Measurability Aspects of the Compactness Theorem for Sample Compression Schemes

TL;DR

This paper investigates the conditions under which sample compression schemes derived via the compactness theorem have measurable hypotheses, focusing on standard Borel spaces with certain properties, and introduces a new variant called copy schemes.

Contribution

It establishes measurability conditions for sample compression schemes in standard Borel spaces and introduces the concept of copy sample compression schemes.

Findings

Measurable hypotheses are guaranteed under specific conditions in standard Borel spaces.

A new variant called copy sample compression scheme is introduced.

Conditions for measurability ensure learnability in concept classes.

Abstract

It was proved in 1998 by Ben-David and Litman that a concept space has a sample compression scheme of size d if and only if every finite subspace has a sample compression scheme of size d. In the compactness theorem, measurability of the hypotheses of the created sample compression scheme is not guaranteed; at the same time measurability of the hypotheses is a necessary condition for learnability. In this thesis we discuss when a sample compression scheme, created from com- pression schemes on finite subspaces via the compactness theorem, have measurable hypotheses. We show that if X is a standard Borel space with a d-maximum and universally separable concept class C, then (X,C) has a sample compression scheme of size d with universally Borel measurable hypotheses. Additionally we introduce a new variant of compression scheme called a copy sample compression scheme.