{
  "abstract": "We develop a quantitative statistical theory of transformers in the large-context regime by adopting the abstraction of contextual flow maps (CFMs): dynamical systems that evolve a distinguished token in the presence of a contextual measure across a stack of attention blocks. Within this framework, the finite-context model approximates an idealized infinite-context system in which the contextual measure is replaced by its underlying population, so that the context length $n$ becomes a statistical resource. Exploiting the McKean--Vlasov structure of the dynamics and the classical machinery of propagation of chaos, we establish a forward bound controlling the deviation between the finite- and infinite-context CFMs uniformly along depth, and a backward bound controlling the deviation between the corresponding training trajectories uniformly across iterations of online gradient descent. Both bounds achieve the optimal Wasserstein rate $n^{-1/d}$ for general CFMs and parametric rate $n^{-1/2}$ for a restricted class of CFMs that includes transformers as a special case. The analysis rests on a new Eulerian adjoint formulation of the loss gradient and stability estimates for the resulting forward--adjoint system, both of which may be of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.16747",
  "author_total": 4,
  "authors": [
    {
      "name": "Shi Chen"
    },
    {
      "name": "Zhengjiang Lin"
    },
    {
      "name": "Kaizhao Liu"
    },
    {
      "name": "Philippe Rigollet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftk8sp5f07wrc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.16747",
  "fulltext_url": null,
  "id": "ftk8sp5f07wrc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.16747",
  "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.AP"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7161682215",
  "pdf_url": "https://arxiv.org/pdf/2605.16747",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ftk8sp5f07wrc.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:09Z",
      "native_id": "2605.16747",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.16747"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Propagation of Chaos in Contextual Flow Maps",
  "title_text": "Propagation of Chaos in Contextual Flow Maps",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/ftk8sp5f07wrc",
  "venue": "arXiv",
  "versions": []
}