Uncertainty quantification for iterative algorithms in linear models with application to early stopping (2404.17856v1)

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.

• The paper introduces novel generalization error estimators that are √n-consistent under Gaussian designs.
• It rigorously analyzes iterative methods like gradient descent and FISTA, detailing their convergence in high-dimensional settings.
• It presents debiasing techniques to construct valid confidence intervals, balancing computational efficiency and statistical accuracy.

Uncertainty Quantification for Iterative Algorithms in High-Dimensional Linear Models

Summary

The paper presents a comprehensive paper on uncertainty quantification for iterative algorithms in high-dimensional linear regression. The focus is on algorithms like Gradient Descent, Proximal Gradient Descent, and their accelerated variants, including Fast Iterative Soft-Thresholding (FISTA). A significant contribution is the development of novel estimators for the generalization error of iterates at any fixed iteration, which are shown to be $\sqrt{n}$-consistent under Gaussian designs. The work also introduces techniques for debiasing corrections and constructing valid confidence intervals for components of the true coefficient vector at any finite iteration.

Analysis of Iterative Algorithms

Iterative algorithms are typically employed where direct solutions are not feasible due to complexity or computational limits. The paper discusses the behavior of such algorithms under the assumption that the feature dimension $p$ is comparable to the sample size $n$, i.e., $p \asymp n$. This setting often leads to challenges in achieving algorithm convergence and estimating error rates effectively. The paper extensively discusses various iterative algorithms, including their convergence properties and limitations in high-dimensional settings.

Estimation of Generalization Error

A novel approach to estimating the generalization error of iterates from iterative algorithms is proposed. The method involves estimating the error at a fixed iteration $t$ using a specifically derived estimator $\hat{r}_t$. The estimator is shown to effectively capture the generalization error, facilitating the selection of an optimal stopping time for the algorithm to minimize error. This is particularly useful in scenarios where iterative processes might converge slowly or oscillate.

Construction of Confidence Intervals

The paper extends the discussion on uncertainty quantification by addressing the construction of confidence intervals for the entries of the coefficient vector in linear models. By leveraging the iterates generated from the algorithms, the proposed method allows for the construction of debiased estimators and confidence intervals without waiting for convergence. This approach is beneficial in practical scenarios where early stopping of the algorithm might be necessary, providing a balance between computational efficiency and statistical accuracy.

Implications and Future Work

The theoretical advancements presented deal with high-dimensional models where traditional assumptions such as $p \ll n$ do not hold, filling a significant gap in statistical theory for such settings. The practical implications are vast, especially in fields where large-scale data analysis is common but computational resources are limited. Future work could explore the extension of these methodologies to other types of models and loss functions, further broadening the applicability of the techniques discussed.

Throughout, the research rigorously adheres to theoretical and methodological robustness, employing extensive simulation studies to validate the theoretical findings. The approach not only augments the current understanding of iterative algorithms in high-dimensional settings but also provides practical tools for data scientists and statisticians working with complex data structures.