The sparsity and bias of the Lasso selection in high-dimensional linear regression
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高维线性回归中 Lasso 变量选择的稀疏性与偏差
- 研究问题
- 当理想模型之外的系数较小但不为零时，Lasso 能否控制所选模型规模和遗漏偏差，并保留足够大的系数？
- 方法与思路
- 作者在设计变量相关性的稀疏 Riesz 条件下研究近似稀疏回归，分析模型规模、选择偏差与估计损失。
- 材料陈述的贡献
- 摘要称证明模型维数阶与选择偏差的控制，并在给定条件下获得均值响应平方误差和系数损失的最优收敛速率。
- 阅读关注点
- 应核对稀疏 Riesz 条件、小系数贡献及阈值偏差的定义，以及大系数入选保证与精确支持恢复的区别。
- 阅读边界
- 仅据版本未注明的索引摘要；具体稀疏度、调参条件和速率表达未完整给出，不能推广为任意设计下的选择保证。
原文摘要
Meinshausen and Buhlmann [Ann. Statist. 34 (2006) 1436–1462] showed that, for neighborhood selection in Gaussian graphical models, under a neighborhood stability condition, the LASSO is consistent, even when the number of variables is of greater order than the sample size. Zhao and Yu [(2006) J. Machine Learning Research 7 2541–2567] formalized the neighborhood stability condition in the context of linear regression as a strong irrepresentable condition. That paper showed that under this condition, the LASSO selects exactly the set of nonzero regression coefficients, provided that these coefficients are bounded away from zero at a certain rate. In this paper, the regression coefficients outside an ideal model are assumed to be small, but not necessarily zero. Under a sparse Riesz condition on the correlation of design variables, we prove that the LASSO selects a model of the correct order of dimensionality, controls the bias of the selected model at a level determined by the contributions of small regression coefficients and threshold bias, and selects all coefficients of greater order than the bias of the selected model. Moreover, as a consequence of this rate consistency of the LASSO in model selection, it is proved that the sum of error squares for the mean response and the ℓα-loss for the regression coefficients converge at the best possible rates under the given conditions. An interesting aspect of our results is that the logarithm of the number of variables can be of the same order as the sample size for certain random dependent designs.