Minimax optimality of rates for the adaptive aggregated KSD test over Sobolev-type alternatives

Determine the minimax separation rates for the adaptive kernel Stein discrepancy (KSD) test that aggregates over Gaussian-kernel bandwidths as proposed by Schrab et al. (2023) when alternatives lie in restricted Sobolev balls, and ascertain whether the separation rates they derive are minimax optimal for those alternative spaces.

Background

The paper constructs an adaptive goodness-of-fit test by aggregating regularized KSD tests across a grid of regularization and kernel parameters and analyzes its minimax separation rates. In contrasting related work on aggregation for KSD across kernel bandwidths, the authors note that prior results established separation boundaries over restricted Sobolev balls but did not establish minimax optimality.

This open question concerns the adaptive aggregated KSD procedure from Schrab et al. (2023), which uses Gaussian kernels and bandwidth aggregation. The unresolved point is whether those obtained separation rates are minimax optimal—i.e., the fastest achievable rates—for the considered Sobolev-type alternatives.