High Dimensional Robust M-Estimation: Asymptotic Variance via Approximate Message Passing

TL;DR

This paper rigorously analyzes high-dimensional robust M-estimators, revealing an additional Gaussian noise component and providing a method to compute and evaluate these estimators using approximate message passing and state evolution.

Contribution

It introduces an AMP algorithm and state evolution framework to analyze the asymptotic variance of high-dimensional robust M-estimators, clarifying the nature of the extra noise phenomenon.

Findings

Extra Gaussian noise component in high-dimensional robust regression

AMP algorithm effectively computes M-estimators in high dimensions

State evolution accurately predicts estimator performance

Abstract

In a recent article (Proc. Natl. Acad. Sci., 110(36), 14557-14562), El Karoui et al. study the distribution of robust regression estimators in the regime in which the number of parameters p is of the same order as the number of samples n. Using numerical simulations and `highly plausible' heuristic arguments, they unveil a striking new phenomenon. Namely, the regression coefficients contain an extra Gaussian noise component that is not explained by classical concepts such as the Fisher information matrix. We show here that that this phenomenon can be characterized rigorously techniques that were developed by the authors to analyze the Lasso estimator under high-dimensional asymptotics. We introduce an approximate message passing (AMP) algorithm to compute M-estimators and deploy state evolution to evaluate the operating characteristics of AMP and so also M-estimates. Our analysis clarifies that the `extra Gaussian noise' encountered in this problem is fundamentally similar to phenomena already studied for regularized least squares in the setting n<p.