Optimality of convergence rates on broader domain classes

Establish whether the upper convergence rates derived for the statistical inverse learning framework are optimal on a larger class of Banach-space domains, and, if possible, derive improved or optimal rates.

Background

The paper derives high-probability convergence rates for Tikhonov-regularized statistical inverse problems on arbitrary Banach spaces, measuring error primarily through the Bregman distance and obtaining norm rates in uniformly convex spaces. The authors explicitly acknowledge that these upper bounds are not generally sharp. They identify the optimality question on a broader class of domains as unresolved and indicate that improved, potentially optimal rates require further investigation.