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.

翻译：我们同时考虑了ℓ₀-惩罚和ℓ₀-约束的分位数回归估计量。针对ℓ₀-惩罚估计量，我们推导了关于超量分位数预测风险尾概率的指数不等式，并将其应用于获得均方参数和回归函数估计误差的非渐近上界。我们还为ℓ₀-约束估计量推导了类似的结果。所得的收敛速率几乎达到极小化最优，且与ℓ₁-惩罚和非凸惩罚估计量相同。此外，我们刻画了ℓ₀-惩罚估计量的期望汉明损失。我们通过混合整数线性规划以及更可扩展的一阶近似算法实现了所提方法。我们在蒙特卡洛实验中展示了该方法在有限样本下的表现，并在关于婴儿出生体重共形预测（n≈10³，p>10³）的实际数据应用中验证了其实用性。总之，与ℓ₁-惩罚和非凸惩罚方法相比，我们的ℓ₀-基方法在保持精度的同时产生了更稀疏的估计结果。