Network alternating direction method of multipliers for ultrahigh‐dimensional decentralised federated learning

Ultrahigh‐dimensional data analysis has received great achievement in recent years. When the data are stored in multiple clients and the clients can be connected only with each other through a network structure, the implementation of ultrahigh‐dimensional analysis can be numerically challenging or even infeasible. In this work, we study decentralised federated learning for ultrahigh‐dimensional data analysis, where the parameters of interest are estimated via a large amount of devices without data sharing by a network structure. In the local machines, each parallel runs gradient ascent to obtain estimators via the sparsity‐restricted constrained methods. Also, we obtain a global model by aggregating each machine's information via an alternating direction method of multipliers (ADMM) using a concave pairwise fusion penalty between different machines through a network structure. The proposed method can mitigate privacy risks from traditional machine learning, recover the sparsity and provide estimates of all regression coefficients simultaneously. Under mild conditions, we show the convergence and estimation consistency of our method. The promising performance of the method is supported by both simulated and real data examples.