  Online convex optimization (OCO) with arbitrary delays, in which gradients or
other information of functions could be arbitrarily delayed, has received
increasing attention recently. Different from previous studies that focus on
stationary environments, this paper investigates the delayed OCO in
non-stationary environments, and aims to minimize the dynamic regret with
respect to any sequence of comparators. To this end, we first propose a simple
algorithm, namely DOGD, which performs a gradient descent step for each delayed
gradient according to their arrival order. Despite its simplicity, our novel
analysis shows that the dynamic regret of DOGD can be automatically bounded by
$O(\sqrt{\bar{d}T}(P_T+1))$ under mild assumptions, and $O(\sqrt{dT}(P_T+1))$
in the worst case, where $\bar{d}$ and $d$ denote the average and maximum delay
respectively, $T$ is the time horizon, and $P_T$ is the path-length of
comparators. Furthermore, we develop an improved algorithm, which reduces those
dynamic regret bounds achieved by DOGD to $O(\sqrt{\bar{d}T(P_T+1)})$ and
$O(\sqrt{dT(P_T+1)})$, respectively. The key idea is to run multiple DOGD with
different learning rates, and utilize a meta-algorithm to track the best one
based on their delayed performance. Finally, we demonstrate that our improved
algorithm is optimal in the worst case by deriving a matching lower bound.
