## Consistent Risk Estimation in High-Dimensional Linear Regression. (arXiv:1902.01753v1 [math.ST])

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 high-dimensional
asymptotic setting $n,p \rightarrow \infty$ and $n/p \rightarrow \delta$
($\delta$ 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 dis查看全文