Personalized federated learning, Row-wise fusion regularization, Multivariate modeling, Sparse estimation

We study personalized federated learning for multivariate responses where client models are heterogeneous yet share variable-level structure. Existing entry-wise penalties ignore cross-response dependence, while matrix-wise fusion over-couples clients. We propose a Sparse Row-wise Fusion (SROF) regularizer that clusters row vectors across clients and induces within-row sparsity, and we develop RowFed, a communication-efficient federated algorithm that embeds SROF into a linearized ADMM framework with privacy-preserving partial participation. Theoretically, we establish an oracle property for SROF-achieving correct variable-level group recovery with asymptotic normality-and prove convergence of RowFed to a stationary solution. Under random client participation, the iterate gap contracts at a rate that improves with participation probability. Empirically, simulations in heterogeneous regimes show that RowFed consistently lowers estimation and prediction error and strengthens variable-level cluster recovery over NonFed, FedAvg, and a personalized matrix-fusion baseline. A real-data study further corroborates these gains while preserving interpretability. Together, our results position row-wise fusion as an effective and transparent paradigm for large-scale personalized federated multivariate learning, bridging the gap between entry-wise and matrix-wise formulations.