  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.
