Pivotal estimation via square-root lasso in nonparametric regression

We propose a self-tuning √ Lasso method that simultaneiously 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 non-asymptotic bounds for √ Lasso including prediction norm rate and sharp sparcity bound. Our analysis is based on new impact factors that are tailored to establish prediction rates. 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 √ Lasso accounting for possible misspecification of the selected model. Under mild conditions the rate of convergence of ols post √ Lasso is no worse than √ Lasso even with a misspecified selected model and possibly better otherwise. As an application, we consider the use of √ Lasso and post √ Lasso as estimators of nuisance parameters in a generic semi-parametric problem (nonlinear instrumental/moment condition or Z-estimation problem).