Fast learning rates with heavy-tailed losses

We study fast learning rates when the losses are not necessarily bounded and may have a distribution with heavy tails. To enable such analyses, we introduce two new conditions: (i) the envelope function supf∈F |` ◦ f |, where ` is the loss function and F is the hypothesis class, exists and is L-integrable, and (ii) ` satisfies the multi-scale Bernstein’s condition on F . Under these assumptions, we prove that learning rate faster than O(n−1/2) can be obtained and, depending on r and the multi-scale Bernstein’s powers, can be arbitrarily close to O(n−1). We then verify these assumptions and derive fast learning rates for the problem of vector quantization by k-means clustering with heavy-tailed distributions. The analyses enable us to obtain novel learning rates that extend and complement existing results in the literature from both theoretical and practical viewpoints.