Consistent Risk Estimation in Moderately High-Dimensional Linear Regression

TL;DR

This paper establishes the theoretical consistency of popular risk estimation methods like LOOCV, ALO, and AMP in moderately high-dimensional linear regression where the ratio of observations to predictors approaches a fixed constant.

Contribution

It provides a unifying theoretical framework and rigorous proofs for the consistency of these risk estimates in high-dimensional settings, filling a gap in existing literature.

Findings

Proves consistency of LOOCV, ALO, and AMP-based risk estimates in high dimensions.

Provides bounds on the discrepancy between AMP and LOOCV residuals.

Offers upper bounds on the convergence rates of the risk estimates.

Abstract

Risk estimation is at the core of many learning systems. The importance of this problem has motivated researchers to propose different schemes, such as cross validation, generalized cross validation, and Bootstrap. The theoretical properties of such estimates have been extensively studied in the low-dimensional settings, where the number of predictors $p$ is much smaller than the number of observations $n$. However, a unifying methodology accompanied with a rigorous theory is lacking in high-dimensional settings. This paper studies the problem of risk estimation under the moderately high-dimensional asymptotic setting $n,p \rightarrow \infty$ and $n/p \rightarrow \delta>1$ ($\delta$ is a fixed number), and proves the consistency of three risk estimates that have been successful in numerical studies, i.e., leave-one-out cross validation (LOOCV), approximate leave-one-out (ALO), and approximate message passing (AMP)-based techniques. A corner stone of our analysis is a bound that we obtain on the discrepancy of the `residuals' obtained from AMP and LOOCV. This connection not only enables us to obtain a more refined information on the estimates of AMP, ALO, and LOOCV, but also offers an upper bound on the convergence rate of each estimate.