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# Fast Algorithms for Distributed k-Clustering with Outliers

*Proceedings of the 40th International Conference on Machine Learning*, PMLR 202:13845-13868, 2023.

#### Abstract

In this paper, we study the $k$-clustering problems with outliers in distributed setting. The current best results for the distributed $k$-center problem with outliers have quadratic local running time with communication cost dependent on the aspect ratio $\Delta$ of the given instance, which may constraint the scalability of the algorithms for handling large-scale datasets. To achieve better communication cost for the problem with faster local running time, we propose an inliers-recalling sampling method, which avoids guessing the optimal radius of the given instance, and can achieve a 4-round bi-criteria $(14(1+\epsilon),1+\epsilon)$-approximation with linear local running time in the data size and communication cost independent of the aspect ratio. To obtain a more practical algorithm for the problem, we propose another space-narrowing sampling method, which automatically adjusts the sample size to adapt to different outliers distributions on each machine, and can achieve a 2-round bi-criteria $(14(1+\epsilon),1+\epsilon)$-approximation with communication cost independent of the number of outliers. We show that, if the data points are randomly partitioned across machines, our proposed sampling-based methods can be extended to the $k$-median/means problems with outliers, and can achieve $(O(\frac{1}{\epsilon^2}),1+\epsilon)$-approximation with communication cost independent of the number of outliers. Empirical experiments suggest that the proposed 2-round distributed algorithms outperform other state-of-the-art algorithms.