Federated learning is a new distributed learning framework to deal with data isolation and privacy protection. This paper focuses on the second-order gradient federated learning algorithm. It carries out the research on the federated learning Newton's algorithm for the more practical scheme that only some users can participate in the federated learning task. The convergence of the algorithm is proved theoretically. By using the McDiarmid inequality, the trade-off of the number of participating users, the convergence speed of the algorithm and the final convergence error are theoretically explained. The experimental results show that the algorithm can converge quickly and efficiently, effectively reduce the communication cost of federated learning, and prove the effectiveness, practicality and theoretical correctness of the algorithm.