  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.
