{
  "abstract": "Language models, once pretrained, are routinely assessed by the holdout method, which estimates their per-token cross-entropy risk---a quantity that may be finite or infinite. We prove possibility and impossibility theorems that characterize the hardness of this estimation problem. The estimand depends on two objects: the unknown data distribution $p$ and the distribution $q$ induced by the language model under study. Our ignorance of $p$ is empirical, whereas our ignorance of $q$, or at least of its tail, is mathematical; so the parameter space {\\em should} range over pairs $(p,q)$---a product space---to fully capture our ignorance. {\\em Negative results}: no estimator is pointwise consistent, even when the parameter space is restricted to full-support models and to data distributions whose expected text length is below a fixed bound; worse, every estimator fails on a dense set of pairs. {\\em Positive results}: the holdout estimator becomes pointwise consistent when the models are further restricted to Transformers or, architecture-neutrally, to full-support models with a finite context window; and, where no such design is in place, it still consistently estimates the risk capped at any fixed threshold, which suffices for screening candidate models in practice. {\\em Lesson}: designs such as the Transformer architecture and finite context windows, although adopted for computational or engineering reasons, pay a {\\em statistical dividend}: they suffice to make pointwise consistency possible---a dividend visible only in how they turn a negative result into positive ones.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.15798",
  "author_total": 1,
  "authors": [
    {
      "name": "Hanti Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7afavg121jbzk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.15798",
  "fulltext_url": null,
  "id": "7afavg121jbzk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.15798",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Topic Modeling"
    ]
  ],
  "openalex": "W7203705162",
  "pdf_url": "https://arxiv.org/pdf/2608.15798",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7afavg121jbzk.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:21Z",
      "native_id": "2608.15798",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.15798"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "The Hardness of Estimating Cross-Entropy Risk for Language Models",
  "title_text": "The Hardness of Estimating Cross-Entropy Risk for Language Models",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/7afavg121jbzk",
  "venue": "arXiv",
  "versions": []
}