{
  "abstract": "Contrastive learning is a successful paradigm for learning $d$-dimensional geometric representations from a collection of ``anchor--positive--negative'' triplets $(i,j^{+},k^{-})$, indicating that ``item $i$ is closer to $j$ than to $k$.'' Despite its success, understanding why contrastive learning leads to representations of high \\textit{generalization} quality---beyond the often pessimistic predictions from PAC-learning---remains a central question. Recently, \\citet*{alon2024optimal} proved that, for PAC-learning $d$-dimensional Euclidean representations of $n$-point datasets, $Θ(\\min(nd, n^2))$ triplets are necessary and sufficient, while they posed as an open question whether their VC dimension bounds for the more realistic setting of \\textit{contrastive learning with a margin} can be improved. For a margin parameter $α>0$, a triplet $(i,j^{+},k^{-})_α$ is satisfied by the embedding $ϕ:[n]\\rightarrow \\mathbb{R}^{d}$, if $\\|ϕ(i)-ϕ(k)\\|_2>(1+α)\\cdot\\|ϕ(i)-ϕ(j)\\|_2$. In this work, we resolve their question by proving that the VC dimension of contrastive learning under any margin $α\\in(0,1)$ is in fact $O(n/α^2)$, improving on the previous bound of $O(n\\log(n)/α^2)$. We also establish that the bounds are optimal up to constant factors, by providing a matching lower bound of $Ω(\\frac{n}{α^2})$ (the previously known lower bound was $Ω(\\frac{n}α)$), for $α\\geq \\max(n^{-1/2},d^{-1/2})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38834",
  "author_total": 4,
  "authors": [
    {
      "name": "Dionysis Arvanitakis"
    },
    {
      "name": "Vaggos Chatziafratis"
    },
    {
      "name": "Yiyuan Luo"
    },
    {
      "name": "Konstantin Makarychev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bkxktw956jj4t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38834",
  "fulltext_url": null,
  "id": "bkxktw956jj4t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38834",
  "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.DS"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Machine Learning and Algorithms"
    ]
  ],
  "openalex": "W7215037831",
  "pdf_url": "https://arxiv.org/pdf/2609.38834",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bkxktw956jj4t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:58Z",
      "native_id": "2609.38834",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38834"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Optimal VC Dimension of Contrastive Learning with Margin",
  "title_text": "Optimal VC Dimension of Contrastive Learning with Margin",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:58Z",
  "url": "https://research.draconitic.com/paper/bkxktw956jj4t",
  "venue": "arXiv",
  "versions": []
}