{
  "abstract": "In offline RL, estimating the optimal action-value function $Q^*$ can be formulated as solving the optimal Bellman equation based solely on offline observations. A fundamental challenge is that the reward function and transition kernel are unknown, so the optimal Bellman operator is not directly observable from data. To address this issue, we propose a novel framework that decouples operator estimation from value function learning. In this approach, we first formulate conditional diffusion models to estimate the reward law and transition kernel, which induces a data-driven approximation of the optimal Bellman operator. We then plug these estimators into the Bellman equation and obtain a deep estimator of $Q^*$ by minimizing the empirical Bellman residual over a neural network function class. Theoretically, we first establish sharp nonasymptotic convergence rates for learning the optimal Bellman operator. We establish the smoothness of $Q^*$ through the Bellman equation using the the Hölder assumptions on the conditional reward and transition densities. With suitable sample sizes, we obtain the oracle value-stage rate $\\widetilde{\\mathcal O}\\bigl(n^{-\\frac{2α}{d_x+d_a+2α}}\\bigr)$ for the excess Bellman residual risk. We then translate this risk bound into an expected $L^2$ estimation error of order $\\widetilde{\\mathcal O}\\bigl(n^{-\\fracα{d_x+d_a+2α}}\\bigr)$ under a concentrability condition. For the total offline budget $\\widetilde n=n+n_r+n_x$, we obtain a value estimation rate of $\\widetilde{\\mathcal O}\\bigl(\\widetilde n^{-\\fracα{2d_x+d_a+2α}}\\bigr)$ and establish an information-theoretic lower bound of order $\\widetilde n^{-\\fracα{d_x+d_a+2α}}$. Importantly, our theoretical analysis does not rely on Bellman completeness assumptions commonly used in deep RL theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.14401",
  "author_total": 5,
  "authors": [
    {
      "name": "Xiaohong Chen"
    },
    {
      "name": "Yuling Jiao"
    },
    {
      "name": "Lican Kang"
    },
    {
      "name": "Jerry Zhijian Yang"
    },
    {
      "name": "Chen Zhong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c6x28bk7q8nmt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.14401",
  "fulltext_url": null,
  "id": "c6x28bk7q8nmt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.14401",
  "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": "W7203650104",
  "pdf_url": "https://arxiv.org/pdf/2608.14401",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c6x28bk7q8nmt.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:21Z",
      "native_id": "2608.14401",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.14401"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Offline Deep Q* Estimation with Diffusion Models",
  "title_text": "Offline Deep Q* Estimation with Diffusion Models",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/c6x28bk7q8nmt",
  "venue": "arXiv",
  "versions": []
}