{
  "abstract": "Why large language models trained via log-loss minimization over massive and diverse datasets exhibit broad capabilities and generalizable skills is not yet fully understood. Grau-Moya et al. (ICML 2024) offer a conceptual explanation through the lens of meta-learning: if sequence-generating environments $ν$ are drawn from a prior $w_ν$, minimizing population log-loss recovers the corresponding Bayesian mixture predictor $ξ=\\sum_νw_νν$. In the idealized limit in which $w$ is a universal prior, the optimal predictor corresponds to Solomonoff universal induction. Practical machine learning algorithms, however, generally produce only \\emph{approximately} optimal predictors. This raises a natural question: \\emph{Does the prediction performance remain robust under approximation}? In this paper, we first provide a tight bound on the worst-case effect of approximation on prediction error. We show any predictor $ρ$ satisfying $D_{KL}(ξ\\|ρ)\\leq η$ retains bounded cumulative-expected-squared prediction error. However, while the Bayesian mixture predictor $ξ$ has error $O(\\log(1/w_μ))$ in an environment $μ$, we establish a tight $Θ(η/w_μ)$ bound on the excess error of $ρ$ relative to $ξ$. Thus, approximation can substantially degrade performance in environments assigned small prior probability. In sharp contrast, when environments are averaged according to prior $w$, we prove that the excess prediction error is only $O(η)$. Furthermore, we establish a distribution-shift theorem for dominated shifts from a training prior $w$ to a test prior $w'$, interpolating our average-case and worst-case guarantees. Finally, we extend our robustness analysis to a finite-sampling setting relevant to learning universal predictors with neural networks and Transformers, where $D_{KL}(ξ\\|ρ)$ could be unbounded. [For the full abstract, see the paper.]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29289",
  "author_total": 2,
  "authors": [
    {
      "name": "Qian Li"
    },
    {
      "name": "Shang-Hua Teng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c1rqa3npm79qd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29289",
  "fulltext_url": null,
  "id": "c1rqa3npm79qd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29289",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1703|Computability, Logic, AI Algorithms"
    ]
  ],
  "openalex": "W7223218380",
  "pdf_url": "https://arxiv.org/pdf/2610.29289",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c1rqa3npm79qd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:29Z",
      "native_id": "2610.29289",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29289"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Approximate Solomonoff Induction: On the Robustness of Learned Universal Predictors",
  "title_text": "Approximate Solomonoff Induction: On the Robustness of Learned Universal Predictors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:29Z",
  "url": "https://research.draconitic.com/paper/c1rqa3npm79qd",
  "venue": "arXiv",
  "versions": []
}