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

Sylvain Arlot (1, 2, 3) , Matthieu Lerasle (4)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 SIERRA - Statistical Machine Learning and Parsimony
3 DI-ENS - Département d'informatique - ENS Paris
4 LJAD - Laboratoire Jean Alexandre Dieudonné

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.

Keywords: penalization; model selection; density estimation; leave-p-out; resampling penalties; V-fold cross-validation; leave-one-out; Monte-Carlo cross-validation

Domains: Statistics [math.ST]; Statistics Theory [stat.TH]; Machine Learning [cs.LG]; Machine Learning [stat.ML]; Methodology [stat.ME]

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