Sparse Quantile Regression
We consider both $\ell _{0}$-penalized and $\ell _{0}$-constrained quantile regression estimators. For the $\ell _{0}$-penalized estimator, we derive an exponential inequality on the tail probability of excess quantile prediction risk and apply it to obtain non-asymptotic upper bounds on the mean-square parameter and regression function estimation errors. We also derive analogous results for the $\ell _{0}$-constrained estimator. The resulting rates of convergence are nearly minimax-optimal and the same as those for $\ell _{1}$-penalized and non-convex penalized estimators. Further, we characterize expected Hamming loss for the $\ell _{0}$-penalized estimator. We implement the proposed procedure via mixed integer linear programming and also a more scalable first-order approximation algorithm. We illustrate the finite-sample performance of our approach in Monte Carlo experiments and its usefulness in a real data application concerning conformal prediction of infant birth weights (with $n\approx 10^{3}$ and up to $p>10^{3}$). In sum, our $\ell _{0}$-based method produces a much sparser estimator than the $\ell _{1}$-penalized and non-convex penalized approaches without compromising precision.