Statistical Robustness of Interval CVaR Based Regression Models under Perturbation and Contamination

TL;DR

This paper provides a comprehensive theoretical analysis of the robustness of interval CVaR-based nonlinear regression models, including neural networks, under contamination and perturbation, demonstrating their advantages over traditional methods.

Contribution

It offers the first rigorous robustness analysis of In-CVaR regression models for nonlinear cases, including neural networks, under contamination and perturbation.

Findings

In-CVaR models have a distributional breakdown point that quantifies robustness.

The estimator is qualitatively robust if the largest portion of losses is trimmed.

In-CVaR outperforms classical methods in robustness in both theory and experiments.

Abstract

Robustness under perturbation and contamination is a prominent issue in statistical learning. We address the robust nonlinear regression based on the so-called interval conditional value-at-risk (In-CVaR), which is introduced to enhance robustness by trimming extreme losses. While recent literature shows that the In-CVaR based statistical learning exhibits superior robustness performance than classical robust regression models, its theoretical robustness analysis for nonlinear regression remains largely unexplored. We rigorously quantify robustness under contamination, with a unified study of distributional breakdown point for a broad class of regression models, including linear, piecewise affine and neural network models with $\ell_1$, $\ell_2$ and Huber losses. Moreover, we analyze the qualitative robustness of the In-CVaR based estimator under perturbation. We show that under several minor assumptions, the In-CVaR based estimator is qualitatively robust in terms of the Prokhorov metric if and only if the largest portion of losses is trimmed. Overall, this study analyzes robustness properties of In-CVaR based nonlinear regression models under both perturbation and contamination, which illustrates the advantages of In-CVaR risk measure over conditional value-at-risk and expectation for robust regression in both theory and numerical experiments.