On Regression in Extreme Regions

TL;DR

This paper develops a theoretical framework for extrapolation in continuous regression, focusing on extreme covariate tails using multivariate regular variation and angular component regression, with finite sample risk bounds and empirical validation.

Contribution

It introduces a novel statistical learning approach for out-of-domain extrapolation in continuous regression based on extreme value theory and angular component analysis.

Findings

Finite sample risk bounds for tail prediction

Bias-variance decomposition in extreme regions

Empirical validation with simulated and real data

Abstract

We establish a statistical learning theoretical framework aimed at extrapolation, or out-of-domain generalization, on the unobserved tails of covariates in continuous regression problems. Our strategy involves performing statistical regression on a subsample of observations with continuous labels that are the furthest away from the origin, focusing specifically on their angular components. The underlying assumptions of our approach are grounded in the theory of multivariate regular variation, a cornerstone of extreme value theory. We address the stylized problem of nonparametric least squares regression with predictors chosen from a Vapnik-Chervonenkis class. This work contributes to a broader initiative to develop statistical learning theoretical foundations for supervised learning strategies that enhance performance on the supposedly heavy tails of covariates. Previous efforts in this area have focused exclusively on binary classification on extreme covariates. Although the continuous target setting necessitates different techniques and regularity assumptions, our main results echo findings from earlier studies. We quantify the predictive performance on tail regions in terms of excess risk, presenting it as a finite sample risk bound with a clear bias-variance decomposition. Numerical experiments with simulated and real data illustrate our theoretical findings.