Estimating the error variance in a high-dimensional linear model

TL;DR

This paper introduces the natural lasso and organic lasso estimators for accurately estimating error variance in high-dimensional linear models, with strong theoretical guarantees and empirical performance.

Contribution

It proposes novel lasso-based estimators for error variance that do not require assumptions on the design matrix or true coefficients, improving estimation in challenging settings.

Findings

The natural lasso estimator has provably good mean squared error performance.

The organic lasso does not require tuning of the regularization parameter.

Both estimators outperform existing methods in empirical tests.

Abstract

The lasso has been studied extensively as a tool for estimating the coefficient vector in the high-dimensional linear model; however, considerably less is known about estimating the error variance in this context. In this paper, we propose the natural lasso estimator for the error variance, which maximizes a penalized likelihood objective. A key aspect of the natural lasso is that the likelihood is expressed in terms of the natural parameterization of the multiparameter exponential family of a Gaussian with unknown mean and variance. The result is a remarkably simple estimator of the error variance with provably good performance in terms of mean squared error. These theoretical results do not require placing any assumptions on the design matrix or the true regression coefficients. We also propose a companion estimator, called the organic lasso, which theoretically does not require tuning of the regularization parameter. Both estimators do well empirically compared to preexisting methods, especially in settings where successful recovery of the true support of the coefficient vector is hard. Finally, we show that existing methods can do well under fewer assumptions than previously known, thus providing a fuller story about the problem of estimating the error variance in high-dimensional linear models.