ℓ1-penalized quantile regression in high-dimensional sparse models

Abstract

We consider median regression and, more generally, a possibly infinite collection of quantile regressions in high-dimensional sparse models. In these models, the number of regressors p is very large, possibly larger than the sample size n, but only at most s regressors have a nonzero impact on each conditional quantile of the response variable, where s grows more slowly than n. Since ordinary quantile regression is not consistent in this case, we consider ℓ₁-penalized quantile regression (ℓ₁-QR), which penalizes the ℓ₁-norm of regression coefficients, as well as the post-penalized QR estimator (post-ℓ₁-QR), which applies ordinary QR to the model selected by ℓ₁-QR. First, we show that under general conditions ℓ₁-QR is consistent at the near-oracle rate $\sqrt{s/n}\sqrt{\log(p\vee n)}$, uniformly in the compact set $\mathcal{U}\subset(0,1)$ of quantile indices. In deriving this result, we propose a partly pivotal, data-driven choice of the penalty level and show that it satisfies the requirements for achieving this rate. Second, we show that under similar conditions post-ℓ₁-QR is consistent at the near-oracle rate $\sqrt{s/n}\sqrt{\log(p\vee n)}$, uniformly over $\mathcal{U}$, even if the ℓ₁-QR-selected models miss some components of the true models, and the rate could be even closer to the oracle rate otherwise. Third, we characterize conditions under which ℓ₁-QR contains the true model as a submodel, and derive bounds on the dimension of the selected model, uniformly over $\mathcal{U}$; we also provide conditions under which hard-thresholding selects the minimal true model, uniformly over $\mathcal{U}$.