The Minimax Lower Bound of Kernel Stein Discrepancy Estimation

Jose Cribeiro-Ramallo; Agnideep Aich; Florian Kalinke; Ashit Baran Aich; Zoltán Szabó

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arXiv · 2510.15058

The Minimax Lower Bound of Kernel Stein Discrepancy Estimation

Jose Cribeiro-Ramallo·Agnideep Aich·Florian Kalinke·Ashit Baran Aich·Zoltán Szabó

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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.

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Jose Cribeiro-Ramallo, Agnideep Aich, Florian Kalinke, Ashit Baran Aich, Zoltán Szabó. 2025-10-16. The Minimax Lower Bound of Kernel Stein Discrepancy Estimation. https://arxiv.org/abs/2510.15058

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