Abstract: Federated learning (FL) is a training paradigm where the clients collaboratively learn models by repeatedly sharing information without compromising much on the privacy of their local sensitive data. In this paper, we introduce \emph{federated $f$-differential privacy}, a new notion specifically tailored to the federated setting, based on the framework of Gaussian differential privacy. Federated $f$-differential privacy operates on \emph{record level}: it provides the privacy guarantee on each individual record of one client's data against adversaries. We then propose a generic private federated learning framework \fedsync that accommodates a large family of state-of-the-art FL algorithms, which provably achieves {federated $f$-differential privacy}. Finally, we empirically demonstrate the trade-off between privacy guarantee and prediction performance for models trained by \fedsync in computer vision tasks.