High‐dimensional quantile regression: Convolution smoothing and concave regularization
High‐dimensional quantile regression: Convolution smoothing and concave regularization
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DOI：
10.1111/rssb.12485
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发表时间：
2021-09
期刊：
影响因子：
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通讯作者：
Kean Ming Tan;Lan Wang;Wen-Xin Zhou
中科院分区：
文献类型：
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作者：
Kean Ming Tan;Lan Wang;Wen-Xin Zhou
惩罚分位数回归（QR）被广泛用于分析具有异质性的高维数据。现在人们认识到，101-惩罚引入了不可忽略的估计偏差，而适当使用凹正则化可能会导致估计器随着信号的增强而具有精确的收敛速度和预言属性。虽然具有强凸损失函数的折叠凹惩罚M-估计已经得到了很好的研究，但关于QR的现有文献相对沉默。主要的困难是分位数损失是分段线性的：它是非光滑的，曲率集中在一个点上。为了克服光滑性和强凸性的不足，我们提出并研究了一种卷积型光滑QR，迭代重加权QR 1正则化。由此产生的平滑经验损失是两次连续可微和（可证明）局部强凸的高概率。我们证明了迭代重加权后的平滑QR估计，经过几次迭代，达到了最佳的收敛速度，而且，在几乎必要和充分的最小信号强度条件下，预言率和强预言性质。大量的数值研究证实了我们的理论结果。
ℓ1 ‐penalized quantile regression (QR) is widely used for analysing high‐dimensional data with heterogeneity. It is now recognized that the ℓ1 ‐penalty introduces non‐negligible estimation bias, while a proper use of concave regularization may lead to estimators with refined convergence rates and oracle properties as the signal strengthens. Although folded concave penalized M‐estimation with strongly convex loss functions have been well studied, the extant literature on QR is relatively silent. The main difficulty is that the quantile loss is piecewise linear: it is non‐smooth and has curvature concentrated at a single point. To overcome the lack of smoothness and strong convexity, we propose and study a convolution‐type smoothed QR with iteratively reweighted ℓ1 ‐regularization. The resulting smoothed empirical loss is twice continuously differentiable and (provably) locally strongly convex with high probability. We show that the iteratively reweighted ℓ1 ‐penalized smoothed QR estimator, after a few iterations, achieves the optimal rate of convergence, and moreover, the oracle rate and the strong oracle property under an almost necessary and sufficient minimum signal strength condition. Extensive numerical studies corroborate our theoretical results.