Adversarial Bandits Robust to $S$-Switch Regret

We study the adversarial bandit problem under $S$ number of switching best arms for unknown $S$. For handling this problem, we adopt the master-base framework using the online mirror descent method (OMD). We first provide a master-base algorithm with basic OMD, achieving $\tilde{O}(S^{1/2}K^{1/3}T^{2/3})$. For improving the regret bound with respect to $T$, we propose to use adaptive learning rates for OMD to control variance of loss estimators, and achieve $\tilde{O}(\min\{\mathbb{E}[\sqrt{SKT\rho_T(h^\dagger)}],S\sqrt{KT}\})$, where $\rho_T(h^\dagger)$ is a variance term for loss estimators.