Uniform Deviation Bounds for k-Means Clustering

Uniform deviation bounds limit the difference between a model’s expected loss and its loss on a random sample uniformly for all models in a learning problem. In this paper, we provide a novel framework to obtain uniform deviation bounds for unbounded loss functions. 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. We further show that this rate also depends on the kurtosis — the normalized fourth moment which measures the “tailedness” of the distribution. We also provide improved rates under progressively stronger assumptions, namely, bounded higher moments, subgaussianity and bounded support of the underlying distribution.