Abstract: 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.