Scale-Free Adversarial Multi-Armed Bandit with Arbitrary Feedback Delays

We consider the Scale-Free Adversarial Multi Armed Bandit (MAB) problem with unrestricted feedback delays. In contrast to the standard assumption that all losses are $[0,1]$-bounded, in our setting, losses can fall in a general bounded interval $[-L, L]$, unknown to the agent before-hand. Furthermore, the feedback of each arm pull can experience arbitrary delays. We propose an algorithm named \texttt{SFBanker} for this novel setting, which combines a recent banker online mirror descent technique and elaborately designed doubling tricks. We show that \texttt{SFBanker} achieves $\mathcal O(\sqrt{K(D+T)}L)\cdot {\rm polylog}(T, L)$ total regret, where $T$ is the total number of steps and $D$ is the total feedback delay. \texttt{SFBanker} also outperforms existing algorithm for non-delayed (i.e., $D=0$) scale-free adversarial MAB problem instances. We also present a variant of \texttt{SFBanker} for problem instances with non-negative losses (i.e., they range in $[0, L]$ for some unknown $L$), achieving an $\tilde{\mathcal O}(\sqrt{K(D+T)}L)$ total regret, which is near-optimal compared to the $\Omega(\sqrt{KT}+\sqrt{D\log K}L)$ lower-bound ([Cesa-Bianchi et al., 2016]).