Observable adjustments in single-index models for regularized M-estimators with bounded p/n
p/n 有界时单指标模型中正则化 M 估计的可观测调整AI 解读 ↓
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p/n 有界时单指标模型中正则化 M 估计的可观测调整
- 研究问题
- 在高斯协变量、未知连接函数且 p/n 趋于有限极限时，如何用可观测量描述估计系数与拟合值的经验分布？
- 方法与思路
- 作者以数据驱动的可观测调整替代均场参数，构造基于近端算子的近似，并据此开展指标分量推断与相关性估计。
- 材料陈述的贡献
- 摘要称该方法无需预先知道指标或连接函数，可构造分量置信区间、估计与真实指标的相关性，并支持相关性导向的调参。
- 阅读关注点
- 应核对调整量的实际计算方式、分量置信区间对应的识别与尺度，以及强凸正则化和无正则化情形各自的条件。
- 阅读边界
- 仅据所给 Crossref 摘要；一般损失与惩罚的适用条件、区间覆盖性质和调参后的推断有效性未说明。
原文摘要
We consider observations (X,y) from single index models with unknown link function, Gaussian covariates and a regularized M-estimator βˆ constructed from convex loss function and regularizer. In the regime where sample size n and dimension p are both increasing such that p/n has a finite limit, the behavior of the empirical distribution of βˆ and the predicted values Xβˆ has been previously characterized in a number of models: The empirical distributions are known to converge to proximal operators of the loss and penalty in a related Gaussian sequence model, which captures the interplay between ratio p n, loss, regularization and the data generating process. This connection between (βˆ,Xβˆ) and the corresponding proximal operators involves mean-field parameters defined as solutions to a nonlinear system of equations. This system typically involve unobservable quantities such as the prior distribution on the index or the link function, so the mean-field parameters need to be estimated. Although estimators for the mean-field parameters have been proposed in specific cases, a general framework that applies simultaneously to a broad class of loss and penalty has so far been missing. This paper develops a different theory to describe the empirical distribution of βˆ and Xβˆ: Approximations of (βˆ,Xβˆ) in terms of proximal operators are provided that only involve observable adjustments in place of the mean-field parameters. These proposed observable adjustments are data-driven, for example, do not require prior knowledge of the index or the link function. These new adjustments yield confidence intervals for individual components of the index, as well as estimators of the correlation of βˆ with the index, enabling parameter tuning to maximize the correlation. The interplay between loss, regularization and the model is captured in a data-driven manner, without relying on the nonlinear systems studied in previous works. The results are proved to hold both strongly convex regularizers and unregularized M-estimation.