{
  "abstract": "In this paper, we study online statistical inference for quantile-based distributional policy evaluation, which aims to characterize the distribution of discounted cumulative rewards under a given policy. To obtain a finite-dimensional representation of the return distribution, we consider the fixed point $\\bmη_m$ of the quantile-projected distributional Bellman equation, parameterized by the quantile vector $\\bmθ_m$. We develop two online inference procedures for $\\bmθ_m$, covering synchronous and asynchronous sampling from a generative model as well as Markovian data. The first method uses the trajectory of quantile temporal-difference learning (QTD). We establish functional central limit theorems for the partial sums of QTD iterates and construct a recursively computable random-scaling statistic. The resulting confidence intervals have asymptotically exact coverage at deterministic horizons, with length of order $T^{-1/2}$ in probability. The second method constructs anytime-valid confidence intervals directly from samples. Exploiting the monotonicity of the quantile Bellman map, we develop a computationally explicit coordinate-shaving algorithm with finite-sample simultaneous coverage. Under positive limiting state-visit frequencies, the resulting confidence intervals have lengths of order $\\sqrt{\\log\\log T/T}$, which we establish as classwise optimal through a matching lower bound. Together, these results provide complementary guarantees for online inference in quantile distributional reinforcement learning, combining asymptotically exact fixed-horizon inference with finite-sample anytime validity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.12973",
  "author_total": 3,
  "authors": [
    {
      "name": "Zijie Cheng"
    },
    {
      "name": "Yang Peng"
    },
    {
      "name": "Zhihua Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cxjscv1hvfsnc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.12973",
  "fulltext_url": null,
  "id": "cxjscv1hvfsnc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.12973",
  "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": "W7203483596",
  "pdf_url": "https://arxiv.org/pdf/2608.12973",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cxjscv1hvfsnc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:43Z",
      "native_id": "2608.12973",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.12973"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Online Inference for Quantile Distributional Reinforcement Learning",
  "title_text": "Online Inference for Quantile Distributional Reinforcement Learning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:38Z",
  "url": "https://research.draconitic.com/paper/cxjscv1hvfsnc",
  "venue": "arXiv",
  "versions": []
}