Convergence of online k-means

02/22/2022

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by   Sanjoy Dasgupta, et al.

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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 cost 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.