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 `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 the previous work of Bateni et al. [12] and Malkomes et al. [31]. As a special case, we achieve the first distributed algorithm for k-median with outliers, answering an open question posed by Malkomes et al. [31]. 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 of Chen et al. which is dimension-agnostic [18]. Furthermore, we give distributed clustering algorithms which return nearly optimal solutions, provided the data satisfies the approximation stability condition of Balcan et al. [8] or the spectral stability condition of Kumar and Kannan [27]. In certain clustering applications where each machine only needs to find a clustering consistent with the global optimum, we show that no communication is necessary if the data satisfies approximation stability. ∗Authors’ addresses: [email protected], [email protected], [email protected]. This work was supported in part by NSF grants CCF-1422910, CCF-1535967, IIS-1618714, a Sloan Research Fellowship, a Microsoft Research Faculty Fellowship, a Google Research Award, and a National Defense Science & Engineering Graduate (NDSEG) fellowship. ar X iv :1 70 3. 00 83 0v 1 [ cs .D S] 2 M ar 2 01 7