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An important question in constructing Cross Validation (CV) estimators of the generalization error is whether rules can be established that allow "optimal" selection of the size of the training set, for fixed sample size $n$. We define the…

Statistics Theory · Mathematics 2015-11-11 Georgios Afendras , Marianthi Markatou

Cross-validation is a widely-used technique to estimate prediction error, but its behavior is complex and not fully understood. Ideally, one would like to think that cross-validation estimates the prediction error for the model at hand, fit…

Methodology · Statistics 2024-03-12 Stephen Bates , Trevor Hastie , Robert Tibshirani

Conformal prediction (CP) is an important tool for distribution-free predictive uncertainty quantification. Yet, a major challenge is to balance computational efficiency and prediction accuracy, particularly for multiple predictions. We…

Machine Learning · Statistics 2025-04-17 Kiljae Lee , Yuan Zhang

Receiver operating characteristic (ROC) analysis is widely used for evaluating diagnostic systems. Recent studies have shown that estimating an area under ROC curve (AUC) with standard cross-validation methods suffers from a large bias. The…

Machine Learning · Statistics 2024-01-25 Ileana Montoya Perez , Antti Airola , Peter J. Boström , Ivan Jambor , Tapio Pahikkala

We consider the parametric learning problem, where the objective of the learner is determined by a parametric loss function. Employing empirical risk minimization with possibly regularization, the inferred parameter vector will be biased…

Machine Learning · Statistics 2017-11-16 Ahmad Beirami , Meisam Razaviyayn , Shahin Shahrampour , Vahid Tarokh

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

Robust estimators for linear regression require non-convex objective functions to shield against adverse affects of outliers. This non-convexity brings challenges, particularly when combined with penalization in high-dimensional settings.…

Computation · Statistics 2025-08-08 David Kepplinger , Siqi Wei

Recursive partitioning approaches producing tree-like models are a long standing staple of predictive modeling, in the last decade mostly as ``sub-learners'' within state of the art ensemble methods like Boosting and Random Forest. However,…

Machine Learning · Statistics 2015-12-14 Amichai Painsky , Saharon Rosset

As a technique that can compactly represent complex patterns, machine learning has significant potential for predictive inference. K-fold cross-validation (CV) is the most common approach to ascertaining the likelihood that a machine…

Machine Learning · Statistics 2026-04-24 Juan M Gorriz , R. Martin Clemente , F Segovia , J Ramirez , A Ortiz , J. Suckling

Leave-one-out cross-validation (LOO) and the widely applicable information criterion (WAIC) are methods for estimating pointwise out-of-sample prediction accuracy from a fitted Bayesian model using the log-likelihood evaluated at the…

Computation · Statistics 2017-12-18 Aki Vehtari , Andrew Gelman , Jonah Gabry

Reinforcement Learning with Verifiable Rewards (RLVR) has become the standard paradigm for LLM mathematical reasoning, where Group Relative Policy Optimization (GRPO) serves as the mainstream algorithm. We point out two understudied…

Artificial Intelligence · Computer Science 2026-05-14 Mingxiong Lin , Zhangquan Gong , Maowen Tang , Qian Li , Chuangchuang Wang , Jian Ma , Sutian Huang , Kai Tang , Haonan Lu

It is crucial to assess the predictive performance of a model to establish its practicality and relevance in real-world scenarios, particularly for high-dimensional data analysis. Among data splitting or resampling methods, cross-validation…

Methodology · Statistics 2025-11-26 Iris Ivy Gauran , Hernando Ombao , Zhaoxia Yu

Statistical machine learning models should be evaluated and validated before putting to work. Conventional k-fold Monte Carlo Cross-Validation (MCCV) procedure uses a pseudo-random sequence to partition instances into k subsets, which…

Machine Learning · Statistics 2019-07-05 Liang Guo , Jianya Liu , Ruodan Lu

This paper studies V-fold cross-validation for model selection in least-squares density estimation. The goal is to provide theoretical grounds for choosing V in order to minimize the least-squares loss of the selected estimator. We first…

Statistics Theory · Mathematics 2015-10-13 Sylvain Arlot , Matthieu Lerasle

K-fold cross validation (CV) is a popular method for estimating the true performance of machine learning models, allowing model selection and parameter tuning. However, the very process of CV requires random partitioning of the data and so…

Computation and Language · Computer Science 2018-06-20 Henry B. Moss , David S. Leslie , Paul Rayson

We propose a simple method for evaluating the model that has been chosen by an adaptive regression procedure, our main focus being the lasso. This procedure deletes each chosen predictor and refits the lasso to get a set of models that are…

Methodology · Statistics 2018-06-06 Leying Guan , Robert Tibshirani

When evaluating and comparing models using leave-one-out cross-validation (LOO-CV), the uncertainty of the estimate is typically assessed using the variance of the sampling distribution. Considering the uncertainty is important, as the…

Methodology · Statistics 2022-02-16 Tuomas Sivula , Måns Magnusson , Aki Vehtari

Used to estimate the risk of an estimator or to perform model selection, cross-validation is a widespread strategy because of its simplicity and its apparent universality. Many results exist on the model selection performances of…

Statistics Theory · Mathematics 2011-02-01 Sylvain Arlot , Alain Celisse

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

Cross-validation is a statistical tool that can be used to improve large covariance matrix estimation. Although its efficiency is observed in practical applications and a convergence result towards the error of the non linear shrinkage is…

Statistics Theory · Mathematics 2025-09-18 Lamia Lamrani , Christian Bongiorno , Marc Potters