When are epsilon-nets small?

11/28/2017

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by   Andrey Kupavskii, et al.

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In many interesting situations the size of epsilon-nets depends only on ϵ 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(1/ϵ) 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.