Robust linear least squares regression

TL;DR

This paper introduces new risk bounds for robust linear regression estimators, including a novel truncated min-max estimator, achieving optimal rates without exponential moment assumptions and demonstrating strong empirical performance.

Contribution

The paper provides improved risk bounds for ridge and least squares estimators and introduces a new robust estimator with better deviation properties under heavy-tailed noise.

Findings

Risk bounds of order d/n without logarithmic factors.

A new estimator with exponential deviation bounds under heavy-tailed noise.

Experimental validation showing the effectiveness of the truncated min-max estimator.

Abstract

We consider the problem of robustly predicting as well as the best linear combination of $d$ given functions in least squares regression, and variants of this problem including constraints on the parameters of the linear combination. For the ridge estimator and the ordinary least squares estimator, and their variants, we provide new risk bounds of order $d/n$ without logarithmic factor unlike some standard results, where $n$ is the size of the training data. We also provide a new estimator with better deviations in the presence of heavy-tailed noise. It is based on truncating differences of losses in a min--max framework and satisfies a $d/n$ risk bound both in expectation and in deviations. The key common surprising factor of these results is the absence of exponential moment condition on the output distribution while achieving exponential deviations. All risk bounds are obtained through a PAC-Bayesian analysis on truncated differences of losses. Experimental results strongly back up our truncated min--max estimator.