论文标题
稀疏的分位数回归
Sparse Quantile Regression
论文作者
论文摘要
我们考虑$ \ ell _ {0} $ - 惩罚和$ \ ell _ {0} $ - 受约束的分数回归估计器。对于$ \ ell _ {0} $ - 惩罚估算器，我们在过量分位数预测风险的尾巴概率上得出了指数不平等，并将其应用于均值上方参数和回归函数函数估计错误上的非质子上限。我们还得出了$ \ ell _ {0} $ - 约束估算器的类似结果。最终的收敛速度几乎是最小的，并且与$ \ ell _ {1} $ - 惩罚和非凸额受惩罚估计量相同。此外，我们表征了$ \ ell _ {0} $的预期锤损失 - 罚款估算器。我们通过混合整数线性编程以及更可扩展的一阶近似算法实现了建议的过程。我们说明了蒙特卡洛实验中方法的有限样本性能及其在有关婴儿出生权重的共形预测的真实数据应用中的有用性（具有$ n \ bif 10^{3} $，最多可达$ p> 10^{3} $）。总而言之，我们的$ \ ell _ {0} $ - 基于$ \ ell _ {1} $ - 惩罚和非注重惩罚的方法而不会损害精度，因此产生的估计器要比$ \ ell _ {1} $ - 惩罚和非注重惩罚的方法。
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.
