We study the problem of out-of-sample risk estimation in the high dimensional regime where both the sample size $n$ and number of features $p$ are large, and $n/p$ can be less than one. Extensive empirical evidence confirms the accuracy of leave-one-out cross validation (LO) for out-of-sample risk estimation. Yet, a unifying theoretical evaluation of the accuracy of LO in high-dimensional problems has remained an open problem. This paper aims to fill this gap for penalized regression in the generalized linear family. With minor assumptions about the data generating process, and without any sparsity assumptions on the regression coefficients, our theoretical analysis obtains finite sample upper bounds on the expected squared error of LO in estimating the out-of-sample error. Our bounds show that the error goes to zero as $n,p \rightarrow \infty$, even when the dimension $p$ of the feature vectors is comparable with or greater than the sample size $n$. One technical advantage of the theory is that it can be used to clarify and connect some results from the recent literature on scalable approximate LO.
For a large class of regularized models, leave-one-out cross-validation can be efficiently estimated with an approximate leave-one-out formula (ALO). We consider the problem of adjusting hyperparameters so as to optimize ALO. We derive efficient formulas to compute the gradient and hessian of ALO and show how to apply a second-order optimizer to find hyperparameters. We demonstrate the usefulness of the proposed approach by finding hyperparameters for regularized logistic regression and ridge regression on various real-world data sets.
Aalto University（阿尔托大学）；Technical University of Denmark（丹麦技术大学）；Chalmers University of Technology（查尔姆斯理工大学）；University of Gothenburg（哥德堡大学）
Model inference, such as model comparison, model checking, and model selection, is an important part of model development. Leave-one-out cross-validation (LOO) is a general approach for assessing the generalizability of a model, but unfortunately, LOO does not scale well to large datasets. We propose a combination of using approximate inference techniques and probability-proportional-to-size-sampling (PPS) for fast LOO model evaluation for large datasets. We provide both theoretical and empirical results showing good properties for large data.
University of California, Merced（加州大学默塞德分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We present a simple algorithm for identifying and correcting real-valued noisy labels from a mixture of clean and corrupted sample points using Gaussian process regression. A heteroscedastic noise model is employed, in which additive Gaussian noise terms with independent variances are associated with each and all of the observed labels. Optimizing the noise model using maximum likelihood estimation leads to the containment of the GPR model's predictive error by the posterior standard deviation in leave-one-out cross-validation. A multiplicative update scheme is proposed for solving the maximum…
SLAC National Accelerator Laboratory（SLAC国家加速器实验室）
One challenge impeding the analysis of terabyte scale x-ray scattering data from the Linac Coherent Light Source LCLS, is determining the number of clusters required for the execution of traditional clustering algorithms. Here we demonstrate that previous work using bi-cross validation (BCV) to determine the number of singular vectors directly maps to the spectral clustering problem of estimating both the number of clusters and hyper parameter values. These results indicate that the process of estimating the number of clusters should not be divorced from the process of estimating other hyper p…
In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for subagged estimators, both for classification and regressor. General loss functions and class of predictors with both finite and infinite VC-dimension are considered. We slightly generalize the formalism introduced by \cite{DUD03} to cover a large variety of cross-validation procedures including leave-one-out cross-validation, $k$-fold cross-validation, hold-out cross-validation (or split sample), and the leave-$\upsilon$-out cross-validation. \bigskip \noindent An interesting…
University of Victoria（维多利亚大学）；Helwan University（赫勒万大学）
First, we analyze the variance of the Cross Validation (CV)-based estimators used for estimating the performance of classification rules. Second, we propose a novel estimator to estimate this variance using the Influence Function (IF) approach that had been used previously very successfully to estimate the variance of the bootstrap-based estimators. The motivation for this research is that, as the best of our knowledge, the literature lacks a rigorous method for estimating the variance of the CV-based estimators. What is available is a set of ad-hoc procedures that have no mathematical foundat…
In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for stable predictors in the context of risk assessment. The notion of stability has been first introduced by \cite{DEWA79} and extended by \cite{KEA95}, \cite{BE01} and \cite{KUNIY02} to characterize class of predictors with infinite VC dimension. In particular, this covers $k$-nearest neighbors rules, bayesian algorithm (\cite{KEA95}), boosting,... General loss functions and class of predictors are considered. We use the formalism introduced by \cite{DUD03} to cover a large var…
The future predictive performance of a Bayesian model can be estimated using Bayesian cross-validation. In this article, we consider Gaussian latent variable models where the integration over the latent values is approximated using the Laplace method or expectation propagation (EP). We study the properties of several Bayesian leave-one-out (LOO) cross-validation approximations that in most cases can be computed with a small additional cost after forming the posterior approximation given the full data. Our main objective is to assess the accuracy of the approximative LOO cross-validation estima…
We investigate the accuracy of the two most common estimators for the maximum expected value of a general set of random variables: a generalization of the maximum sample average, and cross validation. No unbiased estimator exists and we show that it is non-trivial to select a good estimator without knowledge about the distributions of the random variables. We investigate and bound the bias and variance of the aforementioned estimators and prove consistency. The variance of cross validation can be significantly reduced, but not without risking a large bias. The bias and variance of different va…
Institute of Landscape Ecology（景观生态研究所）；Westfälische Wilhelms-Universität Münster（明斯特大学）；Institute for Geoinformatics（地理信息学研究所）
Predictive modelling using machine learning has become very popular for spatial mapping of the environment. Models are often applied to make predictions far beyond sampling locations where new geographic locations might considerably differ from the training data in their environmental properties. However, areas in the predictor space without support of training data are problematic. Since the model has no knowledge about these environments, predictions have to be considered uncertain. Estimating the area to which a prediction model can be reliably applied is required. Here, we suggest a method…
In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for empirical risk minimizers. In the general setting, we prove sanity-check bounds in the spirit of \cite{KR99} \textquotedblleft\textit{bounds showing that the worst-case error of this estimate is not much worse that of training error estimate} \textquotedblright . General loss functions and class of predictors with finite VC-dimension are considered. We closely follow the formalism introduced by \cite{DUD03} to cover a large variety of cross-validation procedures including lea…
Tokyo Institute of Technology（东京工业大学）；Hiroshima University（广岛大学）
Cross-validation (CV) is a technique for evaluating the ability of statistical models/learning systems based on a given data set. Despite its wide applicability, the rather heavy computational cost can prevent its use as the system size grows. To resolve this difficulty in the case of Bayesian linear regression, we develop a formula for evaluating the leave-one-out CV error approximately without actually performing CV. The usefulness of the developed formula is tested by statistical mechanical analysis for a synthetic model. This is confirmed by application to a real-world supernova data set a…
In Bayesian statistics, the marginal likelihood, also known as the evidence, is used to evaluate model fit as it quantifies the joint probability of the data under the prior. In contrast, non-Bayesian models are typically compared using cross-validation on held-out data, either through $k$-fold partitioning or leave-$p$-out subsampling. We show that the marginal likelihood is formally equivalent to exhaustive leave-$p$-out cross-validation averaged over all values of $p$ and all held-out test sets when using the log posterior predictive probability as the scoring rule. Moreover, the log poster…
We consider the problem of estimating the parameters of the covariance function of a Gaussian process by cross-validation. We suggest using new cross-validation criteria derived from the literature of scoring rules. We also provide an efficient method for computing the gradient of a cross-validation criterion. To the best of our knowledge, our method is more efficient than what has been proposed in the literature so far. It makes it possible to lower the complexity of jointly evaluating leave-one-out criteria and their gradients.
Columbia University（哥伦比亚大学）；Massachusetts Institute of Technology（麻省理工学院）；Google Research（谷歌研究院）
Consider the following class of learning schemes: $$\hat{\boldsymbolβ} := \arg\min_{\boldsymbolβ}\;\sum_{j=1}^n \ell(\boldsymbol{x}_j^\top\boldsymbolβ; y_j) + λR(\boldsymbolβ),\qquad\qquad (1) $$ where $\boldsymbol{x}_i \in \mathbb{R}^p$ and $y_i \in \mathbb{R}$ denote the $i^{\text{th}}$ feature and response variable respectively. Let $\ell$ and $R$ be the loss function and regularizer, $\boldsymbolβ$ denote the unknown weights, and $λ$ be a regularization parameter. Finding the optimal choice of $λ$ is a challenging problem in high-dimensional regimes where both $n$ and $p$ are large. We pro…
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…
Cross validation residuals are well known for the ordinary least squares model. Here leave-M-out cross validation is extended to generalised least squares. The relationship between cross validation residuals and Cook's distance is demonstrated, in terms of an approximation to the difference in the generalised residual sum of squares for a model fit to all the data (training and test) and a model fit to a reduced dataset (training data only). For generalised least squares, as for ordinary least squares, there is no need to refit the model to reduced size datasets as all the values for K fold cr…
Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学）
Weighted SVM (or fuzzy SVM) is the most widely used SVM variant owning its effectiveness to the use of instance weights. Proper selection of the instance weights can lead to increased generalization performance. In this work, we extend the span error bound theory to weighted SVM and we introduce effective hyperparameter selection methods for the weighted SVM algorithm. The significance of the presented work is that enables the application of span bound and span-rule with weighted SVM. The span bound is an upper bound of the leave-one-out error that can be calculated using a single trained SVM …
University of Victoria（维多利亚大学）；Helwan University（赫尔旺大学）
Many versions of cross-validation (CV) exist in the literature; and each version though has different variants. All are used interchangeably by many practitioners; yet, without explanation to the connection or difference among them. This article has three contributions. First, it starts by mathematical formalization of these different versions and variants that estimate the error rate and the Area Under the ROC Curve (AUC) of a classification rule, to show the connection and difference among them. Second, we prove some of their properties and prove that many variants are either redundant or "n…
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
Gaussian processes (GPs) are Bayesian non-parametric models useful in a myriad of applications. Despite their popularity, the cost of GP predictions (quadratic storage and cubic complexity with respect to the number of training points) remains a hurdle in applying GPs to large data. We present a fast posterior mean prediction algorithm called FastMuyGPs to address this shortcoming. FastMuyGPs is based upon the MuyGPs hyperparameter estimation algorithm and utilizes a combination of leave-one-out cross-validation, batching, nearest neighbors sparsification, and precomputation to provide scalabl…
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…
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…
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…
Leave-one-out cross-validation (LOOCV) can be particularly accurate among cross-validation (CV) variants for machine learning assessment tasks -- e.g., assessing methods' error or variability. But it is expensive to re-fit a model $N$ times for a dataset of size $N$. Previous work has shown that approximations to LOOCV can be both fast and accurate -- when the unknown parameter is of small, fixed dimension. But these approximations incur a running time roughly cubic in dimension -- and we show that, besides computational issues, their accuracy dramatically deteriorates in high dimensions. Auth…
Gaussian processes (GPs) are non-linear probabilistic models popular in many applications. However, naïve GP realizations require quadratic memory to store the covariance matrix and cubic computation to perform inference or evaluate the likelihood function. These bottlenecks have driven much investment in the development of approximate GP alternatives that scale to the large data sizes common in modern data-driven applications. We present in this manuscript MuyGPs, a novel efficient GP hyperparameter estimation method. MuyGPs builds upon prior methods that take advantage of the nearest neighbo…
Shandong University（山东大学）；Loughborough University（拉夫堡大学）；University of Cambridge（剑桥大学）
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 usually causes subsampling bias, inflates generalization errors and jeopardizes the reliability and effectiveness of cross-validation. Based on ordered systematic sampling theory in statistics and low-discrepancy sequence theory in number theory, we propose a new k-fold cross-validation procedure by replacing a pseudo-random sequence with a best-discrepancy seque…
Georgetown University, Walsh School of Foreign Service（乔治城大学沃尔什外交学院）
Test, Evaluation, Verification, and Validation (TEVV) for Artificial Intelligence (AI) is a challenge that threatens to limit the economic and societal rewards that AI researchers have devoted themselves to producing. A central task of TEVV for AI is estimating brittleness, where brittleness implies that the system functions well within some bounds and poorly outside of those bounds. This paper argues that neither of those criteria are certain of Deep Neural Networks. First, highly touted AI successes (eg. image classification and speech recognition) are orders of magnitude more failure-prone …
