Smoothed Online Convex Optimization with Delayed Feedback

Abstract

Smoothed online convex optimization (SOCO), in which the online player incurs both a hitting cost and a switching cost for changing its decisions, has garnered significant attention in recent years. While existing studies typically assume that the gradient information is revealed immediately, such an assumption may not hold in some real-world applications. To overcome this limitation, we investigate SOCO with delayed feedback, and develop two online algorithms that can minimize the dynamic regret with switching cost. Firstly, we extend Mild-OGD, an existing algorithm that adopts the meta-expert framework for online convex optimization with delayed feedback, to account for switching cost. Specifically, we analyze the switching cost in the expert-algorithm of Mild-OGD, and then modify its meta-algorithm to incorporate this cost when assigning the weight to each expert. We demonstrate that our proposed method, Smelt-DOGD can achieve an O(√(dT(P_T+1))) dynamic regret bound with switching cost, where d is the maximum delay and P_T is the path-length. Secondly, we develop an efficient variant to reduce the number of projections per round from O(log T) to 1, yet maintaining the same theoretical guarantee. The key idea is to construct a new surrogate loss defined over a simpler domain for expert-algorithms so that these experts do not need to perform the complex projection operations in each round. Finally, we conduct experiments to validate the effectiveness and efficiency of our algorithms.