Abstract
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.
