Given a data set of size n in d'-dimensional Euclidean space, the k-means problem asks for a set of k points (called centers) such that the sum of the l_2^2-distances between the data points and the set of centers is minimized. Previous work on this problem in the local differential privacy setting shows how to achieve multiplicative approximation factors arbitrarily close to optimal, but suffers high additive error. The additive error has also been seen to be an issue in implementations of differentially private k-means clustering algorithms in both the central and local settings. In this work, we introduce a new locally private k-means clustering algorithm that achieves near-optimal additive error whilst retaining constant multiplicative approximation factors and round complexity. Concretely, given any c>sqrt(2), our algorithm achieves O(k^(1 + O(1/(2c^2-1))) * sqrt(d' n) * log d' * poly log n) additive error with an O(c^2) multiplicative approximation factor.
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Polylogarithmic Sketches for Clustering
Given n points in 𝓁_p^d, we consider the problem of partitioning points into k clusters with associated centers. The cost of a clustering is the sum of p-th powers of distances of points to their cluster centers. For p ∈ [1,2], we design sketches of size poly(log(nd),k,1/ε) such that the cost of the optimal clustering can be estimated to within factor 1+ε, despite the fact that the compressed representation does not contain enough information to recover the cluster centers or the partition into clusters. This leads to a streaming algorithm for estimating the clustering cost with space poly(log(nd),k,1/ε). We also obtain a distributed memory algorithm, where the n points are arbitrarily partitioned amongst m machines, each of which sends information to a central party who then computes an approximation of the clustering cost. Prior to this work, no such streaming or distributed-memory algorithm was known with sublinear dependence on d for p ∈ [1,2).
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- Award ID(s):
- 2002201
- NSF-PAR ID:
- 10529463
- Editor(s):
- Bojańczyk, Mikołaj; Merelli, Emanuela; Woodruff, David P
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 229
- ISSN:
- 1868-8969
- ISBN:
- 978-3-95977-235-8
- Page Range / eLocation ID:
- 229-229
- Subject(s) / Keyword(s):
- sketching clustering Theory of computation → Sketching and sampling
- Format(s):
- Medium: X Size: 20 pages; 847506 bytes Other: application/pdf
- Size(s):
- 20 pages 847506 bytes
- Right(s):
- Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
- Sponsoring Org:
- National Science Foundation
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