The k-means++ algorithm is commonly restarted multiple times to avoid poor local optima, yet the number of restarts is almost always chosen arbitrarily and applied uniformly regardless of data set difficulty. This undermines any comparison relying on such a choice and wastes computation on easy data sets while potentially under-serving hard ones. We introduce GTRC, a restart criterion combining a Good-Turing estimate, a proven unconditional bound, and a confidence-based bound on the probability that a further restart would improve on the current result, stopping once this probability falls below a user-specified tolerance ε. Across 36 data sets, GTRC reached clustering quality competitive with well-chosen fixed restart counts, while the number of restarts used varied considerably and appropriately with data set difficulty, governed by an interpretable, data-dependent signal rather than a fixed rule. GTRC offers a principled and reportable alternative to fixing the number of k-means++ restarts in advance. Software:https://github.com/RCdeAmorim/Good-Turing-Restart-Criterion.
Big data clustering remains challenging: the Minimum Sum-of-Squares Clustering (MSSC) problem underlying K-means is NP-hard, and existing methods either reach poor local minima or require prohibitive metaheuristic hybrids. We target arbitrarily tall data: a fixed feature space may contain arbitrarily many, possibly infinitely many, observations, while the algorithm accesses only finite random samples. We propose Big-means++, an algorithm achieving scalability and global-search quality by curating inputs to MSSC optimization on big data. It orchestrates local K-means refinements into a data-native global search for big data clustering. Rather than optimizing the full-data MSSC objective, Big-means++ traverses sample-induced surrogate landscapes. Each sample defines a distinct empirical MSSC approximation with a perturbed local-optimum structure, turning sample-to-sample variation into a global-search mechanism. Unlike Big-means, a flowing-incumbent strategy propagates centroid state across empirical landscapes through K-means refinements on fresh samples without rollback to a best-so-far solution. This increases mobility and favors stable, high-quality configurations across approximations of the full-data structure. A new shaking mechanism varies sample size geometrically, broadening the surrogate landscapes explored across resolution scales, accounting for cluster imbalance, and improving solution quality. A competitive multi-agent system asynchronously explores independent sampled landscapes, transforming diverse stochastic trajectories into collective search intelligence. Automatic convergence detection stops each agent after attaining a high-quality solution but before further search risks degrading it, while providing a universal speed-quality control. Experiments on 22 datasets against 11 competing algorithms demonstrate the effectiveness, efficiency, and robustness of Big-means++.
The k-means++ algorithm is a standard and widely used seeding method for k-means clustering, but for a fixed number k of centers its worst-case expected approximation ratio is Θ(logk). We consider the same algorithm when an adversary first fixes the dataset and some K; the number of centers k is then chosen uniformly from {K,…,2K−1}. We prove that k-means++ is an O(1)-approximation with constant probability in this budget-smoothed setup.
Clustering problems such as k-means and k-median are staples of unsupervised learning, and many algorithmic techniques have been developed to tackle their numerous aspects. In this paper, we focus on the class of greedy approximation algorithm, that attracted less attention than local-search or primal-dual counterparts. In particular, we study the recursive greedy algorithm developed by Mettu and Plaxton [SIAM J. Comp 2003]. We provide a simplification of the algorithm, allowing for faster implementation, in graph metrics or in Euclidean space, where our algorithm matches or improves the state-of-the-art.