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Despite a large and significant body of recent work focused on estimating the out-of-sample risk of regularized models in the high dimensional regime, a theoretical understanding of this problem for non-differentiable penalties such as…

Statistics Theory · Mathematics 2024-02-15 Haolin Zou , Arnab Auddy , Kamiar Rahnama Rad , Arian Maleki

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…

Methodology · Statistics 2020-02-12 Kamiar Rahnama Rad , Arian Maleki

The out-of-sample error (OO) is the main quantity of interest in risk estimation and model selection. Leave-one-out cross validation (LO) offers a (nearly) distribution-free yet computationally demanding approach to estimate OO. Recent…

Statistics Theory · Mathematics 2023-10-27 Arnab Auddy , Haolin Zou , Kamiar Rahnama Rad , Arian Maleki

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…

Statistics Theory · Mathematics 2021-01-19 Ji Xu , Arian Maleki , Kamiar Rahnama Rad , Daniel Hsu

Estimating out-of-sample risk for models trained on large high-dimensional datasets is an expensive but essential part of the machine learning process, enabling practitioners to optimally tune hyperparameters. Cross-validation (CV) serves…

Statistics Theory · Mathematics 2025-04-28 Parth Nobel , Daniel LeJeune , Emmanuel J. Candès

Model inference, such as model comparison, model checking, and model selection, is an important part of model development. Leave-one-out cross-validation (LOO) is a general approach for assessing the generalizability of a model, but…

Machine Learning · Statistics 2020-08-12 Måns Magnusson , Michael Riis Andersen , Johan Jonasson , Aki Vehtari

Leave-one-out cross-validation (LOOCV) can be particularly accurate among cross-validation (CV) variants for machine learning assessment tasks -- e.g., assessing methods' error or variability. But it is expensive to re-fit a model $N$ times…

Machine Learning · Statistics 2020-06-24 William T. Stephenson , Tamara Broderick

Recently, new methods for model assessment, based on subsampling and posterior approximations, have been proposed for scaling leave-one-out cross-validation (LOO) to large datasets. Although these methods work well for estimating predictive…

Methodology · Statistics 2020-08-12 Måns Magnusson , Michael Riis Andersen , Johan Jonasson , Aki Vehtari

Leave-one-out cross-validation (LOO-CV) is a popular method for estimating out-of-sample predictive accuracy. However, computing LOO-CV criteria can be computationally expensive due to the need to fit the model multiple times. In the…

Computation · Statistics 2023-09-28 Luca Silva , Giacomo Zanella

We study prediction intervals based on leave-one-out residuals in a linear regression model where the number of explanatory variables can be large compared to sample size. We establish uniform asymptotic validity (conditional on the…

Statistics Theory · Mathematics 2016-02-19 Lukas Steinberger , Hannes Leeb

The lasso procedure is ubiquitous in the statistical and signal processing literature, and as such, is the target of substantial theoretical and applied research. While much of this research focuses on the desirable properties that lasso…

Statistics Theory · Mathematics 2013-08-06 Darren Homrighausen , Daniel J. McDonald

Leave-one-out (LOO) prediction provides a principled, data-dependent measure of generalization, yet guarantees in fully transductive settings remain poorly understood beyond specialized models. We introduce Median of Level-Set Aggregation…

Machine Learning · Computer Science 2026-03-03 Jian Qian , Jiachen Xu

A generic out-of-sample error estimate is proposed for robust $M$-estimators regularized with a convex penalty in high-dimensional linear regression where $(X,y)$ is observed and $p,n$ are of the same order. If $\psi$ is the derivative of…

Statistics Theory · Mathematics 2023-03-31 Pierre C Bellec

We present a weighted version of Leave-One-Out (LOO) cross-validation for estimating the Integrated Squared Error (ISE) when approximating an unknown function by a predictor that depends linearly on evaluations of the function over a finite…

Machine Learning · Statistics 2025-05-27 Luc Pronzato , Maria-João Rendas

Despite the tremendous empirical success of deep learning models to solve various learning tasks, our theoretical understanding of their generalization ability is very limited. Classical generalization bounds based on tools such as the VC…

Machine Learning · Computer Science 2022-03-08 Gregor Bachmann , Thomas Hofmann , Aurélien Lucchi

We analyze the performance of cross-validation (CV) in the density estimation framework with two purposes: (i) risk estimation and (ii) model selection. The main focus is given to the so-called leave-$p$-out CV procedure (Lpo), where $p$…

Statistics Theory · Mathematics 2014-10-02 Alain Celisse

In spite of the wealth of literature on the theoretical properties of the Lasso, there is very little known when the value of the tuning parameter is chosen using the data, even though this is what actually happens in practice. We give a…

Statistics Theory · Mathematics 2016-09-02 Sourav Chatterjee , Jafar Jafarov

We consider the problem of fitting the parameters of a high-dimensional linear regression model. In the regime where the number of parameters $p$ is comparable to or exceeds the sample size $n$, a successful approach uses an…

Statistics Theory · Mathematics 2013-11-04 Adel Javanmard , Andrea Montanari

In recent decades, multilevel regression and poststratification (MRP) has surged in popularity for population inference. However, the validity of the estimates can depend on details of the model, and there is currently little research on…

Methodology · Statistics 2022-09-07 Swen Kuh , Lauren Kennedy , Qixuan Chen , Andrew Gelman

Approximate Leave-One-Out Cross-Validation (ALO-CV) is a method that has been proposed to estimate the generalization error of a regularized estimator in the high-dimensional regime where dimension and sample size are of the same order, the…

Statistics Theory · Mathematics 2026-02-13 Pierre C Bellec
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