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

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 $\widetilde{\mathcal{O}}(\sqrt{(M^2+MP_T)(T+\sum_t \lambda_t)})$ regret, where $P_T$ is the path length of the comparator sequence over $T$ rounds and $M$ is the maximal comparator norm. Our result recovers the optimal adaptive rates for both static and dynamic regret in OCO as the 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.

We study sequential decision problems where the losses are convex functions and there is a "transaction cost" associated with changing decisions. We develop the first algorithms for the unconstrained setting which optimally adapt to non-stationarity (measured in terms of the path-length of an arbitrary benchmark sequence). We also reveal a natural connection between this setting, delayed-feedback problems, and problems in which losses depend on a time-varying history of past decisions, and we extend our results to these settings.