An Equivalence Between Static and Dynamic Regret Minimization (2406.01577v2)

Abstract: We study the problem of dynamic regret minimization in online convex optimization, in which the objective is to minimize the difference between the cumulative loss of an algorithm and that of an arbitrary sequence of comparators. While the literature on this topic is very rich, a unifying framework for the analysis and design of these algorithms is still missing. In this paper we show that for linear losses, dynamic regret minimization is equivalent to static regret minimization in an extended decision space. Using this simple observation, we show that there is a frontier of lower bounds trading off penalties due to the variance of the losses and penalties due to variability of the comparator sequence, and provide a framework for achieving any of the guarantees along this frontier. As a result, we also prove for the first time that adapting to the squared path-length of an arbitrary sequence of comparators to achieve regret $R_{T}(u_{1},\dots,u_{T})\le O(\sqrt{T\sum_{t} |u_{t}-u_{t+1}|^{2}})$ is impossible. However, using our framework we introduce an alternative notion of variability based on a locally-smoothed comparator sequence $\bar u_{1}, \dots, \bar u_{T}$, and provide an algorithm guaranteeing dynamic regret of the form $R_{T}(u_{1},\dots,u_{T})\le \tilde O(\sqrt{T\sum_{i}|\bar u_{i}-\bar u_{i+1}|^{2}})$, while still matching in the worst case the usual path-length dependencies up to polylogarithmic terms.

• The paper presents an equivalence between static and dynamic regret minimization by converting dynamic regret problems into static ones within an extended decision space.
• It employs mathematical tools like Haar wavelet transforms and Fenchel conjugates to decouple variance from comparator variability, achieving O(d log T) computations per round.
• The study bridges static and dynamic optimization theories, opening pathways for advanced algorithm designs and future research on non-stationary data streams.

An Equivalence Between Static and Dynamic Regret Minimization

The paper "An Equivalence Between Static and Dynamic Regret Minimization" by Andrew Jacobsen and Francesco Orabona discusses a significant contribution to the field of online convex optimization (OCO) by presenting a novel equivalence between static and dynamic regret minimization. This work is crucial for understanding how one might unify the analysis and design of algorithms for these traditionally distinct objectives within the OCO framework.

Summary and Core Findings

The authors investigate the problem of dynamic regret minimization, which aims to minimize the difference between the cumulative loss of an algorithm and that of a changing sequence of comparators, as opposed to the more conventional static regret minimization concerning a fixed comparator. They demonstrate that dynamic regret minimization can be converted into a static regret problem by operating within an extended decision space. This innovative reduction allows techniques developed for static regret to be applied to dynamic scenarios.

A primary outcome of this research is the introduction of a lower bound frontier that quantifies trade-offs between penalties attributable to the variance of loss functions and the variability of comparator sequences. Notably, the authors prove that it is infeasible to achieve regret bounds that adapt to the squared path-length of arbitrary comparator sequences. However, they identify an alternative measure related to locally-smoothed squared path-lengths, which admits more favorable adaptation, avoiding worst-case path-length dependency penalties often encountered in prior literature.

Analytical Insights

The authors extensively use mathematical tools like the Kronecker product, Haar wavelet transforms, and Fenchel conjugates to derive their results. Notably, they apply Haar matrices to achieve the reduced variability measure, yielding a bound for dynamic regret that decouples variance and variability penalties. This approach results in computational efficiency, requiring only $O(d\log T)$ computations per round, a feat that aligns with existing dynamic regret algorithms but offers improved theoretical guarantees.

Implications and Future Directions

Practically, this research provides a new framework leaders to design dynamic regret algorithms by selecting suitable norm pairs $(\|\cdot\|, \|\cdot\|_*)$. Theoretically, it challenges the community to explore different notions of comparator variability and encourage the use of infimal convolution in regret analysis, illuminating potential in extracting further decoupled guarantees. The equivalence presents opportunities to transfer rich theoretical developments from static regret studies to the dynamic domain, bridging a gap between two historically discrete areas.

Future research could extend these insights to settings beyond convex optimization, including non-linear or non-convex environments, potentially exploring links with other machine learning paradigms. Furthermore, new advancements in understanding the implications of these results could spur development in handling real-world non-stationary data streams more robustly and efficiently, reflecting the ever-evolving nature of data in practical applications.