Near-Optimal Regret in Adversarial Kernel Bandits

We study the adversarial kernel bandit problem, in which the loss at each round is induced by an arbitrary bounded element of a reproducing kernel Hilbert space (RKHS). We propose an exponential-weights algorithm built on a regularized importance-weighted loss estimator, together with an explicit correction term that cancels the bias introduced by the regularization. Our main result bounds the regret by $\widetilde{{O}}\big(\sqrt{T\, d_*(\lambda)\,\log|{X}|}\big)$, where $d_*(\lambda)$ is a widely-adopted notion of effective dimension that captures the complexity of the kernel. Up to logarithmic factors, this matches the known rate achieved in the related stochastic kernel bandit problem. A notable application is the Mat\'ern$(\nu,d)$ kernel with smoothness parameter $\nu$ on $\mathbb{R}^d$, for which our bound specializes to $\widetilde{{O}}\big(T^{(\nu+d)/(2\nu+d)}\big)$, improving over the best-known prior rate of Chatterji et al. [2019] while simultaneously removing the rank-one adversary assumption required by their analysis. Moreover, this rate is the same as the known optimal rate for stochastic kernel bandits, and also matches a lower bound from concurrent work up to a $\log T$ factor.