Adversarial Bandits against Arbitrary Strategies

Abstract

We study the adversarial bandit problem against arbitrary strategies, where the difficulty is captured by an unknown parameter $S$, which is the number of switches in the best arm in hindsight. To handle this problem, we adopt the master-base framework using the online mirror descent method (OMD). We first provide a master-base algorithm with simple OMD, achieving $\tilde{O}(S^{1/2}{K}^{1/3}{T}^{2/3})$, in which $T^{2/3}$ comes from the variance of loss estimators. To mitigate the impact of the variance, we propose using adaptive learning rates for OMD and achieve $\tilde{O}(\min \{\sqrt{SKT\ddot{I}},S\sqrt{KT}\})$, where $\ddot{I}$ is a variance term for loss estimators.