Uncertainty quantification for iterative algorithms in linear models with application to early stopping

TL;DR

This paper develops novel estimators for the generalization error of iterative algorithms in high-dimensional linear regression, enabling effective early stopping and confidence interval construction with theoretical guarantees.

Contribution

It introduces new estimators for generalization error applicable to various iterative algorithms, with proven $ oot n$-consistency and methods for confidence intervals.

Findings

Estimators accurately predict generalization error in high-dimensional settings.

Early stopping based on these estimators improves model performance.

Confidence intervals for coefficients are valid at finite iterations.

Abstract

This paper investigates the iterates $\hbb^1,\dots,\hbb^T$ obtained from iterative algorithms in high-dimensional linear regression problems, in the regime where the feature dimension $p$ is comparable with the sample size $n$, i.e., $p \asymp n$. The analysis and proposed estimators are applicable to Gradient Descent (GD), proximal GD and their accelerated variants such as Fast Iterative Soft-Thresholding (FISTA). The paper proposes novel estimators for the generalization error of the iterate $\hbb^t$ for any fixed iteration $t$ along the trajectory. These estimators are proved to be $\sqrt n$-consistent under Gaussian designs. Applications to early-stopping are provided: when the generalization error of the iterates is a U-shape function of the iteration $t$, the estimates allow to select from the data an iteration $\hat t$ that achieves the smallest generalization error along the trajectory. Additionally, we provide a technique for developing debiasing corrections and valid confidence intervals for the components of the true coefficient vector from the iterate $\hbb^t$ at any finite iteration $t$. Extensive simulations on synthetic data illustrate the theoretical results.