When are epsilon-nets small?

Given a range space (X,R), where X is a set equipped with probability measure P and the family of measurable subsets R ⊂ 2X , and ǫ > 0, an ǫ-net is a subset of X in the support of P , which intersects each R ∈ R with P (R) ≥ ǫ. In many interesting situations the size of ǫ-nets depends only on ǫ 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 ǫ-nets that generalize/improve the best known general guarantees. In particular, our results work with regimes when small ǫ-nets of size o(1ǫ ) 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 [16].