Necessity of the Vanishing-Robustness Calibration for Bivariate GSW

Determine whether the requirement that the bivariate nonparametric Gram-Schmidt walk design use a robustness parameter satisfying \(\varphi_n\to1\) is necessary for the stated variance-rate guarantee, or whether fixed values of \(\varphi\in(0,1)\) suffice.

Background

The paper studies kernelized Gram-Schmidt walk (GSW) designs for balancing structured nonparametric working models. In the bivariate specification, the design uses a kernel formed by averaging bivariate Matérn kernels over all covariate pairs. Theorem 5.1 establishes a variance bound using a calibration in which $1-\varphi_n$ decreases with sample size and dimension, specifically $1-\varphi_n=1/2\wedge(n^{-1}d^2\log n)^{1/2}$.

The authors note that this calibration makes the robustness parameter converge to one, while simulations suggest that small fixed values of $\varphi$ also perform well. Whether the convergence $\varphi_n\to1$ is merely a technical artifact of the current proof or is genuinely required for the theoretical guarantee remains unresolved.