We propose Banker-OMD, a novel framework generalizing the classical Online Mirror Descent (OMD) technique in online learning algorithm design. Banker-OMD allows algorithms to robustly handle delayed feedback, and offers a general methodology for achieving $\tilde{O}(\sqrt{T} + \sqrt{D})$-style regret bounds in various delayed-feedback online learning tasks, where $T$ is the time horizon length and $D$ is the total feedback delay. We demonstrate the power of Banker-OMD with applications to three important bandit scenarios with delayed feedback, including delayed adversarial Multi-armed bandits (MAB), delayed adversarial linear bandits, and a novel delayed best-of-both-worlds MAB setting. Banker-OMD achieves nearly-optimal performance in all the three settings. In particular, it leads to the first delayed adversarial linear bandit algorithm achieving $\tilde{O}(\text{poly}(n)(\sqrt{T} + \sqrt{D}))$ regret.

翻译：我们提出Banker-OMD，这是一个新颖的框架，推广了在线学习算法设计中的经典在线镜像下降（Online Mirror Descent，OMD）技术。Banker-OMD使算法能够鲁棒地处理延迟反馈，并提供了一种通用方法，在多种延迟反馈在线学习任务中实现$\tilde{O}(\sqrt{T} + \sqrt{D})$形式的遗憾界，其中$T$是时间水平长度，$D$是总反馈延迟。我们通过三个重要的延迟反馈赌博机场景展示了Banker-OMD的威力，包括延迟对抗性多臂赌博机（delayed adversarial Multi-armed bandits，MAB）、延迟对抗性线性赌博机，以及一个新颖的延迟最佳双世界赌博机设置。Banker-OMD在所有三个设置中均实现了接近最优的性能。特别地，它首次提出了一个延迟对抗性线性赌博机算法，实现了$\tilde{O}(\text{poly}(n)(\sqrt{T} + \sqrt{D}))$的遗憾界。