Generalization Error Estimation for Primal--Dual Algorithms in Non-Smooth Regression

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.