Uniform Deviation Bounds for Unbounded Loss Functions like k-Means

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. As such, they are a critical component to empirical risk minimization. In this paper, we provide a novel framework to obtain uniform deviation bounds for loss functions which are unbounded. In our main application, this allows us to obtain 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 further provide improved rates under progressively stronger assumptions, namely, bounded higher moments, subgaussianity and bounded support.