Online Coresets for Clustering with Bregman Divergences
This work addresses the need for efficient online data summarization for clustering tasks, particularly in streaming or large-scale environments, though it is incremental as it builds on prior coreset methods.
The paper tackles the problem of creating coresets for clustering with Bregman divergences in an online setting, achieving small additive error with update time O(d) and coreset sizes that scale polynomially in parameters or logarithmically in the number of points, while also extending to non-parametric versions like DP-Means.
We present algorithms that create coresets in an online setting for clustering problems according to a wide subset of Bregman divergences. Notably, our coresets have a small additive error, similar in magnitude to the lightweight coresets Bachem et. al. 2018, and take update time $O(d)$ for every incoming point where $d$ is dimension of the point. Our first algorithm gives online coresets of size $\tilde{O}(\mbox{poly}(k,d,ε,μ))$ for $k$-clusterings according to any $μ$-similar Bregman divergence. We further extend this algorithm to show existence of a non-parametric coresets, where the coreset size is independent of $k$, the number of clusters, for the same subclass of Bregman divergences. Our non-parametric coresets are larger by a factor of $O(\log n)$ ($n$ is number of points) and have similar (small) additive guarantee. At the same time our coresets also function as lightweight coresets for non-parametric versions of the Bregman clustering like DP-Means. While these coresets provide additive error guarantees, they are also significantly smaller (scaling with $O(\log n)$ as opposed to $O(d^d)$ for points in $R^d$) than the (relative-error) coresets obtained in Bachem et. al. 2015 for DP-Means. While our non-parametric coresets are existential, we give an algorithmic version under certain assumptions.