AI 中文总结
本文针对非光滑回归中原-对偶算法的泛化误差估计问题，开发通用递归框架，构造无协方差的数据驱动校正项，证明有限样本保证并通过数值实验验证其可准确跟踪样本外风险。
AI 中文摘要
本文研究非光滑回归中原-对偶算法的轨迹级泛化误差估计问题。启发性示例包括\boldsymbol{\textit{l}}_1\boldsymbol{-}惩罚最小绝对偏差回归和平方根Lasso回归，其中数据拟合损失不可微，且现有针对梯度型优化路径的风险估计器无法直接应用。我们开发了一个通用递归框架，包含Chambolle–Pock算法及相关原-对偶分裂方法。我们通过用过去对偶迭代的加权组合校正每个样本内拟合值来估计风险，理想权重为Stein导数收缩项，依赖于设计协方差。我们从拟合信号轨迹的可观测导数收缩项构造替代权重，得到无协方差、数据驱动的校正项。针对高维高斯设计和固定有限迭代步数，我们证明了两种估计器的有限样本保证；对于平方根岭回归，我们进一步建立了超越高斯设计的匹配高斯普适性结果。数值实验表明，所提出的估计器能准确跟踪有限优化路径上的样本外风险。
英文摘要
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 primal--dual splitting methods. We estimate risk by correcting each in-sample fitted value with a weighted combination of past dual iterates. The ideal weights are Stein derivative contractions and depend on the design covariance. We construct replacement weights from observable derivative contractions of the fitted-signal trajectory, yielding a covariance-free, data-driven correction. For high-dimensional Gaussian designs and fixed finite iteration horizon, we prove finite-sample guarantees for both estimators. For square-root ridge, we further establish a matched-Gaussian universality result beyond Gaussian designs. Numerical experiments show that the proposed estimators accurately track the out-of-sample risk along finite optimization paths.
