Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD

Abstract:Over the past 20 years, kernel discrepancies have been leveraged as a highly powerful tool for quantifying the disagreement of distributions, with numerous successful applications in two-sample, goodness-of-fit, and independence testing, among others. Their fastest estimators are known to converge at a parametric rate---$n^{-1/2}$---under mild conditions. While this rate is known to be minimax optimal on $\mathbb R^d$ under strict assumptions with bounded kernels, little is known about its optimality beyond the finite-dimensional Euclidean setting with unbounded kernels. In this work, we prove that the minimax lower bound of estimation of the most popular kernel discrepancies (maximum mean discrepancy, Hilbert-Schmidt independence criterion and kernel Stein discrepancy; MMD, HSIC, KSD) is $n^{-1/2}$ on general topological spaces, and under mild assumptions on the kernel; the same rates are shown (as corollaries) to hold for the estimation of the mean embedding and the centered cross-covariance operator. Our results settle the question of optimal estimation of these kernel discrepancies.

Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)

G.3; H.1.1; I.2.6

arXiv:2607.24235 [stat.ML]

(or arXiv:2607.24235v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2607.24235 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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