### Abstract

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, Gine, 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 characterize PAC learnability in the statistical regression framework of probabilistic concepts, solving an open problem posed by Kearns and Schapire. Our characterization shows that the accuracy parameter plays a crucial role in determining the effective complexity of the learner's hypothesis class.

Original language | English (US) |
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Title of host publication | Annual Symposium on Foundatons of Computer Science (Proceedings) |

Editors | Anon |

Publisher | Publ by IEEE |

Pages | 292-301 |

Number of pages | 10 |

ISBN (Print) | 0818643706 |

State | Published - Dec 1 1993 |

Externally published | Yes |

Event | Proceedings of the 34th Annual Symposium on Foundations of Computer Science - Palo Alto, CA, USA Duration: Nov 3 1993 → Nov 5 1993 |

### Publication series

Name | Annual Symposium on Foundatons of Computer Science (Proceedings) |
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ISSN (Print) | 0272-5428 |

### Other

Other | Proceedings of the 34th Annual Symposium on Foundations of Computer Science |
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City | Palo Alto, CA, USA |

Period | 11/3/93 → 11/5/93 |

### All Science Journal Classification (ASJC) codes

- Hardware and Architecture

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## Cite this

*Annual Symposium on Foundatons of Computer Science (Proceedings)*(pp. 292-301). (Annual Symposium on Foundatons of Computer Science (Proceedings)). Publ by IEEE.