Uniform Approximation of Vapnik-Chervonenkis Classes

TL;DR

This paper establishes a uniform approximation property for VC classes, showing they either have infinite VC dimension or can be approximated by finite partitions with small boundary measure, leading to key statistical implications.

Contribution

It proves a dichotomy for VC classes, linking finite VC dimension to uniform approximation by finite partitions and deriving related statistical properties.

Findings

VC classes with finite VC dimension have finite bracketing numbers.

Such classes satisfy uniform laws of large numbers for ergodic processes.

Results extend to VC major and VC graph families.

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.