This work establishes the first framework of federated $\mathcal{X}$-armed bandit, where different clients face heterogeneous local objective functions defined on the same domain and are required to collaboratively figure out the global optimum. We propose the first federated algorithm for such problems, named \texttt{Fed-PNE}. By utilizing the topological structure of the global objective inside the hierarchical partitioning and the weak smoothness property, our algorithm achieves sublinear cumulative regret with respect to both the number of clients and the evaluation budget. Meanwhile, it only requires logarithmic communications between the central server and clients, protecting the client privacy. Experimental results on synthetic functions and real datasets validate the advantages of \texttt{Fed-PNE} over various centralized and federated baseline algorithms.

翻译：本文建立了首个联邦$\mathcal{X}$臂赌博机框架，其中不同客户端面临定义在同一域上且异质的局部目标函数，需协作找出全局最优解。我们针对此类问题提出了首个联邦算法，命名为\texttt{Fed-PNE}。通过利用层级划分中全局目标的拓扑结构及弱光滑性质，该算法在客户端数量和评估预算两个维度上均实现了次线性累积遗憾。同时，算法仅需中央服务器与客户端之间进行对数级别的通信，有效保护了客户端隐私。在合成函数与真实数据集上的实验结果验证了\texttt{Fed-PNE}相较于多种集中式与联邦基线算法的优越性。