A hierarchical Vovk-Azoury-Warmuth forecaster with discounting for online regression in RKHS

TL;DR

This paper introduces a hierarchical, fully adaptive online regression algorithm in RKHS that learns the discount factor and number of features, achieving near-optimal dynamic regret with manageable computational complexity.

Contribution

It extends the discounted Vovk-Azoury-Warmuth framework to non-parametric RKHS using random features, with an adaptive algorithm that learns key parameters.

Findings

Achieves expected dynamic regret of O(T^{2/3} P_T^{1/3} + √T ln T)

Per-iteration complexity is O(T ln T)

Learns both discount factor and number of random features adaptively

Abstract

We study the problem of online regression with the unconstrained quadratic loss against a time-varying sequence of functions from a Reproducing Kernel Hilbert Space (RKHS). Recently, Jacobsen and Cutkosky (2024) introduced a discounted Vovk-Azoury-Warmuth (DVAW) forecaster that achieves optimal dynamic regret in the finite-dimensional case. In this work, we lift their approach to the non-parametric domain by synthesizing the DVAW framework with a random feature approximation. We propose a fully adaptive, hierarchical algorithm, which we call H-VAW-D (Hierarchical Vovk-Azoury-Warmuth with Discounting), that learns both the discount factor and the number of random features. We prove that this algorithm, which has a per-iteration computational complexity of $O(T\ln T)$, achieves an expected dynamic regret of $O(T^{2/3}P_T^{1/3} + \sqrt{T}\ln T)$, where $P_T$ is the functional path length of a comparator sequence.