{
  "abstract": "Compression schemes provide a general framework for deriving data-dependent risk bounds for learning algorithms. For learning rules based on preferent compression, existing results upper-bound the risk in terms of a sample-dependent complexity $\\boldsymbol{k}$, defined as the cardinality of the compression augmented by the training examples on which the reconstructed hypothesis is inappropriate. These results, however, yield a margin of order $O(\\sqrt{\\ln N/N})$ above $\\boldsymbol{k}/N$. In this paper, we establish that the logarithmic factor is not intrinsic. We prove that, for every fixed confidence level $1-δ$, the risk is bounded with probability at least $1-δ$ by $$ \\frac{\\boldsymbol{k}}{N} + O\\left(\\frac{1}{\\sqrt{N}}\\right), $$ uniformly over all possible values of $\\boldsymbol{k}$. Thus, risk estimates for learning algorithms based on preferent compression attain the same law-of-large-numbers rate as the empirical estimate of the probability of a fixed event. We also show that the leading term $\\boldsymbol{k}/N$ is necessary and investigate its relation to the empirical risk. In particular, we identify regimes in which the empirical risk retains the $O(1/\\sqrt{N})$ convergence rate, as well as regimes in which it converges more slowly or ceases to be consistent, while the complexity-based estimator preserves the $O(1/\\sqrt{N})$ rate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13442",
  "author_total": 2,
  "authors": [
    {
      "name": "Simone Garatti"
    },
    {
      "name": "Marco C. Campi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1va5ta5f44f7n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13442",
  "fulltext_url": null,
  "id": "1va5ta5f44f7n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13442",
  "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.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Machine Learning and Algorithms"
    ]
  ],
  "openalex": "W7221529960",
  "pdf_url": "https://arxiv.org/pdf/2610.13442",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1va5ta5f44f7n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13442",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13442"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Learning by Compression: Risk Estimates Attain the Law-of-Large-Numbers Rate",
  "title_text": "Learning by Compression: Risk Estimates Attain the Law-of-Large-Numbers Rate",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:46Z",
  "url": "https://research.draconitic.com/paper/1va5ta5f44f7n",
  "venue": "arXiv",
  "versions": []
}