Uniform Approximation of Vapnik-Chervonenkis Classes

Details Uniform Approximation of Vapnik-Chervonenkis Classes Uniform Approximation of Vapnik-Chervonenkis Classes

2010-10-21

arxiv.org/abs/1010.4515v1

Mathematics

Probability

13 pages, no figures

Scientific paper

For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik-Chervonenkis (VC) dimension or (ii) for every epsilon > 0 there is a finite partition pi such the pi-boundary of each set has measure at most epsilon. Immediate corollaries include the fact that a family with finite VC dimension has finite bracketing numbers, and satisfies uniform laws of large numbers for every ergodic process. From these corollaries, we derive analogous results for VC major and VC graph families of functions.

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