This paper proposes pFedMe, a personalized federated learning algorithm that uses Moreau envelopes as clients' regularized loss functions to address the challenge of statistical diversity among clients in federated learning (FL). pFedMe decouples personalized model optimization from global model learning, enabling parallel optimization of personalized models while updating the global model similarly to standard FL algorithms like FedAvg. Theoretical analysis shows that pFedMe achieves state-of-the-art convergence rates: quadratic speedup for strongly convex objectives and sublinear speedup of order 2/3 for smooth nonconvex objectives. Experimental results demonstrate that pFedMe outperforms existing FL algorithms, including FedAvg and Per-FedAvg, in terms of convergence rate and local accuracy on both real and synthetic datasets. The algorithm's key contributions include a novel bi-level optimization formulation, leveraging the convexity-preserving and smoothness-enabled properties of Moreau envelopes for convergence analysis, and empirical validation of its effectiveness in handling statistical diversity in FL.This paper proposes pFedMe, a personalized federated learning algorithm that uses Moreau envelopes as clients' regularized loss functions to address the challenge of statistical diversity among clients in federated learning (FL). pFedMe decouples personalized model optimization from global model learning, enabling parallel optimization of personalized models while updating the global model similarly to standard FL algorithms like FedAvg. Theoretical analysis shows that pFedMe achieves state-of-the-art convergence rates: quadratic speedup for strongly convex objectives and sublinear speedup of order 2/3 for smooth nonconvex objectives. Experimental results demonstrate that pFedMe outperforms existing FL algorithms, including FedAvg and Per-FedAvg, in terms of convergence rate and local accuracy on both real and synthetic datasets. The algorithm's key contributions include a novel bi-level optimization formulation, leveraging the convexity-preserving and smoothness-enabled properties of Moreau envelopes for convergence analysis, and empirical validation of its effectiveness in handling statistical diversity in FL.