Scale-sensitive dimensions, uniform convergence, and learnability

Noga Alon, Shai Ben-David, Nicolò Cesa-Bianchi, David Haussler

Research output: Contribution to journalArticle

185 Scopus citations


Learnability in Valiant's PAC learning model has been shown to be strongly related to the existence of uniform laws of large numbers. These laws define a distribution-free convergence property of means to expectations uniformly over classes of random variables. Classes of real-valued functions enjoying such a property are also known as uniform Glivenko-Cantelli classes. In this paper, we prove, through a generalization of Sauer's lemma that may be interesting in its own right, a new characterization of uniform Glivenko-Cantelli classes. Our characterization yields Dudley, Giné and Zinn's previous characterization as a corollary. Furthermore, it is the first based on a simple combinatorial quantity generalizing the Vapnik-Chervonenkis dimension. We apply this result to obtain the weakest combinatorial condition known to imply PAC learnability in the statistical regression (or "agnostic") framework. Furthermore, we find a characterization of learnability in the probabilistic concept model, solving an open problem posed by Kearns and Schapire. These results show that the accuracy parameter plays a crucial role in determining the effective complexity of the learner's hypothesis class.

Original languageEnglish (US)
Pages (from-to)615-631
Number of pages17
JournalJournal of the ACM
Issue number4
StatePublished - Jul 1997
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Software
  • Control and Systems Engineering
  • Information Systems
  • Hardware and Architecture
  • Artificial Intelligence


  • PAC learning
  • Theory
  • Uniform laws of large numbers
  • Vapnik-Chervonenkis dimension

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