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

2025-06-27

AI Summary

This paper studies online regression for time-varying function sequences in a reproducing kernel Hilbert space (RKHS) under unconstrained quadratic loss. To address nonstationarity, we propose a hierarchical online prediction algorithm: the first extension of the discounted Vovk–Azoury–Warmuth framework to the nonparametric setting, integrated with random feature approximation and adaptive model selection to jointly optimize the discount factor and feature dimension. The algorithm achieves per-step time complexity of $O(T log T)$ and attains a dynamic regret bound of $O(T^{2/3} P_T^{1/3} + sqrt{T} log T)$, where $P_T$ denotes the total variation of the function sequence. Our key contributions are: (1) the first adaptive discounted nonparametric online regression framework tailored for RKHS; and (2) theoretical guarantees that simultaneously ensure computational efficiency and robust adaptation to dynamic environments.

Technology Category

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

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising

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 $O(Tln T)$, 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.

Problem

• Online regression with unconstrained quadratic loss in RKHS
• Dynamic regret optimization for time-varying functions
• Adaptive hierarchical algorithm for discounting and features

Innovation

• Hierarchical Vovk-Azoury-Warmuth forecaster with discounting
• Random feature approximation in RKHS
• Adaptive learning of discount factor