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arXiv · 1711.10414

When are epsilon-nets small?

Andrey Kupavskii·Nikita Zhivotovskiy
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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.

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BibTeXRIS

Andrey Kupavskii, Nikita Zhivotovskiy. 2017-11-28. When are epsilon-nets small?. https://doi.org/10.1016/j.jcss.2019.12.006

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