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.

翻译：本文研究多用户上下文赌博机问题，其中用户通过图结构相关联，且其奖励函数同时表现出非线性特性与图同质性。我们针对用户奖励函数集合$\{f_u\}$提出了一种基于原则的联合惩罚项，将基于再生核希尔伯特空间距离的图平滑项与个体粗糙度惩罚相结合。我们的核心贡献在于证明该惩罚项等价于单一、统一的\emph{多用户再生核希尔伯特空间}中的平方范数。我们显式推导了其再生核，该核函数将图拉普拉斯矩阵与基础臂核进行了优雅融合。这种统一性使我们能够将问题重新定义为学习单个"提升"函数，从而设计了原则性算法\texttt{LK-GP-UCB}和\texttt{LK-GP-TS}，这些算法利用基于新核的高斯过程后验进行探索。我们提供了高概率遗憾界，其缩放尺度取决于多用户核的\emph{有效维度}，替代了对用户数量或环境维度的依赖。实证结果表明，在非线性场景中我们的方法优于强线性基线与无图感知基线，即使在真实奖励为线性的情况下仍保持竞争力。本研究提出了一个统一、理论坚实且实用的框架，将拉普拉斯正则化与核化赌博机相结合，实现了结构化探索。