Parameter-free Dynamic Regret: Time-varying Movement Costs, Delayed Feedback, and Memory

arXiv:2602.06902v1 Announce Type: cross
Abstract: In this paper, we study dynamic regret in unconstrained online convex optimization (OCO) with movement costs. Specifically, we generalize the standard setting by allowing the movement cost coefficients $lambda_t$ to vary arbitrarily over time. Our main contribution is a novel algorithm that establishes the first comparator-adaptive dynamic regret bound for this setting, guaranteeing $widetilde{mathcal{O}}(sqrt{(1+P_T)(T+sum_t lambda_t)})$ regret, where $P_T$ is the path length of the comparator sequence over $T$ rounds. This recovers the optimal guarantees for both static and dynamic regret in standard OCO as a special case where $lambda_t=0$ for all rounds. To demonstrate the versatility of our results, we consider two applications: OCO with delayed feedback and OCO with time-varying memory. We show that both problems can be translated into time-varying movement costs, establishing a novel reduction specifically for the delayed feedback setting that is of independent interest. A crucial observation is that the first-order dependence on movement costs in our regret bound plays a key role in enabling optimal comparator-adaptive dynamic regret guarantees in both settings.