In this paper, we tackle the problem of distributed K-means in a Byzantine framework where P nodes compute the K centroids at each iteration, which are then aggregated by a server node. Among these P nodes, a fraction \(\alpha \) (where \(\alpha < 1/2\) ) are Byzantine, i.e they compute K erroneous centroids which distort the distributed K-means algorithm. To ensure the convergence of K-means, we propose to use a centroid aggregation rule called Fast Aggregation against Byzantine Attacks (FABA), designed specifically for aggregating gradient vectors in a distributed environment. Distributed K-means is used in federated learning and when data privacy is concerned and even in these cases it is important to be able to provide good clusters of the data. Our research explores the convergence of distributed K-means with Byzantine nodes. By showing that K-means can be seen as a gradient-based method, we conduct a theoretical study on the convergence of distributed K-means by providing upper bounds of the convergence of the latter in the presence of Byzantine centroids. We also show that by applying the FABA to K-means with approx. 50% of byzantine nodes, our distributed K-means converges to clusters fairly close to the centralized K-means.
