We consider observations $(X,y)$ from single index models with unknown link function, Gaussian covariates and a regularized M-estimator $\hat\beta$ 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 $\hat\beta$ and the predicted values $X\hat\beta$ 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$(\hat\beta,X\hat\beta)$ and the corresponding proximal operators require solving fixed-point equations that typically involve unobservable quantities such as the prior distribution on the index or the link function. This paper develops a different theory to describe the empirical distribution of $\hat\beta$ and $X\hat\beta$: Approximations of $(\hat\beta,X\hat\beta)$ in terms of proximal operators are provided that only involve observable adjustments. These proposed observable adjustments are data-driven, e.g., 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 $\hat\beta$ with the index. The interplay between loss, regularization and the model is thus captured in a data-driven manner, without solving the fixed-point equations studied in previous works. The results apply to both strongly convex regularizers and unregularized M-estimation. Simulations are provided for the square and logistic loss in single index models including logistic regression and 1-bit compressed sensing with 20\% corrupted bits.

翻译：我们考虑具有未知链接函数、高斯协变量及由凸损失函数与正则化器构建的正则化M估计量$\hat\beta$的单指标模型观测值$(X,y)$。在样本量$n$与维度$p$同步增长且$p/n$趋于有限极限的框架下，$\hat\beta$的经验分布及其预测值$X\hat\beta$的性态已在多个模型中得到刻画：已知这些经验分布收敛于相关高斯序列模型中损失函数与惩罚项的近端算子，该模型捕捉了比率$p/n$、损失函数、正则化与数据生成过程间的相互作用。建立$(\hat\beta,X\hat\beta)$与对应近端算子之间的关联需要求解通常涉及不可观测量（如指标先验分布或链接函数）的固定点方程。本文发展了一种描述$\hat\beta$与$X\hat\beta$经验分布的新理论：我们给出了仅涉及可观测调整量的$(\hat\beta,X\hat\beta)$近端算子近似表达式。所提出的可观测调整量是数据驱动的（例如，无需指标或链接函数的先验知识）。这些新调整量可构建指标分量的置信区间以及$\hat\beta$与指标相关性的估计量。由此，损失函数、正则化与模型之间的相互作用可通过数据驱动方式被捕获，无需求解先前研究中研究的固定点方程。该结果适用于强凸正则化器与无正则化M估计。我们针对单指标模型中的平方损失与逻辑损失（包括逻辑回归及含20%比特损坏的1比特压缩感知）提供了仿真验证。