Locally Private k-Means Clustering
This work addresses privacy-preserving data analysis for sensitive datasets, offering a significant improvement over prior methods in the local model, though it is incremental within the field of differential privacy.
The paper tackles the Euclidean k-means clustering problem under local differential privacy, reducing the additive error from approximately n^(2/3+a) to n^(1/2+a) while maintaining O(1) multiplicative error, and proves this additive error is nearly optimal with a lower bound of approximately sqrt(n).
We design a new algorithm for the Euclidean $k$-means problem that operates in the local model of differential privacy. Unlike in the non-private literature, differentially private algorithms for the $k$-means objective incur both additive and multiplicative errors. Our algorithm significantly reduces the additive error while keeping the multiplicative error the same as in previous state-of-the-art results. Specifically, on a database of size $n$, our algorithm guarantees $O(1)$ multiplicative error and $\approx n^{1/2+a}$ additive error for an arbitrarily small constant $a>0$. All previous algorithms in the local model had additive error $\approx n^{2/3+a}$. Our techniques extend to $k$-median clustering. We show that the additive error we obtain is almost optimal in terms of its dependency on the database size $n$. Specifically, we give a simple lower bound showing that every locally-private algorithm for the $k$-means objective must have additive error at least $\approx\sqrt{n}$.