# On a Framework for Federated Cluster Analysis
###### Abstract  
Federated learning is becoming increasingly popular to enable automated learning in distributed networks of autonomous partners without sharing raw data. Many works focus on supervised learning, while the area of federated unsupervised learning, similar to federated clustering, is still less explored. In this paper, we introduce a federated clustering framework that solves three challenges: determine the number of global clusters in a federated dataset, obtain a partition of the data via a federated fuzzy \(c\)-means algorithm, and validate the clustering through a federated fuzzy Davies-Bouldin index. The complete framework is evaluated through numerical experiments on artificial and real-world  
medical research [12], social and behavioral sciences [13; 14], strategic management [15], or marketing [16; 17], just to name a few. In all of these areas of application, the data could be initially distributed and hard to centralize. Hence, they are all potential application areas for a federated cluster analysis framework. Ref. [18] described cluster analysis as seven steps that need to be performed in order to derive insights (Section 2.1). In federated clustering, only one step has been (explicitly) addressed, i.e., the clustering method [19; 20; 21; 22]. Other important steps, such as cluster validation or determining the number of clusters, have no federated equivalent yet.  
Federated clustering is an FL setting, where the goal is to group together (local) data points that are globally similar to each other. That is, data points are distributed among multiple clients and are clustered based on a global similarity measure, while all data remain local on client devices. Existing works largely focus on developing and applying a federated clustering method, i.e., partitioning the data given the number of (global) clusters \(K\). While being an important step in the clustering framework, the lack of more comprehensive frameworks (for example, including determination of the number of clusters and cluster validation) might hinder application in practice.  
We contribute to closing this gap by introducing a multistep federated (fuzzy) clustering analysis framework for the first time. In particular, we propose a federated version of the well-known (fuzzy) Davies-Bouldin index for cluster validation, show how to use it for the determination of the number of clusters, and apply a federated fuzzy \(c\)-means algorithm to solve the soft clustering problem. Even though independently developed, we note that our federated fuzzy \(c\)-means algorithm is closely related to other works in the area of federated clustering [19; 20; 21; 22]. It combines local fuzzy \(c\)-means on the client side with \(k\)-means on the global server side. Each idea in itself is not new, but the combination is, and we observe that our formulation is more reliable than other federated fuzzy clustering methods in case the data are non-i.i.d. To the best of our knowledge, there exists no cluster validation index for federated cluster validation yet. Moreover, no work addressing the problem of determining the number of clusters in a federated setting is known to us.  
In the remainder of this section, we demonstrate the need for a federated clustering framework. In subsequent sections, we review relevant works from nonfederated and federated cluster research in Section 2.1. In Section 2.2, we introduce the individual pieces of our framework before fitting them together. Section 3 contains an experimental evaluation on real-world and artificial data to demonstrate the framework's effectiveness and uncover shortcomings. Finally, Section 4 concludes this work.