Pivotal estimation via square-root Lasso in nonparametric regression

TL;DR

This paper introduces a self-tuning square-root Lasso method for high-dimensional regression that effectively handles unknown scale, heteroscedasticity, and non-Gaussian noise, providing sharp nonasymptotic bounds and robust performance even in extreme cases.

Contribution

The paper develops a novel self-tuning square-root Lasso approach with theoretical guarantees under weak conditions, addressing practical issues like heteroscedasticity and design irregularities.

Findings

Provides nonasymptotic bounds for prediction and sparsity.

Achieves Gaussian-like results under weak noise assumptions.

Ensures optimal convergence rates for post-Lasso OLS in model misspecification.

Abstract

We propose a self-tuning $\sqrt{\mathrm {Lasso}}$ method that simultaneously resolves three important practical problems in high-dimensional regression analysis, namely it handles the unknown scale, heteroscedasticity and (drastic) non-Gaussianity of the noise. In addition, our analysis allows for badly behaved designs, for example, perfectly collinear regressors, and generates sharp bounds even in extreme cases, such as the infinite variance case and the noiseless case, in contrast to Lasso. We establish various nonasymptotic bounds for $\sqrt{\mathrm {Lasso}}$ including prediction norm rate and sparsity. Our analysis is based on new impact factors that are tailored for bounding prediction norm. In order to cover heteroscedastic non-Gaussian noise, we rely on moderate deviation theory for self-normalized sums to achieve Gaussian-like results under weak conditions. Moreover, we derive bounds on the performance of ordinary least square (ols) applied to the model selected by $\sqrt{\mathrm {Lasso}}$ accounting for possible misspecification of the selected model. Under mild conditions, the rate of convergence of ols post $\sqrt{\mathrm {Lasso}}$ is as good as $\sqrt{\mathrm {Lasso}}$'s rate. As an application, we consider the use of $\sqrt{\mathrm {Lasso}}$ and ols post $\sqrt{\mathrm {Lasso}}$ as estimators of nuisance parameters in a generic semiparametric problem (nonlinear moment condition or $Z$-problem), resulting in a construction of $\sqrt{n}$-consistent and asymptotically normal estimators of the main parameters.