The Minimax Lower Bound of Kernel Stein Discrepancy Estimation

Jose Cribeiro-Ramallo, Agnideep Aich, Florian Kalinke, ASHIT BARAN AICH, Zoltán Szabó

TL;DR: We derive the minimax lower bound of kernel Stein discrepancy estimation with explicit constants on R^d and on general domains.

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.

Submission Number: 414