Unlabelled Sample Compression Schemes for Intersection-Closed Classes and Extremal Classes

• Main Authors: Rubinstein, J. Hyam, Rubinstein, Benjamin I. P
• Format: Journal Article
• Language: English
• Published: 11-10-2022
• Subjects: Computer Science - Discrete Mathematics; Computer Science - Learning
• DOI: 10.48550/arxiv.2210.05455

The sample compressibility of concept classes plays an important role in
learning theory, as a sufficient condition for PAC learnability, and more
recently as an avenue for robust generalisation in adaptive data analysis.
Whether compression schemes of size $O(d)$ must necessarily exist for all
classes of VC dimension $d$ is unknown, but conjectured to be true by Warmuth.
Recently Chalopin, Chepoi, Moran, and Warmuth (2018) gave a beautiful
unlabelled sample compression scheme of size VC dimension for all maximum
classes: classes that meet the Sauer-Shelah-Perles Lemma with equality. They
also offered a counterexample to compression schemes based on a promising
approach known as corner peeling. In this paper we simplify and extend their
proof technique to deal with so-called extremal classes of VC dimension $d$
which contain maximum classes of VC dimension $d-1$. A criterion is given which
would imply that all extremal classes admit unlabelled compression schemes of
size $d$. We also prove that all intersection-closed classes with VC dimension
$d$ admit unlabelled compression schemes of size at most $11d$.