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arXiv Machine Learning By Chenkai Ma, Jonathan Scarlett

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

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The paper establishes a general minimax regret lower bound of Ω(√{Tγ_T/ log T}) for kernel bandits with non‑constant continuous kernels on compact domains, matching existing upper bounds up to logarithmic factors. It shows that this log factor is unavoidable in general but can be removed under specific conditions. Consequently, the optimal scaling is exactly Θ(√{Tγ_T}) for several common kernels, including Matérn‑ν (ν∈(0,2)), γ‑exponential (γ∈(0,2)), and certain piecewise‑polynomial kernels.

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