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# Convergence of online k-means

*Proceedings of The 25th International Conference on Artificial Intelligence and Statistics*, PMLR 151:8534-8569, 2022.

#### Abstract

We prove asymptotic convergence for a general class of k-means algorithms performed over streaming data from a distribution–the centers asymptotically converge to the set of stationary points of the k-means objective function. To do so, we show that online k-means over a distribution can be interpreted as stochastic gradient descent with a stochastic learning rate schedule. Then, we prove convergence by extending techniques used in optimization literature to handle settings where center-specific learning rates may depend on the past trajectory of the centers.