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非光滑回归中原始—对偶算法路径的泛化误差估计AI 解读

This paper studies trajectory-wise estimation of generalization error for primal--dual algorithms in non-smooth regression. Motivating examples include \(\ell_1\)-penalized least absolute deviations regression and square-root Lasso regression, where the data-fitting loss is non-differentiable and existing risk estimators for gradient-type optimization paths do not apply directly. We develop a general recursive framework that includes the Chambolle--Pock algorithm and related…

单指标模型中正则化 M 估计量的可观测调整AI 解读

We consider observations $(X,y)$ from single index models with unknown link function, Gaussian covariates and a regularized M-estimator $\hatβ$ 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 $\hatβ$ and the predicted values $X\hatβ$ has been previously characterized in a number of models: The empirical…

近似留一交叉验证与平均场推断的同步分析AI 解读

Approximate Leave-One-Out Cross-Validation (ALO-CV) is a method that has been proposed to estimate the generalization error of a regularized estimator in the high-dimensional regime where dimension and sample size are of the same order, the so-called ``proportional regime''. A new analysis is developed to derive the consistency of ALO-CV for non-differentiable regularizers under Gaussian covariates and strong convexity. Using a conditioning argument, the difference between…