Dynamic Regret Reduces to Kernelized Static Regret

TL;DR

This paper presents a novel reduction of dynamic regret minimization in online convex optimization to a static regret problem in a Reproducing Kernel Hilbert Space, enabling optimal guarantees and broad applicability beyond linear losses.

Contribution

It introduces a new dynamic-to-static regret reduction using RKHS, extending guarantees to general loss sequences and enabling practical algorithms in infinite-dimensional spaces.

Findings

Achieves optimal dynamic regret bounds in linear loss settings.

Provides scale-free and directionally-adaptive regret guarantees.

Extends reduction validity to arbitrary loss sequences beyond linear losses.

Abstract

We study dynamic regret in online convex optimization, where the objective is to achieve low cumulative loss relative to an arbitrary benchmark sequence. By observing that competing with an arbitrary sequence of comparators $u_{1},\ldots,u_{T}$ in $\mathcal{W}\subseteq\mathbb{R}^{d}$ is equivalent to competing with a fixed comparator function $u:[1,T]\to \mathcal{W}$, we frame dynamic regret minimization as a static regret problem in a function space. By carefully constructing a suitable function space in the form of a Reproducing Kernel Hilbert Space (RKHS), our reduction enables us to recover the optimal $R_{T}(u_{1},\ldots,u_{T}) = \mathcal{O}(\sqrt{\sum_{t}\|u_{t}-u_{t-1}\|T})$ dynamic regret guarantee in the setting of linear losses, and yields new scale-free and directionally-adaptive dynamic regret guarantees. Moreover, unlike prior dynamic-to-static reductions -- which are valid only for linear losses -- our reduction holds for any sequence of losses, allowing us to recover $\mathcal{O}\big(\|u\|^2_{\mathcal{H}}+d_{\mathrm{eff}}(\lambda)\ln T\big)$ bounds in exp-concave and improper linear regression settings, where $d_{\mathrm{eff}}(\lambda)$ is a measure of complexity of the RKHS. Despite working in an infinite-dimensional space, the resulting reduction leads to algorithms that are computable in practice, due to the reproducing property of RKHSs.