Asymptotically Optimal One- and Two-Sample Testing with Kernels

TL;DR

This paper characterizes the asymptotic performance of nonparametric one- and two-sample tests using kernel methods, establishing conditions under which these tests achieve optimal error decay rates in the universal setting.

Contribution

It provides new theoretical results on the optimality of MMD and KSD based tests, including an extended Sanov's theorem for two-sample testing, advancing the understanding of nonparametric test performance.

Findings

Maximum Mean Discrepancy (MMD) tests attain optimal error exponents on 

Kernel Stein Discrepancy (KSD) tests achieve optimality with asymptotic level constraints

Extended Sanov's theorem enables exact error exponent derivation for two-sample tests

Abstract

We characterize the asymptotic performance of nonparametric one- and two-sample testing. The exponential decay rate or error exponent of the type-II error probability is used as the asymptotic performance metric, and an optimal test achieves the maximum rate subject to a constant level constraint on the type-I error probability. With Sanov's theorem, we derive a sufficient condition for one-sample tests to achieve the optimal error exponent in the universal setting, i.e., for any distribution defining the alternative hypothesis. We then show that two classes of Maximum Mean Discrepancy (MMD) based tests attain the optimal type-II error exponent on $\mathbb R^d$, while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve this optimality with an asymptotic level constraint. For general two-sample testing, however, Sanov's theorem is insufficient to obtain a similar sufficient condition. We proceed to establish an extended version of Sanov's theorem and derive an exact error exponent for the quadratic-time MMD based two-sample tests. The obtained error exponent is further shown to be optimal among all two-sample tests satisfying a given level constraint. Our work hence provides an achievability result for optimal nonparametric one- and two-sample testing in the universal setting. Application to off-line change detection and related issues are also discussed.