We conduct a non asymptotic study of the Cross Validation (CV) estimate of the generalization risk for learning algorithms dedicated to extreme regions of the covariates space. In this Extreme Value Analysis context, the risk function measures the algorithm's error given that the norm of the input exceeds a high quantile. The main challenge within this framework is the negligible size of the extreme training sample with respect to the full sample size and the necessity to re-scale the risk function by a probability tending to zero. We open the road to a finite sample understanding of CV for extreme values by establishing two new results: an exponential probability bound on the \Kfold CV error and a polynomial probability bound on the leave-\textrm{p}-out CV. Our bounds are sharp in the sense that they match state-of-the-art guarantees for standard CV estimates while extending them to encompass a conditioning event of small probability. We illustrate the significance of our results regarding high dimensional classification in extreme regions via a Lasso-type logistic regression algorithm. The tightness of our bounds is investigated in numerical experiments.
The lasso and related sparsity inducing algorithms have been the target of substantial theoretical and applied research. Correspondingly, many results are known about their behavior for a fixed or optimally chosen tuning parameter specified up to unknown constants. In practice, however, this oracle tuning parameter is inaccessible so one must use the data to select one. Common statistical practice is to use a variant of cross-validation for this task. However, little is known about the theoretical properties of the resulting predictions with such data-dependent methods. We consider the high-di…
Cross-validation is one of the most popular model selection methods in statistics and machine learning. Despite its wide applicability, traditional cross validation methods tend to select overfitting models, due to the ignorance of the uncertainty in the testing sample. We develop a new, statistically principled inference tool based on cross-validation that takes into account the uncertainty in the testing sample. This new method outputs a set of highly competitive candidate models containing the best one with guaranteed probability. As a consequence, our method can achieve consistent variable…
Department of Analytics, Marketing and Operations, Imperial Business School, UK（分析、营销与运营系，帝国商业学院，英国）；Department of Industrial and Systems Engineering, Viterbi School of Engineering, University of Southern California, CA, USA（工业与系统工程系，维特里商学院，南加州大学，美国）
This paper deals with prediction of anopheles number using environmental and climate variables. The variables selection is performed by an automatic machine learning method based on Lasso and stratified two levels cross validation. Selected variables are debiased while the predictionis generated by simple GLM (Generalized linear model). Finally, the results reveal to be qualitatively better, at selection, the prediction,and the CPU time point of view than those obtained by B-GLM method.
This paper describes a method for performing inference on models chosen by cross-validation. When the test error being minimized in cross-validation is a residual sum of squares it can be written as a quadratic form. This allows us to apply the inference framework in Loftus et al. (2015) for models determined by quadratic constraints to the model that minimizes CV test error. Our only requirement on the model training pro- cedure is that its selection events are regions satisfying linear or quadratic constraints. This includes both Lasso and forward stepwise, which serve as our main examples t…
Cross-validation (CV) methods are popular for selecting the tuning parameter in the high-dimensional variable selection problem. We show the mis-alignment of the CV is one possible reason of its over-selection behavior. To fix this issue, we propose a version of leave-$n_v$-out cross-validation (CV($n_v$)), for selecting the optimal model among the restricted candidate model set for high-dimensional generalized linear models. By using the same candidate model sequence and a proper order of construction sample size $n_c$ in each CV split, CV($n_v$) avoids the potential hurdles in developing the…
The adaptive lasso refers to a class of methods that use weighted versions of the $L_1$-norm penalty, with weights derived from an initial estimate of the parameter vector to be estimated. Irrespective of the method chosen to compute this initial estimate, the performance of the adaptive lasso critically depends on the value of a hyperparameter, which controls the magnitude of the weighted $L_1$-norm in the penalized criterion. As for other machine learning methods, cross-validation is very popular for the calibration of the adaptive lasso, that this for the selection of a data-driven optimal …
Cross-validation is widely used for selecting among a family of learning rules. This paper studies a related method, called aggregated hold-out (Agghoo), which mixes cross-validation with aggregation; Agghoo can also be related to bagging. According to numerical experiments, Agghoo can improve significantly cross-validation's prediction error, at the same computational cost; this makes it very promising as a general-purpose tool for prediction. We provide the first theoretical guarantees on Agghoo, in the supervised classification setting, ensuring that one can use it safely: at worse, Agghoo …
We propose an adaptive bandwidth selector via cross validation for local M-estimators in locally stationary processes. We prove asymptotic optimality of the procedure under mild conditions on the underlying parameter curves. The results are applicable to a wide range of locally stationary processes such linear and nonlinear processes. A simulation study shows that the method works fairly well also in misspecified situations.
Cross validation is a central tool in evaluating the performance of machine learning and statistical models. However, despite its ubiquitous role, its theoretical properties are still not well understood. We study the asymptotic properties of the cross validated-risk for a large class of models. Under stability conditions, we establish a central limit theorem and Berry-Esseen bounds, which enable us to compute asymptotically accurate confidence intervals. Using our results, we paint a big picture for the statistical speed-up of cross validation compared to a train-test split procedure. A corol…
We consider a one-dimensional Gaussian process having exponential covariance function. Under fixed-domain asymptotics, we prove the strong consistency and asymptotic normality of a cross validation estimator of the microergodic covariance parameter. In this setting, Ying [40] proved the same asymptotic properties for the maximum likelihood estimator. Our proof includes several original or more involved components, compared to that of Ying. Also, while the asymptotic variance of maximum likelihood does not depend on the triangular array of observation points under consideration, that of cross v…
This text is a survey on cross-validation. We define all classical cross-validation procedures, and we study their properties for two different goals: estimating the risk of a given estimator, and selecting the best estimator among a given family. For the risk estimation problem, we compute the bias (which can also be corrected) and the variance of cross-validation methods. For estimator selection, we first provide a first-order analysis (based on expectations). Then, we explain how to take into account second-order terms (from variance computations, and by taking into account the usefulness o…
In model selection, several types of cross-validation are commonly used and many variants have been introduced. While consistency of some of these methods has been proven, their rate of convergence to the oracle is generally still unknown. Until now, an asymptotic analysis of crossvalidation able to answer this question has been lacking. Existing results focus on the ''pointwise'' estimation of the risk of a single estimator, whereas analysing model selection requires understanding how the CV risk varies with the model. In this article, we investigate the asymptotics of the CV risk in the neig…
Cross-validation (CV) is a popular method for model-selection. Unfortunately, it is not immediately obvious how to apply CV to unsupervised or exploratory contexts. This thesis discusses some extensions of cross-validation to unsupervised learning, specifically focusing on the problem of choosing how many principal components to keep. We introduce the latent factor model, define an objective criterion, and show how CV can be used to estimate the intrinsic dimensionality of a data set. Through both simulation and theory, we demonstrate that cross-validation is a valuable tool for unsupervised l…
Comparing Scale Parameter Estimators for Gaussian Process Interpolation with the Brownian Motion Prior: Leave-One-Out Cross Validation and Maximum Likelihood
The asymptotic optimality (a.o.) of various hyper-parameter estimators with different optimality criteria has been studied in the literature for regularized least squares regression problems. The estimators include e.g., the maximum (marginal) likelihood method, $C_p$ statistics, and generalized cross validation method, and the optimality criteria are based on e.g., the inefficiency, the expectation inefficiency and the risk. In this paper, we consider the regularized least squares regression problems with fixed number of regression parameters, choose the optimality criterion based on the risk…
The present work aims at deriving theoretical guaranties on the behavior of some cross-validation procedures applied to the $k$-nearest neighbors ($k$NN) rule in the context of binary classification. Here we focus on the leave-$p$-out cross-validation (L$p$O) used to assess the performance of the $k$NN classifier. Remarkably this L$p$O estimator can be efficiently computed in this context using closed-form formulas derived by \cite{CelisseMaryHuard11}. We describe a general strategy to derive moment and exponential concentration inequalities for the L$p$O estimator applied to the $k$NN classif…
Determining the number of factors is essential to factor analysis. In this paper, we propose {an efficient cross validation (CV)} method to determine the number of factors in approximate factor models. The method applies CV twice, first along the directions of observations and then variables, and hence is referred to hereafter as double cross-validation (DCV). Unlike most CV methods, which are prone to overfitting, the DCV is statistically consistent in determining the number of factors when both dimension of variables and sample size are sufficiently large. Simulation studies show that DCV ha…
Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；Univ. Paris-Sud（巴黎南大学）；CNRS（法国国家科学研究中心）；Université Paris-Saclay（巴黎萨克雷大学）；Univ. Nice Sophia Antipolis（尼斯索菲亚·安提波利斯大学）
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 m…
We study the following three fundamental problems about ridge regression: (1) what is the structure of the estimator? (2) how to correctly use cross-validation to choose the regularization parameter? and (3) how to accelerate computation without losing too much accuracy? We consider the three problems in a unified large-data linear model. We give a precise representation of ridge regression as a covariance matrix-dependent linear combination of the true parameter and the noise. We study the bias of $K$-fold cross-validation for choosing the regularization parameter, and propose a simple bias-c…
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 cross-validation procedures. This survey intends to relate these results to the most recent advances of model selection theory, with a particular emphasis on distinguishing empirical statements from rigorous theoretical results. As a conclusion, guidelines are provided for choosing the best cross-validation procedure according to the particular features of the problem…
This article is an invited discussion of the article by Gronau and Wagenmakers (2018) that can be found at https://dx.doi.org/10.1007/s42113-018-0011-7.
We extend the standard Bayesian multivariate Gaussian generative data classifier by considering a generalization of the conjugate, normal-Wishart prior distribution and by deriving the hyperparameters analytically via evidence maximization. The behaviour of the optimal hyperparameters is explored in the high-dimensional data regime. The classification accuracy of the resulting generalized model is competitive with state-of-the art Bayesian discriminant analysis methods, but without the usual computational burden of cross-validation.
Princeton University（普林斯顿大学）；Harvard University（哈佛大学）；Microsoft Research New England（微软研究院新英格兰分院）
This work develops central limit theorems for cross-validation and consistent estimators of its asymptotic variance under weak stability conditions on the learning algorithm. Together, these results provide practical, asymptotically-exact confidence intervals for $k$-fold test error and valid, powerful hypothesis tests of whether one learning algorithm has smaller $k$-fold test error than another. These results are also the first of their kind for the popular choice of leave-one-out cross-validation. In our real-data experiments with diverse learning algorithms, the resulting intervals and tes…
In this paper, we introduce a new concept of stability for cross-validation, called the $\left( β, \varpi \right)$-stability, and use it as a new perspective to build the general theory for cross-validation. The $\left( β, \varpi \right)$-stability mathematically connects the generalization ability and the stability of the cross-validated model via the Rademacher complexity. Our result reveals mathematically the effect of cross-validation from two sides: on one hand, cross-validation picks the model with the best empirical generalization ability by validating all the alternatives on test sets;…
Cross-validation is a well-known and widely used bandwidth selection method in nonparametric regression estimation. However, this technique has two remarkable drawbacks: (i) the large variability of the selected bandwidths, and (ii) the inability to provide results in a reasonable time for very large sample sizes. To overcome these problems, bagging cross-validation bandwidths are analyzed in this paper. This approach consists in computing the cross-validation bandwidths for a finite number of subsamples and then rescaling the averaged smoothing parameters to the original sample size. Under a …
This note corrects a mistake in the paper "consistent cross-validatory model-selection for dependent data: $hv$-block cross-validation" by Racine (2000). In his paper, he implied that the therein proposed $hv$-block cross-validation is consistent in the sense of Shao (1993). To get this intuition, he relied on the speculation that $hv$-block is a balanced incomplete block design (BIBD). This note demonstrates that this is not the case, and thus the theoretical consistency of $hv$-block remains an open question. In addition, I also provide a Python program counting the number of occurrences of …
We consider the problem of bandwidth selection by cross-validation from a sequential point of view in a nonparametric regression model. Having in mind that in applications one often aims at estimation, prediction and change detection simultaneously, we investigate that approach for sequential kernel smoothers in order to base these tasks on a single statistic. We provide uniform weak laws of large numbers and weak consistency results for the cross-validated bandwidth. Extensions to weakly dependent error terms are discussed as well. The errors may be α-mixing or L2-near epoch dependent, which …
Recent advances in statistical theory, together with advances in the computational power of computers, provide alternative methods to do mass-univariate hypothesis testing in which a large number of univariate tests, can be properly used to compare MEEG data at a large number of time-frequency points and scalp locations. One of the major problematic aspects of this kind of mass-univariate analysis is due to high number of accomplished hypothesis tests. Hence procedures that remove or alleviate the increased probability of false discoveries are crucial for this type of analysis. Here, I propose…
Markov state models (MSMs) are a widely used method for approximating the eigenspectrum of the molecular dynamics propagator, yielding insight into the long-timescale statistical kinetics and slow dynamical modes of biomolecular systems. However, the lack of a unified theoretical framework for choosing between alternative models has hampered progress, especially for non-experts applying these methods to novel biological systems. Here, we consider cross-validation with a new objective function for estimators of these slow dynamical modes, a generalized matrix Rayleigh quotient (GMRQ), which mea…
