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.

我们研究了具有延迟反馈的强凸波段优化问题，通过精细地利用延迟波段反馈的阻塞更新机制，我们的算法改进了损失边界并将其与延迟设置下的传统波段梯度下降（BGD）算法相匹配。

带有延迟反馈的强化学习优化中的改进后悔度