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

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 $\sqrt{T\gamma_T}$ up to log factors, and nearly-matching lower bounds have been derived for specific kernels such as squared exponential and Mat\'ern. 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 $\Omega(\sqrt{T\gamma_T/\log T})$ 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 $\Theta(\sqrt{T\gamma_T})$ (i.e., within constant factors) for the Mat\'ern-$\nu$ kernel with $\nu \in (0,2)$, $\gamma$-exponential kernel with $\gamma \in (0,2)$, and certain piecewise-polynomial kernels.