Kernel Stein discrepancies (KSDs) have emerged as a powerful tool for quantifying goodness-of-fit over the last decade, featuring numerous successful applications. To the best of our knowledge, all existing KSD estimators with known rate achieve $\sqrt n$-convergence. In this work, we present two complementary results (with different proof strategies), establishing that the minimax lower bound of KSD estimation is $n^{-1/2}$ and settling the optimality of these estimators. Our first result focuses on KSD estimation on $\mathbb R^d$ with the Langevin-Stein operator; our explicit constant for the Gaussian kernel indicates that the difficulty of KSD estimation may increase exponentially with the dimensionality $d$. Our second result settles the minimax lower bound for KSD estimation on general domains.

翻译：核斯坦因散度（KSD）在过去十年中已成为量化拟合优度的强大工具，并具有众多成功应用。据我们所知，所有已知收敛速率的KSD估计器均达到$\sqrt n$收敛速度。本文通过两种不同证明策略给出了互补结果，确定了KSD估计的极小极大下界为$n^{-1/2}$，从而证实了这些估计量的最优性。第一个结果聚焦于在$\mathbb R^d$上利用Langevin-Stein算子的KSD估计；针对高斯核的显式常数表明，KSD估计的难度可能随维度$d$呈指数增长。第二个结果则确立了在一般域上KSD估计的极小极大下界。