Choice of V for V-Fold Cross-Validation in Least-Squares Density Estimation

Sylvain Arlot 1, 2, 3 Matthieu Lerasle 4
2 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique de l'École normale supérieure, ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR8548
Abstract : 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 prove a non-asymptotic oracle inequality for V-fold cross-validation and its bias-corrected version (V-fold penalization). In particular, this result implies that V-fold penalization is asymptotically optimal in the nonparametric case. Then, we compute the variance of V-fold cross-validation and related criteria, as well as the variance of key quantities for model selection performance. We show that these variances depend on V like 1+4/(V-1), at least in some particular cases, suggesting that the performance increases much from V=2 to V=5 or 10, and then is almost constant. Overall, this can explain the common advice to take V=5---at least in our setting and when the computational power is limited---, as supported by some simulation experiments. An oracle inequality and exact formulas for the variance are also proved for Monte-Carlo cross-validation, also known as repeated cross-validation, where the parameter V is replaced by the number B of random splits of the data.
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Submitted on : Friday, October 9, 2015 - 10:02:08 PM
Last modification on : Saturday, February 18, 2017 - 1:20:37 AM
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  • HAL Id : hal-00743931, version 3
  • ARXIV : 1210.5830

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Sylvain Arlot, Matthieu Lerasle. Choice of V for V-Fold Cross-Validation in Least-Squares Density Estimation. 2015. <hal-00743931v3>

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