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Local Combinatorial Analogues for Bounded VC Dimension

Author Olga Medrano Martín del Campo

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LIPIcs.LICS.2026.72.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Finite Model Theory
Keywords
  • Littlestone
  • Vapnik-Chervonenkis
  • good sets
  • Regularity Lemma
  • Haussler Packing Lemma
  • Fundamental Theorem of Statistical Learning

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Abstract

Stable graphs, or equivalently Littlestone classes, were characterized by existence of linear-sized "good" sets, a kind of strongly homogeneous set, in work of Malliaris-Shelah and Malliaris-Moran. We prove a parallel result for VC classes, showing these are characterized by existence of linear-sized symmetric or asymmetric good pairs (which we define). We give several proofs, each drawing from methods and results from different areas, and resulting in different kinds of bounds. We finish with a few words on our learning theory motivation for these investigations and state some further research directions.

Cite As Get BibTex

Olga Medrano Martín del Campo. Local Combinatorial Analogues for Bounded VC Dimension. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 72:1-72:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.LICS.2026.72

Author Details

Olga Medrano Martín del Campo
  • University of Chicago, IL, USA

Funding

  • Medrano Martín del Campo, Olga: Research partially supported by Fundación Alberto y Dolores de Andrade, I.A.P.

Acknowledgements

Thanks to Maryanthe Malliaris for the invaluable mentorship as this paper was prepared; to her and to Leonardo Nagami Coregliano for their patient remarks and feedback; to Balder ten Cate, Steffen van Bergerem, Phokion Kolaitis, Caroline Terry, and Gabe Conant for insightful conversations; and to the anonymous referees for their thoughtful and detailed feedback.

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