Robust Lasso Regression Using Tukey's Biweight Criterion

The adaptive lasso is a method for performing simultaneous parameter estimation and variable selection. The adaptive weights used in its penalty term mean that the adaptive lasso achieves the oracle property. In this work, we propose an extension of the adaptive lasso named the Tukey-lasso. By using Tukey's biweight criterion, instead of squared loss, the Tukey-lasso is resistant to outliers in both the response and covariates. Importantly, we demonstrate that the Tukey-lasso also enjoys the oracle property. A fast accelerated proximal gradient (APG) algorithm is proposed and implemented for computing the Tukey-lasso. Our extensive simulations show that the Tukey-lasso, implemented with the APG algorithm, achieves very reliable results, including for high-dimensional data where <i>p</i> &gt; <i>n</i>. In the presence of outliers, the Tukey-lasso is shown to offer substantial improvements in performance compared to the adaptive lasso and other robust implementations of the lasso. Real-data examples further demonstrate the utility of the Tukey-lasso. Supplementary materials for this article are available online.

自适应Lasso（adaptive lasso）是一种可同时完成参数估计与变量选择的方法。其惩罚项中采用的自适应权重，使得自适应Lasso具备神谕性质（oracle property）。本研究中，我们提出了自适应Lasso的一种扩展方法，命名为图基Lasso（Tukey-lasso）。该方法使用Tukey双权重准则（Tukey's biweight criterion）替代平方损失函数，可对响应变量与协变量中的异常值具备鲁棒性。尤为重要的是，我们证明了图基Lasso同样具备神谕性质。我们提出并实现了一种快速加速近端梯度（accelerated proximal gradient, APG）算法，用于求解图基Lasso模型。我们开展的大量模拟实验表明，结合APG算法实现的图基Lasso可获得极为可靠的结果，在变量维度高于样本量（p > n）的高维数据场景下亦表现出色。当存在异常值时，相较于自适应Lasso以及其他Lasso的鲁棒实现方法，图基Lasso的性能提升幅度显著。真实数据集案例进一步验证了图基Lasso的应用价值。本文的补充材料可在线获取。