotics"
    ]
  ],
  "openalex": "W7204574088",
  "pdf_url": "https://arxiv.org/pdf/2608.27313",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/318tdpt9nk357.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:29Z",
      "native_id": "2608.27313",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27313"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "A Finite-Sample Analysis of Quantile Temporal-Difference Learning",
  "title_text": "A Finite-Sample Analysis of Quantile Temporal-Difference Learning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:29Z",
  "url": "https://research.draconitic.com/paper/318tdpt9nk357",
  "venue": "arXiv",
  "versions": []
}