Chang Yao

Yuanyu Wan, Chang Yao, Mingli Song et al.

We investigate bandit convex optimization (BCO) with delayed feedback, where only the loss value of the action is revealed under an arbitrary delay. Let $n,T,\bar{d}$ denote the dimensionality, time horizon, and average delay, respectively. Previous studies have achieved an $O(\sqrt{n}T^{3/4}+(n\bar{d})^{1/3}T^{2/3})$ regret bound for this problem, whose delay-independent part matches the regret of the classical non-delayed bandit gradient descent algorithm. However, there is a large gap between its delay-dependent part, i.e., $O((n\bar{d})^{1/3}T^{2/3})$, and an existing $Ω(\sqrt{\bar{d}T})$ lower bound. In this paper, we illustrate that this gap can be filled in the worst case, where $\bar{d}$ is very close to the maximum delay $d$. Specifically, we first develop a novel algorithm, and prove that it enjoys a regret bound of $O(\sqrt{n}T^{3/4}+\sqrt{dT})$ in general. Compared with the previous result, our regret bound is better for $d=O((n\bar{d})^{2/3}T^{1/3})$, and the delay-dependent part is tight in the worst case. The primary idea is to decouple the joint effect of the delays and the bandit feedback on the regret by carefully incorporating the delayed bandit feedback with a blocking update mechanism. Furthermore, we show that the proposed algorithm can improve the regret bound to $O((nT)^{2/3}\log^{1/3}T+d\log T)$ for strongly convex functions. Finally, if the action sets are unconstrained, we demonstrate that it can be simply extended to achieve an $O(n\sqrt{T\log T}+d\log T)$ regret bound for strongly convex and smooth functions.

Yuanyu Wan, Chang Yao, Yitao Ma et al.

Although online convex optimization (OCO) under arbitrary delays has received increasing attention recently, previous studies focus on stationary environments with the goal of minimizing static regret. In this paper, we investigate the delayed OCO in non-stationary environments, and choose dynamic regret with respect to any sequence of comparators as the performance metric. To this end, we first propose an algorithm called Mild-OGD for the full-information case, where delayed gradients are available. The basic idea is to maintain multiple experts in parallel, each performing a gradient descent step with different learning rates for every delayed gradient according to their arrival order, and utilize a meta-algorithm to track the best one based on their delayed performance. Despite the simplicity of this idea, our novel analysis shows that the dynamic regret of Mild-OGD can be automatically bounded by $O(\sqrt{\bar{d}T(P_T+1)})$ under the in-order assumption 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. Moreover, we demonstrate that the result in the worst case is optimal by deriving a matching lower bound. Finally, we develop a bandit variant of Mild-OGD for a more challenging case with only delayed loss values. Interestingly, we prove that under a relatively large amount of delay, our bandit algorithm even enjoys the best dynamic regret bound of existing non-delayed bandit algorithms.