Necessity of the union-bound dependence on moment order

Determine whether the dependence on the noise moment order introduced by the union bound in the pointwise-to-uniform conversion is unavoidable or merely an artifact of the proof.

Background

The paper derives uniform stochastic-error rates by converting pointwise concentration inequalities into uniform bounds through a union bound over a finite set of regression points. For noise with finite p-th moments, this produces a p-dependent factor in the resulting rate.

The authors explicitly leave unresolved whether this dependence reflects an inherent limitation of uniform regression guarantees or arises only from the proof technique used for the pointwise-to-uniform conversion.

References

It is unclear to us if this is unavoidable or an artifact of the proof.

— User-friendly approximation theory for statistical regression  (2610.01342 - Benning, 1 Oct 2026) in Remark ‘Sample complexity’ in Section 3, subsection ‘Exact quadrature/cubature’

First, the learning rates in the aforementioned works depend explicitly on the moment parameter and deteriorate as the moment condition weakens. Moreover, no matching lower bounds were established, so it is unclear whether these rates are optimal.

— On the Optimality of Random Features under General Conditions: Misspecified Cases and Robust Learning  (2610.18333 - Wang, 13 Oct 2026) in Section 4.3, “Comparison with Robust Kernel Methods”