# Error bounds in estimating the out-of-sample prediction error using leave-one-out cross validation in high-dimensions

arXiv: https://arxiv.org/abs/2003.01770 · HTML: https://mlsanity.com/paper/2003.01770/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Mar 2020.

## Built on by (most cited first, 11 of 11)

- [Subsample Ridge Ensembles: Equivalences and Generalized Cross-Validation](https://mlsanity.com/paper/2304.13016/) (2023)
- [Derivatives and residual distribution of regularized M-estimators with application to adaptive tuning](https://mlsanity.com/paper/2107.05143/) (2021)
- [Certified Data Removal Under High-dimensional Settings](https://mlsanity.com/paper/2505.07640/) (2025)
- [Theoretical Analysis of Leave-one-out Cross Validation for Non-differentiable Penalties under High-dimensional Settings](https://mlsanity.com/paper/2402.08543/) (2024)
- [Extrapolated cross-validation for randomized ensembles](https://mlsanity.com/paper/2302.13511/) (2023)
- [Failures and Successes of Cross-Validation for Early-Stopped Gradient Descent](https://mlsanity.com/paper/2402.16793/) (2024)
- [Approximate Leave-one-out Cross Validation for Regression with ℓ1 Regularizers (extended version)](https://mlsanity.com/paper/2310.17629/) (2023)
- [Newfluence: Boosting Model interpretability and Understanding in High Dimensions](https://mlsanity.com/paper/2507.11895/) (2025)
- [Simultaneous analysis of approximate leave-one-out cross-validation and mean-field inference](https://mlsanity.com/paper/2501.02624/) (2025)
- [Evaluation and Optimization of Leave-one-out Cross-validation for the Lasso](https://mlsanity.com/paper/2508.14368/) (2025)
- [Accelerating Conformal Prediction via Approximate Leave-One-Out](https://mlsanity.com/paper/2606.31915/) (2026)

## Abstract (arXiv)

We study the problem of out-of-sample 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 leave-one-out cross validation (LO) for out-of-sample risk estimation. Yet, a unifying theoretical evaluation of the accuracy of LO in high-dimensional 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 out-of-sample error. Our bounds show that the error goes to zero as n,p rightarrow ∞, 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.