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arXiv · 1702.08249

Uniform Deviation Bounds for Unbounded Loss Functions like k-Means

Olivier Bachem·Mario Lucic·S. Hamed Hassani·Andreas Krause
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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. 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 $\mathcal{O}\left(m^{-\frac12}\right)$ compared to the previously known $\mathcal{O}\left(m^{-\frac14}\right)$ 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.

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Olivier Bachem, Mario Lucic, S. Hamed Hassani, Andreas Krause. 2017-02-27. Uniform Deviation Bounds for Unbounded Loss Functions like k-Means. https://arxiv.org/abs/1702.08249

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