Laplacian Kernelized Bandit

2026-01-01

arXiv.org

AI Summary

This work addresses the challenge of multi-user contextual bandits where users exhibit graph-structured relationships and the reward function is both nonlinear and graph-homophilous. The authors propose a unified learning framework based on joint regularization, introducing a novel regularizer that combines graph smoothness and individual roughness penalties. They establish, for the first time, its equivalence to a norm in a single multi-user reproducing kernel Hilbert space (RKHS) and explicitly construct a composite kernel that integrates the graph Laplacian with the base arm kernel. Building on this, they develop two efficient exploration algorithms, LK-GP-UCB and LK-GP-TS. Theoretical analysis yields high-probability regret bounds dependent only on the effective dimension of the multi-user kernel, eliminating dependence on the number of users or ambient dimensionality. Experiments demonstrate significant superiority over existing baselines in nonlinear settings while maintaining competitive performance in linear cases.

Technology Category

• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Adversarial Learning
• Multiagent Systems: Mechanism Design

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study multi-user contextual bandits where users are related by a graph and their reward functions exhibit both non-linear behavior and graph homophily. We introduce a principled joint penalty for the collection of user reward functions $\{f_u\}$, combining a graph smoothness term based on RKHS distances with an individual roughness penalty. Our central contribution is proving that this penalty is equivalent to the squared norm within a single, unified \emph{multi-user RKHS}. We explicitly derive its reproducing kernel, which elegantly fuses the graph Laplacian with the base arm kernel. This unification allows us to reframe the problem as learning a single''lifted''function, enabling the design of principled algorithms, \texttt{LK-GP-UCB} and \texttt{LK-GP-TS}, that leverage Gaussian Process posteriors over this new kernel for exploration. We provide high-probability regret bounds that scale with an \emph{effective dimension} of the multi-user kernel, replacing dependencies on user count or ambient dimension. Empirically, our methods outperform strong linear and non-graph-aware baselines in non-linear settings and remain competitive even when the true rewards are linear. Our work delivers a unified, theoretically grounded, and practical framework that bridges Laplacian regularization with kernelized bandits for structured exploration.

Problem

• multi-user contextual bandits
• graph homophily
• non-linear rewards
• Laplacian regularization
• structured exploration

Innovation

• Laplacian regularization
• kernelized bandits
• multi-user RKHS
• graph homophily
• Gaussian Process