The Minimax Lower Bound of Kernel Stein Discrepancy Estimation

Jose Cribeiro-Ramallo, Agnideep Aich, Florian Kalinke, ASHIT BARAN AICH, Zoltán Szabó
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:901-909, 2026.

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 base 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-cribeiro-ramallo26a, title = { The Minimax Lower Bound of Kernel Stein Discrepancy Estimation }, author = {Cribeiro-Ramallo, Jose and Aich, Agnideep and Kalinke, Florian and AICH, ASHIT BARAN and Szab{\'o}, Zolt{\'a}n}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {901--909}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/cribeiro-ramallo26a/cribeiro-ramallo26a.pdf}, url = {https://proceedings.mlr.press/v300/cribeiro-ramallo26a.html}, 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 base 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. } }
Endnote
%0 Conference Paper %T The Minimax Lower Bound of Kernel Stein Discrepancy Estimation %A Jose Cribeiro-Ramallo %A Agnideep Aich %A Florian Kalinke %A ASHIT BARAN AICH %A Zoltán Szabó %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-cribeiro-ramallo26a %I PMLR %P 901--909 %U https://proceedings.mlr.press/v300/cribeiro-ramallo26a.html %V 300 %X 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 base 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.
APA
Cribeiro-Ramallo, J., Aich, A., Kalinke, F., AICH, A.B. & Szabó, Z.. (2026). The Minimax Lower Bound of Kernel Stein Discrepancy Estimation . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:901-909 Available from https://proceedings.mlr.press/v300/cribeiro-ramallo26a.html.

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