Scale-sensitive dimensions, uniform convergence, and learnability

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

Research output: Contribution to journalArticlepeer-review

227 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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