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

2026-02-06

AI Summary

This work addresses dynamic regret in unconstrained online convex optimization where the movement cost coefficients $\lambda_t$ vary arbitrarily over time. It proposes the first comparator-adaptive algorithm that unifies the treatment of both static and dynamic regret, while naturally accommodating complex scenarios such as delayed feedback and time-varying memory. By leveraging a path-length-based measure and a problem reduction technique, the method achieves optimal performance without requiring prior knowledge of problem parameters. Specifically, it attains a dynamic regret bound of $\widetilde{O}\left(\sqrt{(1+P_T)(T+\sum \lambda_t)}\right)$ across standard OCO, delayed feedback, and time-varying memory settings, thereby highlighting the critical role of first-order dependence on movement costs.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• dynamic regret
• online convex optimization
• time-varying movement costs
• delayed feedback
• comparator-adaptive

Innovation

• dynamic regret
• comparator-adaptive
• time-varying movement costs
• delayed feedback
• online convex optimization