{
  "abstract": "We study the learning dynamics of fine-tuning a policy model on self-generated and reward-weighted data, with particular focus on a generalized version of REINFORCE -- referred to as RE(S) -- that updates the rollout distribution once every $S \\ge 1$ gradient steps. Prior work in bandits and reinforcement learning has developed rich theory for policy gradient methods, and on-policy sampling (i.e., a small $S$, ideally $1$) is often viewed as crucial to their success; yet in prominent application like post-training large language models, reward-guided self-training has proved to be effective even when the rollout distribution is updated infrequently, but theoretical understanding remains limited for the convergence properties of these off-policy methods. To bridge these gaps, we develop a unified theory for RE(S) that covers the full spectrum of $S \\ge 1$: it can be interpreted as a stage-wise optimization process, where each stage takes $S$ gradient steps for minimizing the Kullback-Leibler distance to a fixed reward-weighted rollout distribution. For multi-arm bandits with softmax policies, our in-depth analysis and numerical experiments reveal three key findings: (1) for any fixed $S$, RE(S) enjoys global convergence to the optimal policy as the number of rollout distribution updates $B = \\lfloor T / S \\rfloor \\rightarrow \\infty$, where $T$ denotes the number of gradient steps; (2) we prove tight two-sided bounds showing that the suboptimality gap of RE(S) achieves an asymptotic $Θ(1 / T)$ convergence rate, while $S$ only affects the length of a burn-in phase; (3) when initialized at a weak policy with a small optimal-action probability, RE(1) gets trapped around suboptimal policies for a long period, whereas RE(S) with a suitable $S$ avoids the detour and achieves significantly faster convergence to the global optimum, highlighting the benefits of off-policyness in this case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36945",
  "author_total": 4,
  "authors": [
    {
      "name": "Zhiwei Wang"
    },
    {
      "name": "Yanxi Chen"
    },
    {
      "name": "Yaliang Li"
    },
    {
      "name": "Bolin Ding"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0kfatjme9vprq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36945",
  "fulltext_url": null,
  "id": "0kfatjme9vprq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36945",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Reinforcement Learning in Robotics"
    ]
  ],
  "openalex": "W7214905810",
  "pdf_url": "https://arxiv.org/pdf/2609.36945",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0kfatjme9vprq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36945",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36945"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Fine-Tuning on Self-Generated and Reward-Weighted Data: Learning Dynamics, Convergence Rates, and Benefits of Off-Policyness",
  "title_text": "Fine-Tuning on Self-Generated and Reward-Weighted Data: Learning Dynamics, Convergence Rates, and Benefits of Off-Policyness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/0kfatjme9vprq",
  "venue": "arXiv",
  "versions": []
}