Error bounds in estimating the outofsample prediction error using leaveoneout cross validation in highdimensions
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Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:40674077, 2020.
Abstract
We study the problem of outofsample risk estimation in the high dimensional regime where both the sample size $n$ and number of features $p$ are large, and $n/p$ can be less than one. Extensive empirical evidence confirms the accuracy of leaveoneout cross validation (LO) for outofsample risk estimation. Yet, a unifying theoretical evaluation of the accuracy of LO in highdimensional problems has remained an open problem. This paper aims to fill this gap for penalized regression in the generalized linear family. With minor assumptions about the data generating process, and without any sparsity assumptions on the regression coefficients, our theoretical analysis obtains finite sample upper bounds on the expected squared error of LO in estimating the outofsample error. Our bounds show that the error goes to zero as $n,p \rightarrow \infty$, even when the dimension $p$ of the feature vectors is comparable with or greater than the sample size $n$. One technical advantage of the theory is that it can be used to clarify and connect some results from the recent literature on scalable approximate LO.
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