A scalable estimate of the extra-sample prediction error via approximate leave-one-out

TL;DR

This paper introduces a computationally efficient approximate leave-one-out (ALO) formula for high-dimensional regularized estimators, providing accurate out-of-sample risk estimates with theoretical guarantees and practical validation.

Contribution

It presents a novel closed-form ALO method that approximates leave-one-out cross-validation in high-dimensional settings without requiring sparsity assumptions.

Findings

ALO closely approximates LO with decreasing error as n,p grow.

Theoretical bounds show the error tends to zero with high probability in large samples.

Numerical experiments confirm the method's excellent finite-sample performance.

Abstract

The paper considers the problem of out-of-sample risk estimation under the high dimensional settings where standard techniques such as $K$-fold cross validation suffer from large biases. Motivated by the low bias of the leave-one-out cross validation (LO) method, we propose a computationally efficient closed-form approximate leave-one-out formula (ALO) for a large class of regularized estimators. Given the regularized estimate, calculating ALO requires minor computational overhead. With minor assumptions about the data generating process, we obtain a finite-sample upper bound for $|\text{LO} - \text{ALO}|$. Our theoretical analysis illustrates that $|\text{LO} - \text{ALO}| \rightarrow 0$ with overwhelming probability, when $n,p \rightarrow \infty$, where the dimension $p$ of the feature vectors may be comparable with or even greater than the number of observations, $n$. Despite the high-dimensionality of the problem, our theoretical results do not require any sparsity assumption on the vector of regression coefficients. Our extensive numerical experiments show that $|\text{LO} - \text{ALO}|$ decreases as $n,p$ increase, revealing the excellent finite sample performance of ALO. We further illustrate the usefulness of our proposed out-of-sample risk estimation method by an example of real recordings from spatially sensitive neurons (grid cells) in the medial entorhinal cortex of a rat.