A General widetilde{Omega}(sqrt{T gamma_T}) Lower Bound for Kernel Bandits

arXiv:2610.11082v1 Announce Type: cross Abstract: The kernel bandit problem consists of sequentially optimizing an unknown function with noisy feedback, where the function has bounded norm in a given Reproducing Kernel Hilbert Space (RKHS). A central quantity in the regret analysis of kernel bandits is the maximum information gain gamma_T. In particular, the best existing upper bounds scale as sqrt{Tgamma_T} up to log factors, and nearly-matching lower bounds have been derived for specific kernels such as squared exponential and Matern. However, lower bounds for general kernels are lacking, thus making it unclear in what generality the upper bounds are near-optimal. In this paper, we establish a general Omega(sqrt{Tgamma_T/log T}) minimax regret lower bound for non-constant continuous kernels on compact domains, establishing near-optimality (within log factors) in a very general sense. We show that the log factor appearing in this bound is unavoidable in general, but that it can be removed under certain conditions. Among other things, our findings imply that the minimax-optimal scaling is exactly Theta(sqrt{Tgamma_T}) (i.e., within constant factors) for the Matern-nu kernel with nu in (0,2), gamma-exponential kernel with gamma in (0,2), and certain piecewise-polynomial kernels.

Source: arXiv cs.LG | 2026-10-09