Error estimation and adaptive tuning for unregularized robust M-estimator

Abstract

We consider unregularized robust M-estimators for linear models under Gaussian design and heavy-tailed noise, in the proportional asymptotics regime where the sample size n and the number of features p are both increasing such that $p/n \to γ\in (0,1)$. An estimator of the out-of-sample error of a robust M-estimator is analyzed and proved to be consistent for a large family of loss functions that includes the Huber loss. As an application of this result, we propose an adaptive tuning procedure of the scale parameter $λ>0$ of a given loss function $ρ$: choosing $\hat λ$ in a given interval $I$ that minimizes the out-of-sample error estimate of the M-estimator constructed with loss $ρ_λ(\cdot) = λ^2 ρ(\cdot/λ)$ leads to the optimal out-of-sample error over $I$. The proof relies on a smoothing argument: the unregularized M-estimation objective function is perturbed, or smoothed, with a Ridge penalty that vanishes as $n\to+\infty$, and shows that the unregularized M-estimator of interest inherits properties of its smoothed version.