Null Measurability at the Symmetrization Interface in VC Learning

2026-04-27

AI Summary

This work addresses the overly restrictive assumption of Borel measurability on the supremum of the ghost gap in classical statistical learning theory, which limits the applicability of VC-dimension-based methods. Focusing on the one-sided ghost-gap envelope actually employed in standard symmetrization arguments, the paper introduces a weaker yet robust “null-measurable” condition by leveraging tools from descriptive set theory—such as Choquet capacities—and real analysis. This new condition is closed under several natural concept classes, and an explicit non-Borel measurable counterexample is constructed to rigorously separate it from the classical assumption. In the realizable setting, the study establishes for the first time that finite VC dimension implies PAC learnability without requiring Borel measurability, thereby significantly relaxing the foundational assumptions of classical learning theory.

Abstract

Recent work revisiting measurability in the fundamental theorem of statistical learning imposes Borel measurability of ghost-gap suprema. We show that, at the one-sided ghost-gap interface actually used by the standard symmetrization proof, this requirement is stronger than necessary. For any Borel-parameterized concept class on a Polish domain, the bad event "there exists a hypothesis whose ghost empirical error exceeds its training empirical error by at least ε/2" is analytic. By Choquet capacitability, it is therefore measurable in the completion of every finite Borel measure. We then construct a concept class whose bad event is null-measurable but not Borel, giving a strict separation from the Borel supremum condition. Finally, we prove closure under patching, fixed and countable interpolation, and fiber-product amalgamation, showing that the weaker regularity level is stable under natural concept-class constructors. In the realizable setting, where targets belong to the class and are measurable, these results weaken the measurability hypothesis needed by the symmetrization route from finite VC dimension to PAC learnability. The main results and the descriptive-set-theoretic infrastructure used by them are formalized in Lean 4.

Problem

• measurability
• symmetrization
• VC dimension
• PAC learnability
• descriptive set theory

Innovation

• null measurability
• symmetrization
• analytic sets
• VC learning
• descriptive set theory