{
  "abstract": "In this paper, we develop an off-policy method for continuous-time reinforcement learning (CTRL), where the system dynamics are governed by an unknown stochastic differential equation (SDE) and only discrete-time trajectory data are available. A central challenge is that the classical discrete-time state--action value function loses its action-dependent signal in the continuous-time limit (Baird, 1994; Jia and Zhou, 2023; Tallec et al., 2019). On the other hand, continuous-time control provides local notions such as the instantaneous advantage function $q(s,a)$, but these typically rely on state value function $V(s)$. To obtain a formulation with a single unknown function, we introduce a new definition of the state--action value function $Q(s,a)$ in CTRL and derive its governing equation. Building on the PhiBE approximation (Zhu, 2024; Zhu et al., 2025), we propose iterative algorithms to approximate the optimal $Q$-function in both model-based and model-free settings using only discrete-time off-policy data. Under linear function approximation, we establish error bounds and explicit convergence rates for the algorithm in the popoluation level. The finite-data algorithm is evaluated numerically.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.21925",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuhua Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5tskbc64dw45f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.21925",
  "fulltext_url": null,
  "id": "5tskbc64dw45f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.21925",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Reinforcement Learning in Robotics"
    ]
  ],
  "openalex": "W7165651724",
  "pdf_url": "https://arxiv.org/pdf/2606.21925",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5tskbc64dw45f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2606.21925",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.21925"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "PhiBE-Q-Learning: Bridging Off-Policy Reinforcement Learning and Continuous-Time Control",
  "title_text": "PhiBE-Q-Learning: Bridging Off-Policy Reinforcement Learning and Continuous-Time Control",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/5tskbc64dw45f",
  "venue": "arXiv",
  "versions": []
}