In this paper, we investigate federated clustering (FedC) problem, that aims to accurately partition unlabeled data samples distributed over massive clients into finite clusters under the orchestration of a parameter server, meanwhile considering data privacy. Though it is an NP-hard optimization problem involving real variables denoting cluster centroids and binary variables denoting the cluster membership of each data sample, we judiciously reformulate the FedC problem into a non-convex optimization problem with only one convex constraint, accordingly yielding a soft clustering solution. Then a novel FedC algorithm using differential privacy (DP) technique, referred to as DP-FedC, is proposed in which partial clients participation and multiple local model updating steps are also considered. Furthermore, various attributes of the proposed DP-FedC are obtained through theoretical analyses of privacy protection and convergence rate, especially for the case of non-identically and independently distributed (non-i.i.d.) data, that ideally serve as the guidelines for the design of the proposed DP-FedC. Then some experimental results on two real datasets are provided to demonstrate the efficacy of the proposed DP-FedC together with its much superior performance over some state-of-the-art FedC algorithms, and the consistency with all the presented analytical results.
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