Tight list replicability bounds via a novel sphere covering theorem

2026-06-04

AI Summary

This work investigates the tight relationship among list size, accuracy parameters, and hypothesis class complexity in the context of list replicability. By introducing a novel spherical covering theorem grounded in the Borsuk–Ulam theorem, it pioneers the systematic application of algebraic topology to statistical learning theory. This approach yields sharp upper and lower bounds on list size as a function of accuracy for VC classes. Notably, for large-margin halfspaces, it achieves optimal list sizes of \(d\) and \(\lceil d/2 \rceil + 1\) under distinct margin regimes, substantially advancing the understanding of the fundamental limits of list replicability.

Abstract

In recent years, list replicability has emerged as a framework for formalizing reproducibility in learning theory. A central question is how the required list size relates to the accuracy parameter and natural complexity measures of the hypothesis class. To achieve sharp bounds on list replicability, we prove a novel topological sphere covering theorem, derived from the Borsuk-Ulam theorem. Specifically, if the $d$-sphere is covered by open sets, each of which lies in an open hemisphere, then $d+1$ of these sets must have a common intersection. Using this result, we obtain a sharp bound on the relationship between list size and accuracy for VC classes. We also show that for large-margin half-spaces, provided the margin is not too large, the optimal list size equals the ambient dimension. However, when the margin is taken to be very large, we devise a replicable algorithm achieving the minimal list size of $\lceil d/2 \rceil + 1$.

Problem

• list replicability
• reproducibility
• VC classes
• margin
• hypothesis class

Innovation

• list replicability
• sphere covering theorem
• Borsuk-Ulam theorem
• VC classes
• large-margin half-spaces