Replicable Clustering
Hossein Esfandiari, Amin Karbasi, Vahab Mirrokni, Grigoris Velegkas, Felix Zhou
TL;DR
The paper studies replicable clustering under the distributional learning framework, introducing replicable algorithms for statistical $k$-means, $k$-medians, and $k$-centers. It builds replicable coresets via a replicable quad-tree, ensuring the cost on the coreset approximates the population cost, and leverages black-box approximation oracles to obtain near-optimal partitions with high probability. For Euclidean data, it further employs Johnson-Lindenstrauss dimensionality reduction to achieve poly$(d)$ sample complexity and outputs a clustering function that labels points rather than explicitly outputting centers. The results establish a principled path to replicable clustering with strong utility guarantees, supported by experiments on synthetic 2D distributions. This advances reliable, repeatable unsupervised learning in high dimensions by combining coresets, stable discretizations, and dimensionality reduction with robust probabilistic guarantees.
Abstract
We design replicable algorithms in the context of statistical clustering under the recently introduced notion of replicability from Impagliazzo et al. [2022]. According to this definition, a clustering algorithm is replicable if, with high probability, its output induces the exact same partition of the sample space after two executions on different inputs drawn from the same distribution, when its internal randomness is shared across the executions. We propose such algorithms for the statistical $k$-medians, statistical $k$-means, and statistical $k$-centers problems by utilizing approximation routines for their combinatorial counterparts in a black-box manner. In particular, we demonstrate a replicable $O(1)$-approximation algorithm for statistical Euclidean $k$-medians ($k$-means) with $\operatorname{poly}(d)$ sample complexity. We also describe an $O(1)$-approximation algorithm with an additional $O(1)$-additive error for statistical Euclidean $k$-centers, albeit with $\exp(d)$ sample complexity. In addition, we provide experiments on synthetic distributions in 2D using the $k$-means++ implementation from sklearn as a black-box that validate our theoretical results.