Dynamic Regret for Online Regression in RKHS via Discounted VAW and Subspace Approximation

2026-04-27

AI Summary

This work addresses dynamic regret minimization in online regression over a reproducing kernel Hilbert space (RKHS) when the optimal comparator sequence varies over time. The authors extend the finite-dimensional discounted Vovk–Azoury–Warmuth (VAW) algorithm to RKHS by running an ensemble of discounted VAW predictors on adaptively constructed finite-dimensional subspaces. A unified orthogonal truncation framework is introduced to accommodate various approximation strategies—including explicit feature expansions, Mercer spectral truncation, and kernel section projections—under a single analysis. The approach applies to kernels such as Gaussian, analytic dot-product, and Matérn, and establishes dynamic regret bounds that depend on the comparator’s path length and the decay rate of kernel eigenvalues, with distinct guarantees under both fast and slow learning rates.

Abstract

We study online regression with the square loss in a reproducing kernel Hilbert space under a dynamic regret criterion. The learner is compared with a time-varying comparator sequence, and the bounds depend on its path length in the RKHS norm. The proposed method transfers the finite-dimensional discounted Vovk--Azoury--Warmuth approach of Jacobsen \& Cutkosky (2024) to the RKHS setting by means of finite-dimensional subspace approximations. For a fixed subspace, we run a VAW-based ensemble of discounted VAW forecasters over a geometric grid of discount factors. The additional approximation error is controlled by the uniform projection error of kernel sections. We then introduce a general orthogonal truncation method: starting from a feature expansion of the kernel, we construct the associated RKHS by introducing an inner product that makes the feature functions orthonormal, and then use the spans of the first basis functions as finite-dimensional approximation spaces. The resulting subspace reduction is applied to several approximation schemes. Explicit feature expansions yield fast-regime bounds for Gaussian and analytic dot-product kernels. Mercer truncations provide a spectral approximation method and lead to dynamic regret bounds in fast and slow regimes, depending on the eigenvalue decay. Finally, we study subspaces spanned by kernel sections and apply this construction to Matérn kernels.

Problem

• online regression
• dynamic regret
• RKHS
• time-varying comparator
• subspace approximation

Innovation

• dynamic regret
• online regression
• RKHS
• subspace approximation
• discounted VAW