In the field of online sequential decision-making, we address the problem with delays utilizing the framework of online convex optimization (OCO), where the feedback of a decision can arrive with an unknown delay. Unlike previous research that is limited to Euclidean norm and gradient information, we propose three families of delayed algorithms based on approximate solutions to handle different types of received feedback. Our proposed algorithms are versatile and applicable to universal norms. Specifically, we introduce a family of Follow the Delayed Regularized Leader algorithms for feedback with full information on the loss function, a family of Delayed Mirror Descent algorithms for feedback with gradient information on the loss function and a family of Simplified Delayed Mirror Descent algorithms for feedback with the value information of the loss function's gradients at corresponding decision points. For each type of algorithm, we provide corresponding regret bounds under cases of general convexity and relative strong convexity, respectively. We also demonstrate the efficiency of each algorithm under different norms through concrete examples. Furthermore, our theoretical results are consistent with the current best bounds when degenerated to standard settings.

翻译：在在线序列决策领域，我们利用在线凸优化（OCO）框架处理带延迟的问题，其中决策的反馈可能以未知延迟到达。与先前局限于欧几里得范数和梯度信息的研究不同，我们基于近似解提出了三类延迟算法，以处理不同类型的接收反馈。所提出的算法具有通用性，适用于任意范数。具体而言，我们针对具有损失函数完整信息的反馈引入了一类“跟随延迟正则化领导者”算法，针对具有损失函数梯度信息的反馈提出了一类“延迟镜像下降”算法，并针对仅具有对应决策点损失函数梯度值信息的反馈提出了一类“简化延迟镜像下降”算法。对于每类算法，我们分别在一般凸性和相对强凸性的情形下给出了相应的遗憾界，并通过具体实例展示了不同范数下各类算法的有效性。此外，当退化至标准设置时，我们的理论结果与当前最优界一致。