Cross-validation is a widely-used technique to estimate prediction error, but its behavior is complex and not fully understood. Ideally, one would like to think that cross-validation estimates the prediction error for the model at hand, fit to the training data. We prove that this is not the case for the linear model fit by ordinary least squares; rather it estimates the average prediction error of models fit on other unseen training sets drawn from the same population. We further show that this phenomenon occurs for most popular estimates of prediction error, including data splitting, bootstrapping, and Mallow's Cp. Next, the standard confidence intervals for prediction error derived from cross-validation may have coverage far below the desired level. Because each data point is used for both training and testing, there are correlations among the measured accuracies for each fold, and so the usual estimate of variance is too small. We introduce a nested cross-validation scheme to estimate this variance more accurately, and we show empirically that this modification leads to intervals with approximately correct coverage in many examples where traditional cross-validation intervals fail.
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This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves from an inhomogeneous medium in two dimensions. We propose a deep learning-based iterative reconstruction algorithm for recovering the contrast of the inhomogeneous medium from the far-field data. The proposed algorithm is given by repeated applications of the Landweber method, the iteratively regularized Gauss-Newton method (IRGNM) and a deep neural network. The Landweber method is used to generate initial guesses for the exact contrast, and the IRGNM is employed to make further improvements to the estimated contrast. Our deep neural network (called the learned projector in this paper) mainly focuses on learning the a priori information of the shape of the unknown contrast by using a normalization technique in the training process and is trained to act like a projector which is expected to make the estimated contrast obtained by the Landweber method or the IRGNM closer to the exact contrast. It is believed that the application of the normalization technique can release the burden of training the deep neural network and lead to good performance of the proposed algorithm. Furthermore, the learned projector is expected to provide good initial guesses for IRGNM and be helpful for accelerating the proposed algorithm. Extensive numerical experiments show that our inversion algorithm has a satisfactory reconstruction capacity and good generalization ability.
We study the problem of variance estimation in general graph-structured problems. First, we develop a linear time estimator for the homoscedastic case that can consistently estimate the variance in general graphs. We show that our estimator attains minimax rates for the chain and 2D grid graphs when the mean signal has a total variation with canonical scaling. Furthermore, we provide general upper bounds on the mean squared error performance of the fused lasso estimator in general graphs under a moment condition and a bound on the tail behavior of the errors. These upper bounds allow us to generalize for broader classes of distributions, such as sub-Exponential, many existing results on the fused lasso that are only known to hold with the assumption that errors are sub-Gaussian random variables. Exploiting our upper bounds, we then study a simple total variation regularization estimator for estimating the signal of variances in the heteroscedastic case. Our results show that the variance estimator attains minimax rates for estimating signals of bounded variation in grid graphs, $K$-nearest neighbor graphs with very mild assumptions, and it is consistent for estimating the variances in any connected graph. In addition, extensive numerical results show that our proposed estimators perform reasonably well in a variety of graph-structured models.
Constructing a differentially private (DP) estimator requires deriving the maximum influence of an observation, which can be difficult in the absence of exogenous bounds on the input data or the estimator, especially in high dimensional settings. This paper shows that standard notions of statistical depth, i.e., halfspace depth and regression depth, are particularly advantageous in this regard, both in the sense that the maximum influence of a single observation is easy to analyze and that this value is typically low. This is used to motivate new approximate DP location and regression estimators using the maximizers of these two notions of statistical depth. A more computationally efficient variant of the approximate DP regression estimator is also provided. Also, to avoid requiring that users specify a priori bounds on the estimates and/or the observations, variants of these DP mechanisms are described that satisfy random differential privacy (RDP), which is a relaxation of differential privacy provided by Hall, Wasserman, and Rinaldo (2013). We also provide simulations of the two DP regression methods proposed here. The proposed estimators appear to perform favorably relative to the existing DP regression methods we consider in these simulations when either the sample size is at least 100-200 or the privacy-loss budget is sufficiently high.
The observation that stochastic gradient descent (SGD) favors flat minima has played a fundamental role in understanding implicit regularization of SGD and guiding the tuning of hyperparameters. In this paper, we provide a quantitative explanation of this striking phenomenon by relating the particular noise structure of SGD to its \emph{linear stability} (Wu et al., 2018). Specifically, we consider training over-parameterized models with square loss. We prove that if a global minimum $\theta^*$ is linearly stable for SGD, then it must satisfy $\|H(\theta^*)\|_F\leq O(\sqrt{B}/\eta)$, where $\|H(\theta^*)\|_F, B,\eta$ denote the Frobenius norm of Hessian at $\theta^*$, batch size, and learning rate, respectively. Otherwise, SGD will escape from that minimum \emph{exponentially} fast. Hence, for minima accessible to SGD, the flatness -- as measured by the Frobenius norm of the Hessian -- is bounded independently of the model size and sample size. The key to obtaining these results is exploiting the particular geometry awareness of SGD noise: 1) the noise magnitude is proportional to loss value; 2) the noise directions concentrate in the sharp directions of local landscape. This property of SGD noise provably holds for linear networks and random feature models (RFMs) and is empirically verified for nonlinear networks. Moreover, the validity and practical relevance of our theoretical findings are justified by extensive numerical experiments.
Modern statistical analyses often encounter datasets with massive sizes and heavy-tailed distributions. For datasets with massive sizes, traditional estimation methods can hardly be used to estimate the extreme value index directly. To address the issue, we propose here a subsampling-based method. Specifically, multiple subsamples are drawn from the whole dataset by using the technique of simple random subsampling with replacement. Based on each subsample, an approximate maximum likelihood estimator can be computed. The resulting estimators are then averaged to form a more accurate one. Under appropriate regularity conditions, we show theoretically that the proposed estimator is consistent and asymptotically normal. With the help of the estimated extreme value index, we can estimate high-level quantiles and tail probabilities of a heavy-tailed random variable consistently. Extensive simulation experiments are provided to demonstrate the promising performance of our method. A real data analysis is also presented for illustration purpose.
When training a machine learning classifier on data where one of the classes is intrinsically rare, the classifier will often assign too few sources to the rare class. To address this, it is common to up-weight the examples of the rare class to ensure it isn't ignored. It is also a frequent practice to train on restricted data where the balance of source types is closer to equal for the same reason. Here we show that these practices can bias the model toward over-assigning sources to the rare class. We also explore how to detect when training data bias has had a statistically significant impact on the trained model's predictions, and how to reduce the bias's impact. While the magnitude of the impact of the techniques developed here will vary with the details of the application, for most cases it should be modest. They are, however, universally applicable to every time a machine learning classification model is used, making them analogous to Bessel's correction to the sample variance.
The median absolute deviation is a widely used robust measure of statistical dispersion. Using a scale constant, we can use it as an asymptotically consistent estimator for the standard deviation under normality. For finite samples, the scale constant should be corrected in order to obtain an unbiased estimator. The bias-correction factor depends on the sample size and the median estimator. When we use the traditional sample median, the factor values are well known, but this approach does not provide optimal statistical efficiency. In this paper, we present the bias-correction factors for the median absolute deviation based on the Harrell-Davis quantile estimator and its trimmed modification which allow us to achieve better statistical efficiency of the standard deviation estimations. The obtained estimators are especially useful for samples with a small number of elements.
Model selection in machine learning (ML) is a crucial part of the Bayesian learning procedure. Model choice may impose strong biases on the resulting predictions, which can hinder the performance of methods such as Bayesian neural networks and neural samplers. On the other hand, newly proposed approaches for Bayesian ML exploit features of approximate inference in function space with implicit stochastic processes (a generalization of Gaussian processes). The approach of Sparse Implicit Processes (SIP) is particularly successful in this regard, since it is fully trainable and achieves flexible predictions. Here, we expand on the original experiments to show that SIP is capable of correcting model bias when the data generating mechanism differs strongly from the one implied by the model. We use synthetic datasets to show that SIP is capable of providing predictive distributions that reflect the data better than the exact predictions of the initial, but wrongly assumed model.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
In recent years, DBpedia, Freebase, OpenCyc, Wikidata, and YAGO have been published as noteworthy large, cross-domain, and freely available knowledge graphs. Although extensively in use, these knowledge graphs are hard to compare against each other in a given setting. Thus, it is a challenge for researchers and developers to pick the best knowledge graph for their individual needs. In our recent survey, we devised and applied data quality criteria to the above-mentioned knowledge graphs. Furthermore, we proposed a framework for finding the most suitable knowledge graph for a given setting. With this paper we intend to ease the access to our in-depth survey by presenting simplified rules that map individual data quality requirements to specific knowledge graphs. However, this paper does not intend to replace our previously introduced decision-support framework. For an informed decision on which KG is best for you we still refer to our in-depth survey.
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