Banker Online Mirror Descent: A Universal Approach for Delayed Online Bandit Learning

We propose `Banker-OMD`, a novel framework generalizing the classical Online Mirror Descent (OMD) technique in the online learning literature. The `Banker-OMD` framework almost completely decouples feedback delay handling and the task-specific OMD algorithm design, thus allowing the easy design of new algorithms capable of easily and robustly handling feedback delays. Specifically, it offers a general methodology for achieving $\tilde{\mathcal O}(\sqrt{T} + \sqrt{D})$-style regret bounds in online bandit learning tasks with delayed feedback, where $T$ is the number of rounds and $D$ is the total feedback delay. We demonstrate the power of \texttt{Banker-OMD} by applications to two important bandit learning scenarios with delayed feedback, including delayed scale-free adversarial Multi-Armed Bandits (MAB) and delayed adversarial linear bandits. `Banker-OMD` leads to the first delayed scale-free adversarial MAB algorithm achieving $\tilde{\mathcal O}(\sqrt{K(D+T)}L)$ regret and the first delayed adversarial linear bandit algorithm achieving $\tilde{\mathcal O}(\text{poly}(n)(\sqrt{T} + \sqrt{D}))$ regret. As a corollary, the first application also implies $\tilde{\mathcal O}(\sqrt{KT}L)$ regret for non-delayed scale-free adversarial MABs, which is the first to match the $\Omega(\sqrt{KT}L)$ lower bound up to logarithmic factors and can be of independent interest.