On Tail Decay Rate Estimation of Loss Function Distributions

Etrit Haxholli; Marco Lorenzi

The study of loss-function distributions is critical to characterize a model's behaviour on a given machine-learning problem. While model quality is commonly measured by the average loss assessed on a testing set, this quantity does not ascertain the existence of the mean of the loss distribution. Conversely, the existence of a distribution's statistical moments can be verified by examining the thickness of its tails. Cross-validation schemes determine a family of testing loss distributions conditioned on the training sets. By marginalizing across training sets, we can recover the overall (marginal) loss distribution, whose tail-shape we aim to estimate. Small sample-sizes diminish the reliability and efficiency of classical tail-estimation methods like Peaks-Over-Threshold, and we demonstrate that this effect is notably significant when estimating tails of marginal distributions composed of conditional distributions with substantial tail-location variability. We mitigate this problem by utilizing a result we prove: under certain conditions, the marginal-distribution's tail-shape parameter is the maximum tail-shape parameter across the conditional distributions underlying the marginal. We label the resulting approach as `cross-tail estimation (CTE)'. We test CTE in a series of experiments on simulated and real data showing the improved robustness and quality of tail estimation as compared to classical approaches.

On Tail Decay Rate Estimation of Loss Function Distributions

Abstract