In this paper, we formulate a Collaborative Pure Exploration in Kernel Bandit problem (CoPE-KB), which provides a novel model for multi-agent multi-task decision making under limited communication and general reward functions, and is applicable to many online learning tasks, e.g., recommendation systems and network scheduling. We consider two settings of CoPE-KB, i.e., Fixed-Confidence (FC) and Fixed-Budget (FB), and design two optimal algorithms CoopKernelFC (for FC) and CoopKernelFB (for FB). Our algorithms are equipped with innovative and efficient kernelized estimators to simultaneously achieve computation and communication efficiency. Matching upper and lower bounds under both the statistical and communication metrics are established to demonstrate the optimality of our algorithms. The theoretical bounds successfully quantify the influences of task similarities on learning acceleration and only depend on the effective dimension of the kernelized feature space. Our analytical techniques, including data dimension decomposition, linear structured instance transformation and (communication) round-speedup induction, are novel and applicable to other bandit problems. Empirical evaluations are provided to validate our theoretical results and demonstrate the performance superiority of our algorithms.

翻译：本文提出了核化赌博机中的协同纯探索问题（CoPE-KB），该模型为有限通信与通用奖励函数下的多智能体多任务决策提供了新颖框架，可应用于推荐系统、网络调度等在线学习任务。我们考虑了CoPE-KB的两种设定，即固定置信度（FC）与固定预算（FB），并分别设计了最优算法CoopKernelFC（用于FC）与CoopKernelFB（用于FB）。所提算法配备了创新高效的核化估计器，可同时实现计算效率与通信效率的提升。我们在统计和通信两种度量下建立了匹配的上界与下界，证明了算法的最优性。理论界限成功量化了任务相似性对学习加速的影响，且仅依赖于核化特征空间的有效维度。我们的分析技术，包括数据维度分解、线性结构化实例变换和（通信）轮次加速归纳法，具有创新性且可推广至其他赌博机问题。通过实证评估验证了理论结果，并展示了算法性能的优越性。