A General $\widetildeΩ(\sqrt{T γ_T})$ Lower Bound for Kernel Bandits

2026-10-07

AI Summary

This study addresses the long-standing absence of minimax regret lower bounds in kernel bandit problems, which has precluded determining the universal optimality of existing algorithmic upper bounds. By leveraging reproducing kernel Hilbert space (RKHS) theory and information gain analysis within a minimax game-theoretic framework, this work investigates function optimization over RKHSs. It establishes the first $\Omega(\sqrt{T\gamma_T}/\log T)$ regret lower bound for general continuous non-constant kernels and proves that the optimal scaling for specific kernels, such as Matérn, is precisely $\Theta(\sqrt{T\gamma_T})$. These results confirm the near-optimality of existing kernel bandit algorithms, characterize the exact order of minimax regret for specific kernel classes, and reveal the theoretical necessity of the logarithmic factor.

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 $γ_T$. In particular, the best existing upper bounds scale as $\sqrt{Tγ_T}$ up to log factors, and nearly-matching lower bounds have been derived for specific kernels such as squared exponential and Matérn. 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 $Ω(\sqrt{Tγ_T/\log T})$ 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 $Θ(\sqrt{Tγ_T})$ (i.e., within constant factors) for the Matérn-$ν$ kernel with $ν\in (0,2)$, $γ$-exponential kernel with $γ\in (0,2)$, and certain piecewise-polynomial kernels.

Problem

• Kernel Bandits
• Regret Lower Bound
• Maximum Information Gain
• Reproducing Kernel Hilbert Space

Innovation

• Kernel Bandits
• Minimax Regret Lower Bound
• Maximum Information Gain
• Reproducing Kernel Hilbert Space
• Matérn Kernel