Title: Personalized federated composite learning with forward-backward envelopes. Abstract: Federated composite optimization (FCO) is an optimization problem in federated learning whose loss function  contains a non-smooth regularizer. It arises naturally in the applications of federated learning (FL) that involve  requirements such as sparsity, low rankness, and monotonicity. In this study, we propose a personalization method, called pFedFBE, for FCO by using forward-backward envelope (FBE) as clients’ loss functions. With FBE, we not only decouple the personalized model from the global model, but also allow personalized models to be smooth and easily optimized. In spite of the nonsmoothness of FCO, pFedFBE shows the same convergence complexity results as FedAvg for FL with unconstrained smooth objectives. Numerical experiments are shown to demonstrate the effectiveness of our proposed method.