When are epsilon-nets small?

TL;DR

This paper systematically studies the size of epsilon-nets in geometry and learning, providing new bounds and bridging the gap between these fields, especially in regimes with very small epsilon-nets.

Contribution

It offers a unified treatment of complexity measures influencing epsilon-net sizes and introduces improved upper bounds applicable to small epsilon-nets.

Findings

New upper bounds on epsilon-net sizes generalizing previous results

Existence of small epsilon-nets of size o(1/epsilon) in certain regimes

Short proof of Haussler's upper bound on packing numbers

Abstract

In many interesting situations the size of epsilon-nets depends only on $\epsilon$ together with different complexity measures. The aim of this paper is to give a systematic treatment of such complexity measures arising in Discrete and Computational Geometry and Statistical Learning, and to bridge the gap between the results appearing in these two fields. As a byproduct, we obtain several new upper bounds on the sizes of epsilon-nets that generalize/improve the best known general guarantees. In particular, our results work with regimes when small epsilon-nets of size $o(\frac{1}{\epsilon})$ exist, which are not usually covered by standard upper bounds. Inspired by results in Statistical Learning we also give a short proof of the Haussler's upper bound on packing numbers.