Adapting to Delays and Data in Adversarial Multi-Armed Bandits

We consider the adversarial multi-armed bandit problem under delayed feedback. We analyze variants of the Exp3 algorithm that tune their step size using only information (about the losses and delays) available at the time of the decisions, and obtain regret guarantees that adapt to the observed (rather than the worst-case) sequences of delays and/or losses. First, through a remarkably simple proof technique, we show that with proper tuning of the step size, the algorithm achieves an optimal (up to logarithmic factors) regret of order $\sqrt{\log(K)(TK + D)}$ both in expectation and in high probability, where $K$ is the number of arms, $T$ is the time horizon, and $D$ is the cumulative delay. The high-probability version of the bound, which is the first high-probability delay-adaptive bound in the literature, crucially depends on the use of implicit exploration in estimating the losses. Then, following Zimmert and Seldin (2019), we extend these results so that the algorithm can ``skip'' rounds with large delays, resulting in regret bounds of order $\sqrt{TK\log(K)} + |R| + \sqrt{D_{\bar{R}}\log(K)}$, where $R$ is an arbitrary set of rounds (which are skipped) and $D_{\bar{R}}$ is the cumulative delay of the feedback for other rounds. Finally, we present another, data-adaptive (AdaGrad-style) version of the algorithm for which the regret adapts to the observed (delayed) losses instead of only adapting to the cumulative delay (this algorithm requires an a priori upper bound on the maximum delay, or the advance knowledge of the delay for each decision when it is made). The resulting bound can be orders of magnitude smaller on benign problems, and it can be shown that the delay only affects the regret through the loss of the best arm.