Sharpness of the functional m^{-1/2} convergence rate

Determine whether the m^{-1/2} convergence rate in the supremum-norm quadratic Wasserstein bound for a one-hidden-layer randomly initialized neural network with polynomial activation and inputs in a compact domain is sharp.

Background

Theorem 1 establishes an upper bound of order m{-1/2} for the quadratic Wasserstein distance between a one-hidden-layer neural network and its neural-network Gaussian-process limit, viewed as random continuous functions under the supremum norm, when the activation is polynomial and the input domain is compact. The discussion explains that a single-input lower bound is only of order m{-1}, so it does not determine whether the functional rate m{-1/2} is optimal for general finite or dense input sets.

For finite input sets, the best available rate depends on the covariance structure: an m{-1} rate is available when the limiting covariance matrix is invertible, whereas the singular-kernel setting treated in the paper yields m{-1/2}. Establishing a matching lower bound in the functional setting would therefore resolve whether the paper’s upper rate is genuinely sharp.

References

Whether the rate in Theorem~\ref{thm:functional} is sharp remains open.

— Lifting Finite-Sample Gaussian Approximation of Wide Neural Networks to Functional Convergence  (2610.18166 - Wiederhold et al., 13 Oct 2026) in Section 4, Discussion, paragraph “Sharpness of the rate”

Our bound controls the function values but not their derivatives. The only stronger metric in Table~\ref{tab:related} is the $\mathcal C{k}$ norm of \citet{DBLP:journals/corr/abs-2307-06092}, obtained at the slower rate $m{-1/8}$, and whether $m{-1/2}$ is attainable in such a norm is open in general.

— Lifting Finite-Sample Gaussian Approximation of Wide Neural Networks to Functional Convergence  (2610.18166 - Wiederhold et al., 13 Oct 2026) in Section 4, Discussion, paragraph “The metric”