In this article we prove that estimator stability, together with a gradient-growth condition on the expected fitted loss, is enough to provide concentration bounds for leave-one-out cross-validation in a general framework. In particular, we provide concentration bounds beyond Lipschitz continuity assumptions on the loss or on the estimator. We obtain our results by relying on random variables with distribution satisfying the logarithmic Sobolev inequality, providing us a relatively rich class of distributions. We illustrate our method by considering several interesting examples, including linear regression, kernel density estimation, and stabilized/truncated estimators such as stabilized kernel regression.
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…
We demonstrate the consistency of cross validation for comparing multiple density estimators using simple inequalities on the likelihood ratio. In nonparametric problems, the splitting of data does not require the domination of test data over the training/estimation data, contrary to Shao (1993). The result is complementary to that of Yang (2005) and Yang (2006).
A general framework is that the estimators of a distribution are obtained by minimizing a function (the estimating function) and they are assessed through another function (the assessment function). The estimating and assessment functions generally estimate risks. A classical case is that both functions estimate an information risk (specifically cross entropy); in that case Akaike information criterion (AIC) is relevant. In more general cases, the assessment risk can be estimated by leave-one-out crossvalidation. Since leave-one-out crossvalidation is computationally very demanding, an approxi…
Columbia University（哥伦比亚大学）；Baruch College, City University of New York（纽约城市大学巴鲁克学院）
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 properties of such estimates have been extensively studied in the low-dimensional settings, where the number of predictors $p$ is much smaller than the number of observations $n$. However, a unifying methodology accompanied with a rigorous theory is lacking in high-dimensional settings. This paper studies the problem of risk estimation under the moderately high-dimen…
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 $p$ denotes the cardinality of the test set. Closed-form expressions are settled for the Lpo estimator of the risk of projection estimators. These expressions provide a great improvement upon $V$-fold cross-validation in terms of variability and computational complexity. From a theoretical point of view, closed-form expressions also enable to study the Lpo perform…
University of Granada（格拉纳达大学）；University of Cambridge（剑桥大学）
In the 70s a novel branch of statistics emerged focusing its effort in selecting a function in the pattern recognition problem, which fulfils a definite relationship between the quality of the approximation and its complexity. These data-driven approaches are mainly devoted to problems of estimating dependencies with limited sample sizes and comprise all the empirical out-of sample generalization approaches, e.g. cross validation (CV) approaches. Although the latter are \emph{not designed for testing competing hypothesis or comparing different models} in neuroimaging, there are a number of the…
Decoding, ie prediction from brain images or signals, calls for empirical evaluation of its predictive power. Such evaluation is achieved via cross-validation, a method also used to tune decoders' hyper-parameters. This paper is a review on cross-validation procedures for decoding in neuroimaging. It includes a didactic overview of the relevant theoretical considerations. Practical aspects are highlighted with an extensive empirical study of the common decoders in within-and across-subject predictions, on multiple datasets --anatomical and functional MRI and MEG-- and simulations. Theory and e…
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.
The benefits of the wavelet approach for density estimation are well established in the literature, especially when the density to estimate is irregular or heterogeneous in smoothness. However, wavelet density estimates are typically not bona fide densities. In Aya-Moreno et al (2018), a `shape-preserving' wavelet density estimator was introduced, including as main step the estimation of the square-root of the density. A natural concept involving square-root of densities is the Hellinger distance - or equivalently, the Bhattacharyya affinity coefficient. In this paper, we deliver a fully data-…
In the multiple testing context, a challenging problem is the estimation of the proportion $π_0$ of true-null hypotheses. A large number of estimators of this quantity rely on identifiability assumptions that either appear to be violated on real data, or may be at least relaxed. Under independence, we propose an estimator $\hatπ_0$ based on density estimation using both histograms and cross-validation. Due to the strong connection between the false discovery rate (FDR) and $π_0$, many multiple testing procedures (MTP) designed to control the FDR may be improved by introducing an estimator of $…
Although the leave-subject-out cross-validation (CV) has been widely used in practice for tuning parameter selection for various nonparametric and semiparametric models of longitudinal data, its theoretical property is unknown and solving the associated optimization problem is computationally expensive, especially when there are multiple tuning parameters. In this paper, by focusing on the penalized spline method, we show that the leave-subject-out CV is optimal in the sense that it is asymptotically equivalent to the empirical squared error loss function minimization. An efficient Newton-type…
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 possesses---predictive risk consistency, sign consistency, correct model selection---all of it has assumes that the tuning parameter is chosen in an oracle fashion. Yet, this is impossible in practice. Instead, data analysts must use the data twice, once to choose the tuning parameter and again to estimate the model. But only heuristics have ever justified suc…
Comparing Scale Parameter Estimators for Gaussian Process Interpolation with the Brownian Motion Prior: Leave-One-Out Cross Validation and Maximum Likelihood
The present paper provides a new generic strategy leading to non-asymptotic theoretical guarantees on the Leave-one-Out procedure applied to a broad class of learning algorithms. This strategy relies on two main ingredients: the new notion of $L^q$ stability, and the strong use of moment inequalities. $L^q$ stability extends the ongoing notion of hypothesis stability while remaining weaker than the uniform stability. It leads to new PAC exponential generalisation bounds for Leave-one-Out under mild assumptions. In the literature, such bounds are available only for uniform stable algorithms und…
This paper proposes a new method of bandwidth selection in kernel estimation of density and distribution functions motivated by the connection between maximisation of the entropy of probability integral transforms and maximum likelihood in classical parametric models. The proposed estimators are designed to indirectly maximise the entropy of the leave-one-out kernel estimates of a distribution function, which are the analogues of the parametric probability integral transforms. The estimators based on minimisation of the Cramer-von Mises discrepancy, near-solution of the moment-based estimating…
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 …
We investigate the optimality for model selection of the so-called slope heuristics, $V$-fold cross-validation and $V$-fold penalization in a heteroscedastic with random design regression context. We consider a new class of linear models that we call strongly localized bases and that generalize histograms, piecewise polynomials and compactly supported wavelets. We derive sharp oracle inequalities that prove the asymptotic optimality of the slope heuristics---when the optimal penalty shape is known---and $V$ -fold penalization. Furthermore, $V$-fold cross-validation seems to be suboptimal for a…
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…
Kriging-based surrogate models have become very popular during the last decades to approximate a computer code output from few simulations. In practical applications, it is very common to sequentially add new simulations to obtain more accurate approximations. We propose in this paper a method of kriging-based sequential design which combines both the error evaluation providing by the kriging model and the observed errors of a Leave-One-Out cross-validation procedure. This method is proposed in two versions, the first one selects points one at-a-time. The second one allows us to parallelize th…
We present a new family of model selection algorithms based on the resampling heuristics. It can be used in several frameworks, do not require any knowledge about the unknown law of the data, and may be seen as a generalization of local Rademacher complexities and $V$-fold cross-validation. In the case example of least-square regression on histograms, we prove oracle inequalities, and that these algorithms are naturally adaptive to both the smoothness of the regression function and the variability of the noise level. Then, interpretating $V$-fold cross-validation in terms of penalization, we e…
Aggregated hold-out (Agghoo) is a method which averages learning rules selected by hold-out (that is, cross-validation with a single split). We provide the first theoretical guarantees on Agghoo, ensuring that it can be used safely: Agghoo performs at worst like the hold-out when the risk is convex. The same holds true in classification with the 0-1 risk, with an additional constant factor. For the hold-out, oracle inequalities are known for bounded losses, as in binary classification. We show that similar results can be proved, under appropriate assumptions, for other risk-minimization proble…
We consider Bayesian inference in inverse regression problems where the objective is to infer about unobserved covariates from observed responses and covariates. We establish posterior consistency of such unobserved covariates in Bayesian inverse regression problemsunder appropriate priors in a leave-one-out cross-validation setup. We relate this to posterior consistency of inverse reference distributions (Bhattacharya (2013)) for assessing model adequacy. We illustrate our theory and methods with various examples of Bayesian inverse regression, along with adequate simulation experiments.
