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The paper considers the problem of out-of-sample risk estimation under thehigh dimensional settings where standard techniques such as $K$-fold crossvalidation suffer from large biases. Motivated by the low bias of theleave-one-out cross validation (LO) method, we propose a computationallyefficient closed-form approximate leave-one-out formula (ALO) for a large classof regularized estimators. Given the regularized estimate, calculating ALOrequires minor computational overhead. With minor assumptions about the datagenerating 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 withor even greater than the number of observations, $n$. Despite thehigh-dimensionality of the problem, our theoretical results do not require anysparsity assumption on the vector of regression coefficients. Our extensivenumerical experiments show that $|\text{LO} - \text{ALO}|$ decreases as $n,p$increase, revealing the excellent finite sample performance of ALO. We furtherillustrate the usefulness of our proposed out-of-sample risk estimation methodby an example of real recordings from spatially sensitive neurons (grid cells)in the medial entorhinal cortex of a rat.

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