Robust Estimation of High-Dimensional Mean Regression
高维均值回归的稳健估计AI 解读 ↓
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高维均值回归的稳健估计
- 研究问题
- 在误差重尾、仅有二阶矩且维度可随样本量指数增长时，如何稳健估计条件均值，避免将条件中位数误当作均值目标？
- 方法与思路
- 作者提出采用发散参数的惩罚 Huber 损失，即 RA-Lasso，以减小传统 Huber 损失的偏差，并研究复合梯度下降算法。
- 材料陈述的贡献
- 摘要称 RA-Lasso 可达到轻尾情形下的最优收敛率，充分迭代后的算法解也具有该速率，并提供均值集中不等式及模拟比较。
- 阅读关注点
- 应核对稀疏性与设计矩阵条件、Huber 参数的增长方式，以及统计误差与迭代误差如何衔接；摘要未说明具体误差度量。
- 阅读边界
- 这是 arXiv 预印本，同行评审未核实。仅有二阶矩并非摘要列出的全部条件，最优速率和有限样本表现仍需核对全文。
原文摘要
Data subject to heavy-tailed errors are commonly encountered in various scientific fields, especially in the modern era with explosion of massive data. To address this problem, procedures based on quantile regression and Least Absolute Deviation (LAD) regression have been devel- oped in recent years. These methods essentially estimate the conditional median (or quantile) function. They can be very different from the conditional mean functions when distributions are asymmetric and heteroscedastic. How can we efficiently estimate the mean regression functions in ultra-high dimensional setting with existence of only the second moment? To solve this problem, we propose a penalized Huber loss with diverging parameter to reduce biases created by the traditional Huber loss. Such a penalized robust approximate quadratic (RA-quadratic) loss will be called RA-Lasso. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, our results reveal that the RA-lasso estimator produces a consistent estimator at the same rate as the optimal rate under the light-tail situation. We further study the computational convergence of RA-Lasso and show that the composite gradient descent algorithm indeed produces a solution that admits the same optimal rate after sufficient iterations. As a byproduct, we also establish the concentration inequality for estimat- ing population mean when there exists only the second moment. We compare RA-Lasso with other regularized robust estimators based on quantile regression and LAD regression. Extensive simulation studies demonstrate the satisfactory finite-sample performance of RA-Lasso.