Generalized Linear Models with Structured Sparsity Estimators
广义线性模型中的结构化稀疏估计AI 解读 ↓
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广义线性模型中的结构化稀疏估计
- 研究问题
- 如何将结构化稀疏估计及去偏方法推广到广义线性模型，并为初始估计和二阶导数矩阵的近似逆建立误差界？
- 方法与思路
- 作者采用带结构范数的惩罚损失，证明两类 oracle 不等式，通过可行加权逐节点回归构造近似逆并去偏。
- 材料陈述的贡献
- 摘要称，结果覆盖非次高斯误差及解释变量，扩展至凸锥生成范数，并明确逐节点回归范数与初始惩罚范数的关系。
- 阅读关注点
- 应核对样本单点 margin 条件的证明、尾部与稀疏性要求、两类范数的强弱关系，以及去偏结果实际保证的性质。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核实；摘要未列出完整条件、误差界或最终推断定理，无法判断适用维度。
原文摘要
In this paper, we introduce structured sparsity estimators in Generalized Linear Models. Structured sparsity estimators in the least squares loss are introduced by Stucky and van de Geer (2018) recently for fixed design and normal errors. We extend their results to debiased structured sparsity estimators with Generalized Linear Model based loss. Structured sparsity estimation means penalized loss functions with a possible sparsity structure used in the chosen norm. These include weighted group lasso, lasso and norms generated from convex cones. The significant difficulty is that it is not clear how to prove two oracle inequalities. The first one is for the initial penalized Generalized Linear Model estimator. Since it is not clear how a particular feasible-weighted nodewise regression may fit in an oracle inequality for penalized Generalized Linear Model, we need a second oracle inequality to get oracle bounds for the approximate inverse for the sample estimate of second-order partial derivative of Generalized Linear Model. Our contributions are fivefold: 1. We generalize the existing oracle inequality results in penalized Generalized Linear Models by proving the underlying conditions rather than assuming them. One of the key issues is the proof of a sample one-point margin condition and its use in an oracle inequality. 2. Our results cover even non sub-Gaussian errors and regressors. 3. We provide a feasible weighted nodewise regression proof which generalizes the results in the literature from a simple l_1 norm usage to norms generated from convex cones. 4. We realize that norms used in feasible nodewise regression proofs should be weaker or equal to the norms in penalized Generalized Linear Model loss. 5. We can debias the first step estimator via getting an approximate inverse of the singular-sample second order partial derivative of Generalized Linear Model loss.