Greedy bi-criteria approximations for k-medians and k-means

This paper investigates the following natural greedy procedure for clustering in the bi-criterion setting: iteratively grow a set of centers, in each round adding the center from a candidate set that maximally decreases clustering cost. In the case of k-medians and k-means, the key results are as follows. • When the method considers all data points as candidate centers, then selecting O(k log(1/ε)) centers achieves cost at most 2 + ε times the optimal cost with k centers. • Alternatively, the same guarantees hold if each round samples O(k/ε) candidate centers proportionally to their cluster cost (as with kmeans++, but holding centers fixed). • In the case of k-means, considering an augmented set of nd1/εe candidate centers gives 1 + ε approximation withO(k log(1/ε)) centers, the entire algorithm takingO(dk log(1/ε)n1+d1/εe) time, where n is the number of data points in R. • In the case of Euclidean k-medians, generating a candidate set via nO(1/ε2) executions of stochastic gradient descent with adaptively determined constraint sets will once again give approximation 1 + ε with O(k log(1/ε)) centers in dk log(1/ε)nO(1/ε2) time. Ancillary results include: guarantees for cluster costs based on powers of metrics; a brief, favorable empirical evaluation against kmeans++; data-dependent bounds allowing 1 + ε in the first two bullets above, for example with k-medians over finite metric spaces. ar X iv :1 60 7. 06 20 3v 1 [ cs .D S] 2 1 Ju l 2 01 6