Determine the dimension dependence of the gamma-VC dimension of decision stumps

Determine the exact dependence on the ambient dimension d of the gamma-VC dimension of the class of decision stumps in R^d, beyond the established bounds Omega(1/gamma) and O(d/gamma).

Background

The paper studies the gamma-VC dimension as a measure of the expressive power obtained by aggregating weak hypotheses. For d-dimensional decision stumps, prior work established a lower bound of Omega(1/gamma) and an upper bound of O(d/gamma), which determine the dependence on the edge parameter gamma but leave the dependence on the ambient dimension d unresolved. The authors subsequently address this problem by proving sharper bounds, but the quoted passage explicitly identifies the dependence as open at the point where it is introduced.

References

These bounds determine the dependence on $\gamma$, but leave open the exact dependence on the ambient dimension $d$.

— Boosting and the Expressive Power of Simple Weak Learners via the $γ$-VC Dimension  (2610.10383 - Cunha et al., 7 Oct 2026) in Section 1, subsection “Expressivity of Simple Concept Classes”