To address the challenges of statistical heterogeneity across clients and within-client data contamination in federated learning, we propose a robust and sparse personalized federated learning method based on the gamma-divergence, termed Gamma-PFL. Specifically, we combine a loss function based on the gamma-divergence with fusion and sparse regularization terms to exploit similarities in regression coefficients across clients while enabling robust estimation of personalized models and variable selection. Furthermore, to efficiently solve the estimation problem in the federated server--client system, we develop an optimization algorithm that combines the Alternating Direction Method of Multipliers with the Majorization--Minimization algorithm. Numerical simulations showed that Gamma-PFL maintained relatively stable estimation accuracy as the contamination proportion or outlier magnitude increased, demonstrating its robustness to within-client data contamination. In addition, a real data analysis showed that Gamma-PFL achieved predictive performance comparable to or better than that of an existing personalized federated learning method.

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