Pivotal Estimation via Square-root Lasso in Nonparametric Regression

We propose a self-tuning square-root 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 non-asymptotic bounds for square-root 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 square-root Lasso accounting for possible misspecification of the selected model. Under mild conditions the rate of convergence of ols post square-root Lasso is as good as square-root Lasso's rate. As an application, we consider the use of square-root Lasso and ols post square-root Lasso as estimators of nuisance parameters in a generic semi-parametric problem (nonlinear moment condition or Z-problem), resulting in a construction of $\sqrt{n}$-consistent and asymptotically normal estimators of the main parameters.