General and Robust Communication-Efficient Algorithms for Distributed Clustering

As datasets become larger and more distributed, algorithms for distributed clustering have become more and more important. In this work, we present a general framework for designing distributed clustering algorithms that are robust to outliers. Using our framework, we give a distributed approximation algorithm for k-means, k-median, or generally any L_p objective, with z outliers and/or balance constraints, using O(m(k+z)(d+log n)) bits of communication, where m is the number of machines, n is the size of the point set, and d is the dimension. This generalizes and improves over previous work of Bateni et al. and Malkomes et al. As a special case, we achieve the first distributed algorithm for k-median with outliers, answering an open question posed by Malkomes et al. For distributed k-means clustering, we provide the first dimension-dependent communication complexity lower bound for finding the optimal clustering. This improves over the lower bound from Chen et al. which is dimension-agnostic. Furthermore, we give distributed clustering algorithms which return nearly optimal solutions, provided the data satisfies the approximation stability condition of Balcan et al. or the spectral stability condition of Kumar and Kannan.