False discoveries occur early on the Lasso path
Lasso 路径上的错误发现会较早出现AI 解读 ↓
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Lasso 路径上的错误发现会较早出现
- 研究问题
- 在线性稀疏的高维回归中，即使解释变量独立且真实效应很强，Lasso 路径能否先可靠选出真实变量而几乎不引入无效变量？
- 方法与思路
- 作者借助近似消息传递理论分析 Lasso 路径上真实与无效变量的交错，并加入处理正则化参数可能被自适应选择的分析工具。
- 材料陈述的贡献
- 摘要称在线性稀疏渐近框架下，真实与无效变量始终交错，并给出发现与错误指标间的尖锐权衡，限制同时保持两类错误较低的可能性。
- 阅读关注点
- 应核对线性稀疏、设计分布和效应分布的条件，严格区分错误发现与错误选择指标的分母，并查看自适应参数选择下结论的范围。
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- 仅据版本未注明的 OpenAlex 索引摘要；具体设计条件与权衡公式未提供，不能推广至所有稀疏制度、相关设计或变量选择方法。
原文摘要
In regression settings where explanatory variables have very low correlations and there are relatively few effects, each of large magnitude, we expect the Lasso to find the important variables with few errors, if any. This paper shows that in a regime of linear sparsity—meaning that the fraction of variables with a nonvanishing effect tends to a constant, however small—this cannot really be the case, even when the design variables are stochastically independent. We demonstrate that true features and null features are always interspersed on the Lasso path, and that this phenomenon occurs no matter how strong the effect sizes are. We derive a sharp asymptotic trade-off between false and true positive rates or, equivalently, between measures of type I and type II errors along the Lasso path. This trade-off states that if we ever want to achieve a type II error (false negative rate) under a critical value, then anywhere on the Lasso path the type I error (false positive rate) will need to exceed a given threshold so that we can never have both errors at a low level at the same time. Our analysis uses tools from approximate message passing (AMP) theory as well as novel elements to deal with a possibly adaptive selection of the Lasso regularizing parameter.