When are epsilon-nets small?
This work addresses foundational problems in theoretical computer science and machine learning by improving general guarantees for epsilon-nets, which are incremental but with specific applications in geometry and learning.
The paper systematically examines complexity measures that determine the size of epsilon-nets in discrete/computational geometry and statistical learning, bridging gaps between these fields and providing new upper bounds, including for regimes where small epsilon-nets of size o(1/ε) exist.
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 $o(\frac{1}ε)$ 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.