Distributed K-Clustering with Exponential Convergence

L. Wang, Roberto Tron

Abstract: We consider the problem of distributed K-clustering in a network of N agents where each agent holds a part of a dataset. The goal for the agents is to agree, in a distributed manner, on a set of K cluster centers that effectively summarize the entire dataset (e.g., by identifying points where the data is most concentrated), without explicitly exchanging all the data between all the agents or with a central node.First, we consider the special case when there is only one cluster (K = 1). We propose the Distributed 1-clustering Multiplier Algorithm (Di1MA), which ensures exponential convergence of all the copies of the cluster center at each agent to a common value by combining the properties of consensus algorithms and of the augmented Lagrangian method.Second, based on the previous case, we propose an algorithm called Distributed K-clustering Multiplier Algorithm (DiKMA) to solve the case with K > 1 cluster centers. For this case, using ideas from switching algorithms, our solution alternates q steps of updates of the cluster centers with one update of the local assignments of data to cluster centers. We show that there must exist a q large enough such that the algorithm ensures the exponential convergence of the K cluster centers.We provide simulations to verify the proposed algorithms, and illustrate some of their properties.