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 likelihood estimation problem under non-negative constraints. While we provide proof of convergence for certain special cases, the multiplicative scheme has empirically demonstrated monotonic convergence behavior in virtually all our numerical experiments. We show that the presented method can pinpoint corrupted sample points and lead to better regression models when trained on synthetic and real-world scientific 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.
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 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…
Cross-validation can be used to measure a model's predictive accuracy for the purpose of model comparison, averaging, or selection. Standard leave-one-out cross-validation (LOO-CV) requires that the observation model can be factorized into simple terms, but a lot of important models in temporal and spatial statistics do not have this property or are inefficient or unstable when forced into a factorized form. We derive how to efficiently compute and validate both exact and approximate LOO-CV for any Bayesian non-factorized model with a multivariate normal or Student-t distribution on the outcom…
Recently, new methods for model assessment, based on subsampling and posterior approximations, have been proposed for scaling leave-one-out cross-validation (LOO) to large datasets. Although these methods work well for estimating predictive performance for individual models, they are less powerful in model comparison. We propose an efficient method for estimating differences in predictive performance by combining fast approximate LOO surrogates with exact LOO subsampling using the difference estimator and supply proofs with regards to scaling characteristics. The resulting approach can be orde…
When evaluating and comparing models using leave-one-out cross-validation (LOO-CV), the uncertainty of the estimate is typically assessed using the variance of the sampling distribution. Considering the uncertainty is important, as the variability of the estimate can be high in some cases. An important result by Bengio and Grandvalet (2004) states that no general unbiased variance estimator can be constructed, that would apply for any utility or loss measure and any model. We show that it is possible to construct an unbiased estimator considering a specific predictive performance measure and m…
Comparing Scale Parameter Estimators for Gaussian Process Interpolation with the Brownian Motion Prior: Leave-One-Out Cross Validation and Maximum Likelihood
Leave-one-out cross-validation (LOO) and the widely applicable information criterion (WAIC) are methods for estimating pointwise out-of-sample prediction accuracy from a fitted Bayesian model using the log-likelihood evaluated at the posterior simulations of the parameter values. LOO and WAIC have various advantages over simpler estimates of predictive error such as AIC and DIC but are less used in practice because they involve additional computational steps. Here we lay out fast and stable computations for LOO and WAIC that can be performed using existing simulation draws. We introduce an eff…
Predicting the evolution of mortality rates plays a central role for life insurance and pension funds.Various stochastic frameworks have been developed to model mortality patterns taking into account the main stylized facts driving these patterns. However, relying on the prediction of one specific model can be too restrictive and lead to some well documented drawbacks including model misspecification, parameter uncertainty and overfitting. To address these issues we first consider mortality modelling in a Bayesian Negative-Binomial framework to account for overdispersion and the uncertainty ab…
Comparison of competing statistical models is an essential part of psychological research. From a Bayesian perspective, various approaches to model comparison and selection have been proposed in the literature. However, the applicability of these approaches strongly depends on the assumptions about the model space $\mathcal{M}$, the so-called model view. Furthermore, traditional methods like leave-one-out cross-validation (LOO-CV) estimate the expected log predictive density (ELPD) of a model to investigate how the model generalises out-of-sample, which quickly becomes computationally ineffici…
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…
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…
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.
Recent literature provides many computational and modeling approaches for covariance matrices estimation in a penalized Gaussian graphical models but relatively little study has been carried out on the choice of the tuning parameter. This paper tries to fill this gap by focusing on the problem of shrinkage parameter selection when estimating sparse precision matrices using the penalized likelihood approach. Previous approaches typically used K-fold cross-validation in this regard. In this paper, we first derived the generalized approximate cross-validation for tuning parameter selection which …
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…
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.
Consider the problem of modeling hysteresis for finite-state random walks using higher-order Markov chains. This Letter introduces a Bayesian framework to determine, from data, the number of prior states of recent history upon which a trajectory is statistically dependent. The general recommendation is to use leave-one-out cross validation, using an easily-computable formula that is provided in closed form. Importantly, Bayes factors using flat model priors are biased in favor of too-complex a model (more hysteresis) when a large amount of data is present and the Akaike information criterion (…
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…
Probabilistic regression models typically use the Maximum Likelihood Estimation or Cross-Validation to fit parameters. These methods can give an advantage to the solutions that fit observations on average, but they do not pay attention to the coverage and the width of Prediction Intervals. A robust two-step approach is used to address the problem of adjusting and calibrating Prediction Intervals for Gaussian Processes Regression. First, the covariance hyperparameters are determined by a standard Cross-Validation or Maximum Likelihood Estimation method. A Leave-One-Out Coverage Probability is i…
Baruch College City University of New York（纽约市立大学巴鲁克学院）；Columbia University（哥伦比亚大学）
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 spa…
We consider comparisons of statistical learning algorithms using multiple data sets, via leave-one-in cross-study validation: each of the algorithms is trained on one data set; the resulting model is then validated on each remaining data set. This poses two statistical challenges that need to be addressed simultaneously. The first is the assessment of study heterogeneity, with the aim of identifying a subset of studies within which algorithm comparisons can be reliably carried out. The second is the comparison of algorithms using the ensemble of data sets. We address both problems by integrati…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
As the main workhorse for model selection, Cross Validation (CV) has achieved an empirical success due to its simplicity and intuitiveness. However, despite its ubiquitous role, CV often falls into the following notorious dilemmas. On the one hand, for small data cases, CV suffers a conservatively biased estimation, since some part of the limited data has to hold out for validation. On the other hand, for large data cases, CV tends to be extremely cumbersome, e.g., intolerant time-consuming, due to the repeated training procedures. Naturally, a straightforward ambition for CV is to validate th…
California State Polytechnic University, Pomona（加州州立理工大学波莫纳分校）；University of California, Davis（加州大学戴维斯分校）
The large data size and dimensionality of hyperspectral data demands complex processing and data analysis. Multispectral data do not suffer the same limitations, but are normally restricted to blue, green, red, red edge, and near infrared bands. This study aimed to identify the optimal set of spectral bands for nitrogen detection in grape leaves using ensemble feature selection on hyperspectral data from over 3,000 leaves from 150 Flame Seedless table grapevines. Six machine learning base rankers were included in the ensemble: random forest, LASSO, SelectKBest, ReliefF, SVM-RFE, and chaotic cr…
The roles played by learning and memorization represent an important topic in deep learning research. Recent work on this subject has shown that the optimization behavior of DNNs trained on shuffled labels is qualitatively different from DNNs trained with real labels. Here, we propose a novel permutation approach that can differentiate memorization from learning in deep neural networks (DNNs) trained as usual (i.e., using the real labels to guide the learning, rather than shuffled labels). The evaluation of weather the DNN has learned and/or memorized, happens in a separate step where we compa…
Fast Leave-One-Out Approximation from Fragment-Target Prevalence Vectors (molFTP) : From Dummy Masking to Key-LOO for Leakage-Free Feature Construction
Memorial Sloan Kettering Cancer Center（纪念斯隆凯特琳癌症中心）；Rochester Institute of Technology（罗切斯特理工学院）；Institute of Research and Medical Care (IRCCS) SDN（研究与医疗护理研究所（IRCCS）SDN）；University of Cagliari（卡利亚里大学）
Artificial intelligence (AI) has been successful at solving numerous problems in machine perception. In radiology, AI systems are rapidly evolving and show progress in guiding treatment decisions, diagnosing, localizing disease on medical images, and improving radiologists' efficiency. A critical component to deploying AI in radiology is to gain confidence in a developed system's efficacy and safety. The current gold standard approach is to conduct an analytical validation of performance on a generalization dataset from one or more institutions, followed by a clinical validation study of the s…
Random projections and Kernelised Leave One Cluster Out Cross-Validation: Universal baselines and evaluation tools for supervised machine learning for materials properties
University of Liverpool（利物浦大学）；Materials Innovation Factory, University of Liverpool（利物浦大学材料创新工厂）；Leverhulme Research Centre for Functional Materials Design, University of Liverpool（利物浦大学勒沃霍尔姆功能材料设计研究中心）；Department of Chemistry, University of Liverpool（利物浦大学化学系）
Deep neural networks (DNNs) are notorious for their vulnerability to adversarial attacks, which are small perturbations added to their input images to mislead their prediction. Detection of adversarial examples is, therefore, a fundamental requirement for robust classification frameworks. In this work, we present a method for detecting such adversarial attacks, which is suitable for any pre-trained neural network classifier. We use influence functions to measure the impact of every training sample on the validation set data. From the influence scores, we find the most supportive training sampl…
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 derive information theoretic generalization bounds for supervised learning algorithms based on a new measure of leave-one-out conditional mutual information (loo-CMI). Contrary to other CMI bounds, which are black-box bounds that do not exploit the structure of the problem and may be hard to evaluate in practice, our loo-CMI bounds can be computed easily and can be interpreted in connection to other notions such as classical leave-one-out cross-validation, stability of the optimization algorithm, and the geometry of the loss-landscape. It applies both to the output of training algorithms as…
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…
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…
In regular statistical models, the leave-one-out cross-validation is asymptotically equivalent to the Akaike information criterion. However, since many learning machines are singular statistical models, the asymptotic behavior of the cross-validation remains unknown. In previous studies, we established the singular learning theory and proposed a widely applicable information criterion, the expectation value of which is asymptotically equal to the average Bayes generalization loss. In the present paper, we theoretically compare the Bayes cross-validation loss and the widely applicable informati…
We explore the use of neural networks trained with dropout in predicting epileptic seizures from electroencephalographic data (scalp EEG). The input to the neural network is a 126 feature vector containing 9 features for each of the 14 EEG channels obtained over 1-second, non-overlapping windows. The models in our experiments achieved high sensitivity and specificity on patient records not used in the training process. This is demonstrated using leave-one-out-cross-validation across patient records, where we hold out one patient's record as the test set and use all other patients' records for …
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…
Charging Phase Health Indicators for Battery State-of-Health Estimation: A Systematic Comparison of CC, CV, and Combined Approaches under Cross-Battery Validation
