Concentration bounds for the empirical angular measure with statistical learning applications

Abstract

The angular measure on the unit sphere characterizes the firstorder dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation when the components of the vector have different distributions, the rank transformation offers a convenient and robust way of standardizing data in order to build an empirical version of the angular measure based on the most extreme observations. However, the study of the sampling distribution of the resulting empirical angular measure is challenging. It is the purpose of the paper to establish finite-sample bounds for the maximal deviations between the empirical and true angular measures, uniformly over classes of Borel sets of controlled combinatorial complexity. The bounds are valid with high probability and scale essentially as the square root of the effective sample size, up to a logarithmic factor. Discarding the most extreme observations yields a truncated version of the empirical angular measure for which the logarithmic factor in the concentration bound is replaced by a factor depending on the truncation level. The bounds are applied to provide performance guarantees for two statistical learning procedures tailored to extreme regions of the input space and built upon the empirical angular measure: binary classification in extreme regions through empirical risk minimization and unsupervised anomaly detection through minimum-volume sets of the sphere.

Cross Validation for Rare Events

“…Similar rates are obtained in an unsupervised framework e.g. in Goix et al 2015;Drees and Sabourin 2021;Clémençon et al 2021. All these results match the asymptotic rate of convergence for tail empirical processes, see e.g.…”

“…error, from Goix et al 2015 for incorporating the probability of the considered class within the upper bounds. In particular we use a Bernstein-type inequality proved in McDiarmid 1998 which is also exploited in Goix et al 2015;Clémençon et al 2021;Drees and Sabourin 2021. Limitations and further work. Sanity check bounds are difficult to improve upon without further assumptions regarding algorithm stability, e.g.…”