Universal Hypothesis Testing with Kernels: Asymptotically Optimal Tests for Goodness of Fit
This provides foundational asymptotic guarantees for kernel-based hypothesis testing, impacting statistical machine learning and nonparametric inference.
The paper tackles nonparametric goodness-of-fit testing by characterizing asymptotic optimality using exponential decay rates of type-II errors, showing that Maximum Mean Discrepancy (MMD) based tests achieve optimality on ℝᵈ, and Kernel Stein Discrepancy (KSD) tests achieve optimal rates under relaxed constraints, with quadratic-time MMD tests also optimal for two-sample problems.
We characterize the asymptotic performance of nonparametric goodness of fit testing. The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, and a test is optimal if it achieves the maximum rate subject to a constant level constraint on the type-I error probability. We show that two classes of Maximum Mean Discrepancy (MMD) based tests attain this optimality on $\mathbb R^d$, while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve the maximum exponential decay rate under a relaxed level constraint. Under the same performance metric, we proceed to show that the quadratic-time MMD based two-sample tests are also optimal for general two-sample problems, provided that kernels are bounded continuous and characteristic. Key to our approach are Sanov's theorem from large deviation theory and the weak metrizable properties of the MMD and KSD.