## Abstract

For any class of binary functions on [n] = {1,..., n} a classical result by Sauer states a sufficient condition for its VC-dimension to be at least d: its cardinality should be at least O(n^{d-1}). A necessary condition is that its cardinality be at least 2^{d} (which is O(1) with respect to n). How does the size of a 'typical' class of VC-dimension d compare to these two extreme thresholds? To answer this, we consider classes generated randomly by two methods, repeated biased coin flips on the n-dimensional hypercube or uniform sampling over the space of all possible classes of cardinality k on [n]. As it turns out, the typical behavior of such classes is much more similar to the necessary condition; the cardinality k need only be larger than a threshold of 2^{d} for its VC-dimension to be at least d with high probability. If its expected size is greater than a threshold of O(log n) (which is still significantly smaller than the sufficient size of O(n^{d-1})) then it shatters every set of size d with high probability. The behavior in the neighborhood of these thresholds is described by the asymptotic probability distribution of the VC-dimension and of the largest d such that all sets of size d are shattered.

Original language | English |
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Pages (from-to) | 113-128 |

Number of pages | 16 |

Journal | Discrete Mathematics and Theoretical Computer Science |

Volume | 10 |

Issue number | 1 |

State | Published - 2008 |

## Keywords

- Poisson approximation
- Random binary functions
- Vapnik-Chervonenkis dimension