Non-stationary Online Convex Optimization with Arbitrary Delays

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 DOGD can attain an $O(\sqrt{dT}(P_T+1)$ dynamic regret bound in the worst case, where $d$ is the maximum delay, $T$ is the time horizon, and $P_T$ is the path length of comparators. More importantly, in case delays do not change the arrival order of gradients, it can automatically reduce the dynamic regret to $O(\sqrt{S}(1+P_T))$, where $S$ is the sum of delays. Furthermore, we develop an improved algorithm, which can reduce those dynamic regret bounds achieved by DOGD to $O(\sqrt{dT(P_T+1)})$ and $O(\sqrt{S(1+P_T)})$, respectively. The essential 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 both cases by deriving a matching lower bound.