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

Key points

• A central quantity in the regret analysis of kernel bandits is the maximum information gain $γT$.1
• In particular, the best existing upper bounds scale as $\sqrt{TγT}$ up to log factors, and nearly-matching lower bounds have been derived for specific kernels such as squared exponential and Matérn.1
• In this paper, we establish a general $Ω(\sqrt{TγT/\log T})$ minimax regret lower bound for non-constant continuous kernels on compact domains, establishing near-optimality (within log factors) in a very general sense.1
• Among other things, our findings imply that the minimax-optimal scaling is exactly $Θ(\sqrt{TγT})$ (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.1

Sources (1)

• [1]A General $\widetildeΩ(\sqrt{T γ_T})$ Lower Bound for Kernel Bandits
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 8, 01:44 AM

  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 $γ_T$.

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