Communication-Constrained Distributed Quantile Regression with Optimal Statistical Guarantees
Communication-Constrained Distributed Quantile Regression with Optimal Statistical Guarantees
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发表时间：
2021-10
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通讯作者：
Kean Ming Tan;H. Battey;Wen-Xin Zhou
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作者：
Kean Ming Tan;H. Battey;Wen-Xin Zhou
我们解决的问题，如何实现最佳的推理分布分位数回归没有严格的缩放条件。由于分位数回归（QR）损失函数的非平滑性质，这是具有挑战性的，这使得现有方法的使用无效。通过一个双平滑的方法，适用于本地（在每个数据源）和全球目标函数的困难得到解决。尽管依赖于局部和全局平滑参数的微妙组合，分位数回归模型是完全参数化的，从而便于解释。在低维的制度，我们建立了一个有限样本的理论框架，顺序定义的分布QR估计。这揭示了通信成本和统计误差之间的权衡。我们进一步讨论和比较几种替代的置信度集的建设，基于反演的沃尔德和分数类型的测试和resternation技术，详细介绍了一种改进，是有效的更极端的分位数系数。在高维度，稀疏的框架被采用，其中建议的双重平滑的目标函数是补充与$\ell_1$-罚款。我们表明，相应的分布式惩罚QR估计达到了全局收敛速度后，一个接近常数的通信轮数。一个全面的模拟研究进一步阐明了我们的研究结果。
We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. This is challenging due to the non-smooth nature of the quantile regression (QR) loss function, which invalidates the use of existing methodology. The difficulties are resolved through a double-smoothing approach that is applied to the local (at each data source) and global objective functions. Despite the reliance on a delicate combination of local and global smoothing parameters, the quantile regression model is fully parametric, thereby facilitating interpretation. In the low-dimensional regime, we establish a finite-sample theoretical framework for the sequentially defined distributed QR estimators. This reveals a trade-off between the communication cost and statistical error. We further discuss and compare several alternative confidence set constructions, based on inversion of Wald and score-type tests and resampling techniques, detailing an improvement that is effective for more extreme quantile coefficients. In high dimensions, a sparse framework is adopted, where the proposed doubly-smoothed objective function is complemented with an $\ell_1$-penalty. We show that the corresponding distributed penalized QR estimator achieves the global convergence rate after a near-constant number of communication rounds. A thorough simulation study further elucidates our findings.