Improved Regret for Bandit Convex Optimization with Delayed Feedback

We investigate bandit convex optimization (BCO) with delayed feedback, where only the loss value of the action is revealed under an arbitrary delay. Previous studies have established a regret bound of $O(T^{3/4}+d^{1/3}T^{2/3})$ for this problem, where $d$ is the maximum delay, by simply feeding delayed loss values to the classical bandit gradient descent (BGD) algorithm. In this paper, we develop a novel algorithm to enhance the regret, which carefully exploits the delayed bandit feedback via a blocking update mechanism. Our analysis first reveals that the proposed algorithm can decouple the joint effect of the delays and bandit feedback on the regret, and improve the regret bound to $O(T^{3/4}+\sqrt{dT})$ for convex functions. Compared with the previous result, our regret matches the $O(T^{3/4})$ regret of BGD in the non-delayed setting for a larger amount of delay, i.e., $d=O(\sqrt{T})$, instead of $d=O(T^{1/4})$. Furthermore, we consider the case with strongly convex functions, and prove that the proposed algorithm can enjoy a better regret bound of $O(T^{2/3}\log^{1/3}T+d\log T)$. Finally, we show that in a special case with unconstrained action sets, it can be simply extended to achieve a regret bound of $O(\sqrt{T\log T}+d\log T)$ for strongly convex and smooth functions.