Online Convex Optimization with Continuous Switching Constraint
This addresses sequential decision-making problems where switching costs matter, such as in server management, but it is incremental as it extends existing online convex optimization frameworks.
The paper tackles online convex optimization with a continuous switching constraint to control costs from decision changes, achieving minimax optimal regret bounds of order Ω(√T) for certain budgets and O(log T) for strongly convex functions.
In many sequential decision making applications, the change of decision would bring an additional cost, such as the wear-and-tear cost associated with changing server status. To control the switching cost, we introduce the problem of online convex optimization with continuous switching constraint, where the goal is to achieve a small regret given a budget on the \emph{overall} switching cost. We first investigate the hardness of the problem, and provide a lower bound of order $Ω(\sqrt{T})$ when the switching cost budget $S=Ω(\sqrt{T})$, and $Ω(\min\{\frac{T}{S},T\})$ when $S=O(\sqrt{T})$, where $T$ is the time horizon. The essential idea is to carefully design an adaptive adversary, who can adjust the loss function according to the cumulative switching cost of the player incurred so far based on the orthogonal technique. We then develop a simple gradient-based algorithm which enjoys the minimax optimal regret bound. Finally, we show that, for strongly convex functions, the regret bound can be improved to $O(\log T)$ for $S=Ω(\log T)$, and $O(\min\{T/\exp(S)+S,T\})$ for $S=O(\log T)$.