Minimax optimal goodness-of-fit testing with kernel Stein discrepancy

We explore the minimax optimality of goodness-of-fit tests on general domains using the kernelized Stein discrepancy (KSD). The KSD framework offers a flexible approach for goodness-of-fit testing, avoiding strong distributional assumptions, accommodating diverse data structures beyond Euclidean spaces, and relying only on partial knowledge of the reference distribution, while maintaining computational efficiency. Although KSD is a powerful framework for goodness-of-fit testing, only the consistency of the corresponding tests has been established so far, and their statistical optimality remains largely unexplored. In this paper, we develop a general framework and an operator-theoretic representation of the KSD, encompassing many existing KSD tests in the literature, which vary depending on the domain. Building on this representation, we propose a modified discrepancy by applying the concept of spectral regularization to the KSD framework. We establish the minimax optimality of the proposed regularized test for a wide range of the smoothness parameter θ under a specific alternative space, defined over general domains, using the χ²-divergence as the separation metric. In contrast, we demonstrate that the unregu-larized KSD test fails to achieve the minimax separation rate for the considered alternative space. Additionally, we introduce an adaptive test capable of achieving minimax optimality up to a logarithmic factor by adapting to unknown parameters. Through numerical experiments, we illustrate the superior performance of our proposed tests across various domains compared to their unregularized counterparts.