Uniform Approximation of Vapnik-Chervonenkis Classes
Probability
2010-10-22 v1 Statistics Theory
Machine Learning
Statistics Theory
Abstract
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.
Cite
@article{arxiv.1010.4515,
title = {Uniform Approximation of Vapnik-Chervonenkis Classes},
author = {Terrence M. Adams and Andrew B. Nobel},
journal= {arXiv preprint arXiv:1010.4515},
year = {2010}
}
Comments
13 pages, no figures