  In this work, we study the personalized federated $\mathcal{X}$-armed bandit
problem, where the heterogeneous local objectives of the clients are optimized
simultaneously in the federated learning paradigm. We propose the
\texttt{PF-PNE} algorithm with a unique double elimination strategy, which
safely eliminates the non-optimal regions while encouraging federated
collaboration through biased but effective evaluations of the local objectives.
The proposed \texttt{PF-PNE} algorithm is able to optimize local objectives
with arbitrary levels of heterogeneity, and its limited communications protects
the confidentiality of the client-wise reward data. Our theoretical analysis
shows the benefit of the proposed algorithm over single-client algorithms.
Experimentally, \texttt{PF-PNE} outperforms multiple baselines on both
synthetic and real life datasets.
