Sharp non-asymptotic performance bounds for $$\ell _1$$ ℓ 1 and Huber robust regression estimators

Abstract

A quantitative study of the robustness properties of the \(\ell _1\) and the Huber M-estimator on finite samples is presented. The focus is on the linear model involving a fixed design matrix and additive errors restricted to the dependent variables consisting of noise and sparse outliers. We derive sharp error bounds for the \(\ell _1\) estimator in terms of the leverage constants of a design matrix introduced here. A similar analysis is performed for Huber’s estimator using an equivalent problem formulation of independent interest. Our analysis considers outliers of arbitrary magnitude, and we recover breakdown point results as particular cases when outliers diverge. The practical implications of the theoretical analysis are discussed on two real datasets.