Given a range space (X , R), where X is a set equipped with probability measure P and R ⊂ 2 X is a family of measurable subsets, 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. Some of our results deal with improvements in logarithmic factors (which is a subject of several classical problems in Learning Theory and Computational Geometry), while others work in regimes when ǫ-nets of size o( 1 ǫ ) exist. Inspired by results in Statistical Learning, we also give a short proof of the Haussler's upper bound on packing numbers [20].
ǫ-nets. Combinatorial and geometric point of viewConsider a set X equipped with probability measure P and a family of measurable subsets R ⊂ 2 X , where 2 X is the power set of X . For simplicity and to avoid potential measurability issues we assume that the set X is finite. The pair (X , R) is called a range space. For a fixed ǫ > 0, a subset S ⊂ X ∩ supp(P) is an ǫ-net, 1 if R ∩ S = ∅ for each R ∈ R with P(R) ≥ ǫ. ǫ-nets often arise in the context of Computational