Consistent Risk Estimation in Moderately High-Dimensional Linear Regression
This work addresses a gap in theoretical understanding for risk estimation in high-dimensional settings, which is important for statisticians and machine learning practitioners, though it is incremental as it extends existing methods with new analysis.
The paper tackles the problem of risk estimation in moderately high-dimensional linear regression, where the number of predictors grows with observations, and proves the consistency of three existing risk estimates (LOOCV, ALO, and AMP-based techniques) under this setting, providing an upper bound on their convergence rates.
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 $n/p \rightarrow δ>1$ ($δ$ 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.