AI 中文总结
针对大型分布式数据下惩罚分位数回归的计算难题，提出分布式随机平滑ADMM算法，通过Huber平滑检查损失实现并行计算，在同质与异质数据分区下均表现良好，阐明了相关权衡关系。
AI 中文摘要
分位数回归非常适合处理异质性和重尾数据，但由于检查损失函数非光滑，对于大型分布式数据集而言计算颇具挑战性。我们提出了一种用于水平分区惩罚分位数回归的分布式随机平滑交替方向乘子法（DSS-ADMM）。每个工作节点计算Huber平滑检查损失的小批量梯度，协调节点对正则项执行一次近端聚合步骤。原始观测值保持本地化，工作节点更新并行运行，且该方法无需矩阵求逆。对于正常、闭且凸的惩罚项，将本地系数向量堆叠可得到标准的双块随机ADMM公式。在固定平滑参数下，我们建立了联合目标-可行性的期望$O(\frac{\text{log}K}{\text{sqrt}K})$界，以及原始检查损失目标的显式$\frac{\text{eps}}{4}$近似项；当平滑块为强凸时，该界改进为$O(\frac{\text{log}K}{K})$。我们还确定了其扩展到极小极大凹惩罚项和光滑绝对偏差惩罚项的适用范围。可复现模拟考虑了两种工作节点分区：同质分区（各工作节点上的观测值独立同分布）和异质分区（各工作节点的协变量分布不同）。敏感性研究及对糖尿病数据和Engel数据的分析，阐明了逐观测梯度评估、通信、一致性、稀疏性与预测之间的权衡关系。
英文摘要
Quantile regression is well suited to heterogeneous and heavy-tailed data, but computation becomes challenging for large, distributed data sets because the check loss is nonsmooth. We propose a distributed stochastic smoothing alternating direction method of multipliers (DSS-ADMM) for horizontally partitioned penalized quantile regression. Each worker computes a mini-batch gradient of a Huber-smoothed check loss, and a coordinator performs a single proximal aggregation step for the regularizer. Raw observations remain local, worker updates run in parallel, and the method requires no matrix inversion. For proper, closed, and convex penalties, stacking the local coefficient vectors yields a standard two-block stochastic ADMM formulation. With fixed smoothing, we establish an expected $O(\log K/\sqrt K)$ joint objective-feasibility bound and an explicit $\eps/4$ approximation term for the original check-loss objective; when the smooth block is strongly convex, the bound improves to $O(\log K/K)$. We also characterize the scope of an extension to the minimax concave penalty and the smoothly clipped absolute deviation penalty. Reproducible simulations consider both homogeneous worker partitions, in which observations are independently and identically distributed across workers, and heterogeneous partitions, in which worker-specific covariate distributions differ. Sensitivity studies and analyses of the diabetes and Engel data illustrate the trade-offs among per-observation gradient evaluations, communication, consensus, sparsity, and prediction.
