Uniform Deviation Bounds for k-Means Clustering

Abstract

Uniform deviation bounds limit the difference between a model’s expected loss and its loss on an empirical sample uniformly for all models in a learning problem. In this paper, we provide a novel framework to obtain uniform deviation bounds for loss functions which are unbounded. As a result, we obtain competitive uniform deviation bounds for k-Means clustering under weak assumptions on the underlying distribution. If the fourth moment is bounded, we prove a rate of O(m^-1/2) compared to the previously known O(m^-1/4) rate. Furthermore, we show that the rate also depends on the kurtosis – the normalized fourth moment which measures the “tailedness” of a distribution. We also provide improved rates under progressively stronger assumptions, namely, bounded higher moments, subgaussianity and bounded support of the underlying distribution.

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