In this paper, the content popularity prediction problem in fog radio access networks (F-RANs) is investigated. In order to obtain accurate prediction with low complexity, we propose a novel context-aware popularity prediction policy based on federated learning. Firstly, user preference learning is applied by considering that users prefer to request the contents they are interested in. Then, users' context information is utilized to cluster users efficiently by adaptive context space partitioning. After that, we formulate a popularity prediction optimization problem to learn the local model parameters using the stochastic variance reduced gradient (SVRG) algorithm. Finally, federated learning based model integration is proposed to construct the global popularity prediction model based on local models by combining the distributed approximate Newton (DANE) algorithm with SVRG. Our proposed popularity prediction policy not only predicts content popularity accurately, but also significantly reduces computational complexity. Simulation results show that our proposed policy increases the cache hit rate by up to 21.5 % compared to the traditional policies.