Abstract: The talk will explore properties of the iterates obtained from iterative algorithms in high-dimensional linear regression problems, in the regime where the feature dimension is comparable with the sample size. Examples of common iterative algorithms covered by the analysis include Gradient Descent (GD), proximal GD and their accelerated variants such as Fast Iterative Soft-Thresholding (FISTA), as well as Stochastic Gradient Descent (SDG). For these estimators, we will introduce estimators for the generalization error of the iterate for any fixed iteration along the trajectory. These estimators are proved to be root-n consistent under Gaussian designs. Applications to early-stopping are provided: when the generalization error of the iterates is a U-shape function of the iterations, the estimates allow to select from the data an iteration that achieves the smallest generalization error along the trajectory. Time permitting, we will introduce debiasing corrections and valid confidence intervals for the components of the true coefficient vector from the iterate at any finite iteration.