Estimating Generalization Error and Constructing Confidence Intervals for Iterative Algorithms in High-Dimensional Linear Regression

Core Concepts

This paper proposes novel estimators for the generalization error of iterates obtained from iterative algorithms such as Gradient Descent, Accelerated Gradient Descent, and FISTA in high-dimensional linear regression problems. The estimators are proved to be √n-consistent under Gaussian designs. The paper also provides a technique for developing debiasing corrections and valid confidence intervals for the components of the true coefficient vector from the iterates.

Abstract

The paper investigates the iterates obtained from iterative algorithms in high-dimensional linear regression problems, where the feature dimension p is comparable to the sample size n. The analysis and proposed estimators are applicable to a broad range of algorithms, including Gradient Descent (GD), proximal GD, and their accelerated variants such as Fast Iterative Soft-Thresholding (FISTA).
The key contributions are:

Introduction of a reliable method for estimating the generalization error of each iterate b_t obtained from the algorithms. The estimator takes the form of a weighted average of the residual vectors, with carefully chosen data-driven weights.

Demonstration of how the proposed estimator of the generalization error can be used to identify the iteration index t that minimizes the true generalization error, enabling an effective early stopping strategy.

Development of a method for constructing valid confidence intervals for the elements of the true coefficient vector b^* based on a given iterate b_t, without requiring the algorithm to converge.

Detailed expressions of the weights for several popular algorithms, including GD, Accelerated GD, ISTA, and FISTA, along with extensive simulation studies corroborating the theoretical findings.

The paper provides a practical solution for determining the optimal stopping iteration of the algorithm to enjoy the smallest generalization error, and enables uncertainty quantification of the iterates without waiting for convergence.

Stats

The following sentences contain key metrics or important figures used to support the author's key logics:
The sample size n and predictor dimension p satisfy p/n ≤ γ for a constant γ ∈ (0, ∞).
The design matrix X has i.i.d. rows from N_p(0, Σ) for some positive definite matrix Σ satisfying 0 < λ_min(Σ) ≤ 1 ≤ λ_max(Σ) and ∥Σ∥_op∥Σ^(-1)∥_op ≤ κ.
The noise ε is independent of X and has i.i.d. entries from N(0, σ^2).

Quotes

"The paper proposes novel estimators for the generalization error of the iterate b_t for any fixed iteration t along the trajectory. These estimators are proved to be √n-consistent under Gaussian designs."
"The proposed estimator of the generalization error serves as a practical proxy for minimizing the true generalization error. To identify the iteration index t such that b_t has the lowest generalization error, one can choose the t that minimizes the estimated generalization error."
"We develop a method for constructing valid confidence intervals for the elements of b^* based on a given iteration t and iterate b_t. Consequently, one needs not wait for the convergence of the algorithm to construct valid confidence intervals for b^*."

Key Insights Distilled From

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

by Pierre C. Be... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.17856.pdf

Deeper Inquiries

How can the proposed framework be extended to handle more general data distributions beyond the Gaussian assumption

To extend the proposed framework to handle more general data distributions beyond the Gaussian assumption, one approach would be to incorporate robust estimation techniques. Robust statistics can handle data that deviates from Gaussian assumptions by using methods that are less sensitive to outliers and non-normality. Techniques such as M-estimation, which minimizes a robust loss function, or quantile regression, which focuses on estimating conditional quantiles, can be integrated into the framework. By adapting the estimation procedures to be robust to different data distributions, the framework can be made more versatile and applicable to a wider range of scenarios.

What are the limitations of the current approach, and how can it be improved to handle non-convex optimization problems more effectively

One limitation of the current approach is its focus on convex optimization problems, which may not fully capture the complexities of non-convex optimization. To improve the handling of non-convex optimization problems, techniques from non-convex optimization theory can be incorporated. This may involve exploring different initialization strategies, such as random restarts or adaptive initialization schemes, to escape local optima. Additionally, incorporating regularization techniques that promote sparsity or low-rank structures can help navigate the non-convex landscape more effectively. By leveraging insights from non-convex optimization theory and incorporating tailored strategies, the framework can better address the challenges posed by non-convex problems.

Can the insights from this work on uncertainty quantification of iterates be leveraged to develop new optimization algorithms that explicitly account for statistical properties during the iterative process

The insights gained from uncertainty quantification of iterates can indeed be leveraged to develop new optimization algorithms that explicitly account for statistical properties during the iterative process. By integrating statistical considerations into the optimization algorithm, one can design algorithms that not only optimize for the objective function but also consider the uncertainty in the estimates. This can lead to algorithms that are more robust, provide confidence intervals for the solutions, and make decisions based on both optimization performance and statistical reliability. By combining optimization and statistical principles, new algorithms can be developed that offer a more comprehensive approach to solving high-dimensional problems.