The Minimax Lower Bound of Kernel Stein Discrepancy Estimation
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908919849287680 |
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| author | Cribeiro-Ramallo, Jose Aich, Agnideep Kalinke, Florian Aich, Ashit Baran Szabó, Zoltán |
| author_facet | Cribeiro-Ramallo, Jose Aich, Agnideep Kalinke, Florian Aich, Ashit Baran Szabó, Zoltán |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15058 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Minimax Lower Bound of Kernel Stein Discrepancy Estimation Cribeiro-Ramallo, Jose Aich, Agnideep Kalinke, Florian Aich, Ashit Baran Szabó, Zoltán Machine Learning Statistics Theory 62C20 (Primary) 46E22, 62B10 (Secondary) G.3; H.1.1; I.2.6 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. |
| title | The Minimax Lower Bound of Kernel Stein Discrepancy Estimation |
| topic | Machine Learning Statistics Theory 62C20 (Primary) 46E22, 62B10 (Secondary) G.3; H.1.1; I.2.6 |
| url | https://arxiv.org/abs/2510.15058 |