Size sensitive packing number for Hamming cube and its consequences

TL;DR

This paper extends Haussler's Packing lemma to size-sensitive set-systems with bounded primal shatter dimension, leading to improved discrepancy bounds and better sizes for relative approximations and samples.

Contribution

It introduces a size-sensitive version of Haussler's Packing lemma for set-systems with bounded primal shatter dimension, addressing open questions and improving discrepancy bounds.

Findings

Proves a size-sensitive packing lemma for specific set-systems.

Improves discrepancy bounds for these set-systems.

Enhances sizes of relative approximations and samples.

Abstract

We prove a size-sensitive version of Haussler's Packing lemma~\cite{Haussler92spherepacking} for set-systems with bounded primal shatter dimension, which have an additional {\em size-sensitive property}. This answers a question asked by Ezra~\cite{Ezra-sizesendisc-soda-14}. We also partially address another point raised by Ezra regarding overcounting of sets in her chaining procedure. As a consequence of these improvements, we get an improvement on the size-sensitive discrepancy bounds for set systems with the above property. Improved bounds on the discrepancy for these special set systems also imply an improvement in the sizes of {\em relative $(\varepsilon, \delta)$-approximations} and $(\nu, \alpha)$-samples.