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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Differentially Private k-Means via Exponential Mechanism and Max Cover
We introduce a new (ϵₚ, δₚ)-differentially private algorithm for the k-means clustering problem. Given a dataset in Euclidean space, the k-means clustering problem requires one to find k points in that space such that the sum of squares of Euclidean distances between each data point and its closest respective point among the k returned is minimised. Although there exist privacy-preserving methods with good theoretical guarantees to solve this problem, in practice it is seen that it is the additive error which dictates the practical performance of these methods. By reducing the problem to a sequence of instances of maximum coverage on a grid, we are able to derive a new method that achieves lower additive error than previous works. For input datasets with cardinality n and diameter Δ, our algorithm has an O(Δ² (k log² n log(1/δₚ)/ϵₚ + k √(d log(1/δₚ))/ϵₚ)) additive error whilst maintaining constant multiplicative error. We conclude with some experiments and find an improvement over previously implemented work for this problem.
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- Award ID(s):
- 1750716
- NSF-PAR ID:
- 10316109
- Date Published:
- Journal Name:
- Proceedings of the AAAI Conference on Artificial Intelligence
- Volume:
- 35
- Issue:
- 8
- ISSN:
- 2159-5399
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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