This paper studies kernelized bandits (also known as Gaussian process bandits) in an adversarial environment, where the reward functions in a known reproducing kernel Hilbert space (RKHS) may be adversarially chosen at each round. We show that the exponential-weight algorithm achieves $\tilde{O}(\sqrt{T γ_T})$ adversarial regret, where $T$ and $γ_T$ denote the number of total rounds and the maximum information gain, respectively. For squared exponential (SE) and $ν$-Matérn kernels, we also show algorithm-independent lower bounds that guarantee the optimality of our algorithm up to polylogarithmic factors. Furthermore, we present a computationally efficient variant of our algorithm using Nyström approximation while maintaining nearly optimal regret guarantees.

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