# The Minimax Lower Bound of Kernel Stein Discrepancy Estimation

arXiv: https://arxiv.org/abs/2510.15058 · HTML: https://mlsanity.com/paper/2510.15058/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 76 references that are in the map.
- First posted on arXiv 16 Oct 2025.

## Builds on (most cited first, 30 of 33)

- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [A Non-Asymptotic Analysis for Stein Variational Gradient Descent](https://mlsanity.com/paper/2006.09797/) (2020)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Stein Points](https://mlsanity.com/paper/1803.10161/) (2018)
- [Characteristic and Universal Tensor Product Kernels](https://mlsanity.com/paper/1708.08157/) (2017)
- [Kernel-based Tests for Joint Independence](https://mlsanity.com/paper/1603.00285/) (2016)
- [Stein's Method Meets Computational Statistics: A Review of Some Recent Developments](https://mlsanity.com/paper/2105.03481/) (2021)
- [Stein Point Markov Chain Monte Carlo](https://mlsanity.com/paper/1905.03673/) (2019)
- [KSD Aggregated Goodness-of-fit Test](https://mlsanity.com/paper/2202.00824/) (2022)
- [A Kernel Stein Test for Comparing Latent Variable Models](https://mlsanity.com/paper/1907.00586/) (2019)
- [Stein Variational Gradient Descent as Moment Matching](https://mlsanity.com/paper/1810.11693/) (2018)
- [Targeted Separation and Convergence with Kernel Discrepancies](https://mlsanity.com/paper/2209.12835/) (2022)
- [A Stein Goodness-of-fit Test for Directional Distributions](https://mlsanity.com/paper/2002.06843/) (2020)
- [A Kernel Test for Three-Variable Interactions](https://mlsanity.com/paper/1306.2281/) (2013)
- [Bayesian posterior approximation via greedy particle optimization](https://mlsanity.com/paper/1805.07912/) (2018)
- [The reproducing Stein kernel approach for post-hoc corrected sampling](https://mlsanity.com/paper/2001.09266/) (2020)
- [Kernel Stein Tests for Multiple Model Comparison](https://mlsanity.com/paper/1910.12252/) (2019)
- [A kernel Stein test of goodness of fit for sequential models](https://mlsanity.com/paper/2210.10741/) (2022)
- [Minimax Optimal Goodness-of-Fit Testing with Kernel Stein Discrepancy](https://mlsanity.com/paper/2404.08278/) (2024)
- [Debiased Distribution Compression](https://mlsanity.com/paper/2404.12290/) (2024)
- [Interpretable Stein Goodness-of-fit Tests on Riemannian Manifolds](https://mlsanity.com/paper/2103.00895/) (2021)
- [Kernelized Cumulants: Beyond Kernel Mean Embeddings](https://mlsanity.com/paper/2301.12466/) (2023)
- [Handling Hard Affine SDP Shape Constraints in RKHSs](https://mlsanity.com/paper/2101.01519/) (2021)
- [Stein's method of normal approximation: Some recollections and reflections](https://mlsanity.com/paper/2104.08302/) (2021)
- [Nystr"om Kernel Stein Discrepancy](https://mlsanity.com/paper/2406.08401/) (2024)

## Built on by (most cited first, 3 of 3)

- [Minimax Estimation of Kernel Stein Discrepancy: Trace versus Hilbert-Schmidt Scales](https://mlsanity.com/paper/2607.03367/) (2026)
- [Nystr"om Kernel Stein Discrepancy Tests](https://mlsanity.com/paper/2605.25173/) (2026)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)

## Abstract (arXiv)

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 √ 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 Rd 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.