🤖 AI Summary
This work studies differentially private (DP) $k$-means and $k$-median clustering over data streams, requiring single-pass processing, sublinear space complexity $operatorname{poly}(k,d,log T)$, and continual output of clusterings at each time step. We propose the first plug-and-play DP streaming clustering framework, enabling black-box integration of offline DP clustering or coreset algorithms, and achieving end-to-end differential privacy for the first time. Our algorithm attains a constant multiplicative approximation factor and $operatorname{poly}(k,d,log T)$ additive error. Its space complexity is $ ilde{O}(operatorname{poly}(k,d,log T))$ or $ ilde{O}(k^{1.5} cdot operatorname{poly}(d,log T))$, making it suitable for real-time, privacy-preserving clustering of high-dimensional Euclidean data streams.
📝 Abstract
The streaming model is an abstraction of computing over massive data streams, which is a popular way of dealing with large-scale modern data analysis. In this model, there is a stream of data points, one after the other. A streaming algorithm is only allowed one pass over the data stream, and the goal is to perform some analysis during the stream while using as small space as possible. Clustering problems (such as $k$-means and $k$-median) are fundamental unsupervised machine learning primitives, and streaming clustering algorithms have been extensively studied in the past. However, since data privacy becomes a central concern in many real-world applications, non-private clustering algorithms are not applicable in many scenarios. In this work, we provide the first differentially private streaming algorithms for $k$-means and $k$-median clustering of $d$-dimensional Euclidean data points over a stream with length at most $T$ using $poly(k,d,log(T))$ space to achieve a constant multiplicative error and a $poly(k,d,log(T))$ additive error. In particular, we present a differentially private streaming clustering framework which only requires an offline DP coreset or clustering algorithm as a blackbox. By plugging in existing results from DP clustering Ghazi, Kumar, Manurangsi 2020 and Kaplan, Stemmer 2018, we achieve (1) a $(1+gamma)$-multiplicative approximation with $ ilde{O}_gamma(poly(k,d,log(T)))$ space for any $gamma>0$, and the additive error is $poly(k,d,log(T))$ or (2) an $O(1)$-multiplicative approximation with $ ilde{O}(k^{1.5} cdot poly(d,log(T)))$ space and $poly(k,d,log(T))$ additive error. In addition, our algorithmic framework is also differentially private under the continual release setting, i.e., the union of outputs of our algorithms at every timestamp is always differentially private.
