Gradient-enhanced Kriging (GE-Kriging) is a well-established surrogate modelling technique for approximating expensive computational models. However, it tends to get impractical for high-dimensional problems due to the size of the inherent correlation matrix and the associated high-dimensional hyper-parameter tuning problem. To address these issues, a new method, called sliced GE-Kriging (SGE-Kriging), is developed in this paper for reducing both the size of the correlation matrix and the number of hyper-parameters. We first split the training sample set into multiple slices, and invoke Bayes' theorem to approximate the full likelihood function via a sliced likelihood function, in which multiple small correlation matrices are utilized to describe the correlation of the sample set rather than one large one. Then, we replace the original high-dimensional hyper-parameter tuning problem with a low-dimensional counterpart by learning the relationship between the hyper-parameters and the derivative-based global sensitivity indices. The performance of SGE-Kriging is finally validated by means of numerical experiments with several benchmarks and a high-dimensional aerodynamic modeling problem. The results show that the SGE-Kriging model features an accuracy and robustness that is comparable to the standard one but comes at much less training costs. The benefits are most evident for high-dimensional problems with tens of variables.

We consider observations $(X,y)$ from single index models with unknown link function, Gaussian covariates and a regularized M-estimator $\hat\beta$ constructed from convex loss function and regularizer. In the regime where sample size $n$ and dimension $p$ are both increasing such that $p/n$ has a finite limit, the behavior of the empirical distribution of $\hat\beta$ and the predicted values $X\hat\beta$ has been previously characterized in a number of models: The empirical distributions are known to converge to proximal operators of the loss and penalty in a related Gaussian sequence model, which captures the interplay between ratio $p/n$, loss, regularization and the data generating process. This connection between$(\hat\beta,X\hat\beta)$ and the corresponding proximal operators require solving fixed-point equations that typically involve unobservable quantities such as the prior distribution on the index or the link function. This paper develops a different theory to describe the empirical distribution of $\hat\beta$ and $X\hat\beta$: Approximations of $(\hat\beta,X\hat\beta)$ in terms of proximal operators are provided that only involve observable adjustments. These proposed observable adjustments are data-driven, e.g., do not require prior knowledge of the index or the link function. These new adjustments yield confidence intervals for individual components of the index, as well as estimators of the correlation of $\hat\beta$ with the index. The interplay between loss, regularization and the model is thus captured in a data-driven manner, without solving the fixed-point equations studied in previous works. The results apply to both strongly convex regularizers and unregularized M-estimation. Simulations are provided for the square and logistic loss in single index models including logistic regression and 1-bit compressed sensing with 20\% corrupted bits.

This paper discusses pairing double/debiased machine learning (DDML) with stacking, a model averaging method for combining multiple candidate learners, to estimate structural parameters. We introduce two new stacking approaches for DDML: short-stacking exploits the cross-fitting step of DDML to substantially reduce the computational burden and pooled stacking enforces common stacking weights over cross-fitting folds. Using calibrated simulation studies and two applications estimating gender gaps in citations and wages, we show that DDML with stacking is more robust to partially unknown functional forms than common alternative approaches based on single pre-selected learners. We provide Stata and R software implementing our proposals.

The neural tangent kernel (NTK) theory (Jacot et al., 2018), which shows that the gradient kernel regression approximates the over-parametrized neural network trained by gradient descent well (Jacot et al., 2018; Allen-Zhu et al., 2019; Lee et al., 2019), brings us a natural surrogate to understand the generalization behavior of the neural networks in certain circumstances. This surrogate has led to recent renaissance of the study of kernel methods. For example, one would ask whether overfitting could harm the generalization (Bartlett et al., 2020), how the smoothness of the underlying regression function would affect the generalization error (Li et al., 2023), or if one can determine the lower bound of the generalization error at a specific function? All these problems can be answered by the generalization error curve which aims at determining the exact generalization error of a certain kernel regression method with respect to the kernel, the regression function, the noise level and the choice of the regularization parameter. It is clear that such a generalization error curve would provide a comprehensive picture of the generalization ability of the corresponding kernel regression method (Bordelon et al., 2020; Cui et al., 2021; Li et al., 2023).

Uncertainty quantification is an important task in machine learning - a task in which standardneural networks (NNs) have traditionally not excelled. This can be a limitation for safety-critical applications, where uncertainty-aware methods like Gaussian processes or Bayesian linear regression are often preferred. Bayesian neural networks are an approach to address this limitation. They assume probability distributions for all parameters and yield distributed predictions. However, training and inference are typically intractable and approximations must be employed. A promising approximation is NNs with Bayesian last layer (BLL). They assume distributed weights only in the linear output layer and yield a normally distributed prediction. To approximate the intractable Bayesian neural network, point estimates of the distributed weights in all but the last layer should be obtained by maximizing the marginal likelihood. This has previously been challenging, as the marginal likelihood is expensive to evaluate in this setting. We present a reformulation of the log-marginal likelihood of a NN with BLL which allows for efficient training using backpropagation. Furthermore, we address the challenge of uncertainty quantification for extrapolation points. We provide a metric to quantify the degree of extrapolation and derive a method to improve the uncertainty quantification for these points. Our methods are derived for the multivariate case and demonstrated in a simulation study. In comparison to Bayesian linear regression with fixed features, and a Bayesian neural network trained with variational inference, our proposed method achieves the highest log-predictive density on test data.

In recent years, privacy-preserving machine learning algorithms have attracted increasing attention because of their important applications in many scientific fields. However, in the literature, most privacy-preserving algorithms demand learning objectives to be strongly convex and Lipschitz smooth, which thus cannot cover a wide class of robust loss functions (e.g., quantile/least absolute loss). In this work, we aim to develop a fast privacy-preserving learning solution for a sparse robust regression problem. Our learning loss consists of a robust least absolute loss and an $\ell_1$ sparse penalty term. To fast solve the non-smooth loss under a given privacy budget, we develop a Fast Robust And Privacy-Preserving Estimation (FRAPPE) algorithm for least absolute deviation regression. Our algorithm achieves a fast estimation by reformulating the sparse LAD problem as a penalized least square estimation problem and adopts a three-stage noise injection to guarantee the $(\epsilon,\delta)$-differential privacy. We show that our algorithm can achieve better privacy and statistical accuracy trade-off compared with the state-of-the-art privacy-preserving regression algorithms. In the end, we conduct experiments to verify the efficiency of our proposed FRAPPE algorithm.

The Evidential Regression Network (ERN) represents a novel approach that integrates deep learning with Dempster-Shafer's theory to predict a target and quantify the associated uncertainty. Guided by the underlying theory, specific activation functions must be employed to enforce non-negative values, which is a constraint that compromises model performance by limiting its ability to learn from all samples. This paper provides a theoretical analysis of this limitation and introduces an improvement to overcome it. Initially, we define the region where the models can't effectively learn from the samples. Following this, we thoroughly analyze the ERN and investigate this constraint. Leveraging the insights from our analysis, we address the limitation by introducing a novel regularization term that empowers the ERN to learn from the whole training set. Our extensive experiments substantiate our theoretical findings and demonstrate the effectiveness of the proposed solution.

We propose and evaluate an automated pipeline for discovering significant topics from legal decision texts by passing features synthesized with topic models through penalised regressions and post-selection significance tests. The method identifies case topics significantly correlated with outcomes, topic-word distributions which can be manually-interpreted to gain insights about significant topics, and case-topic weights which can be used to identify representative cases for each topic. We demonstrate the method on a new dataset of domain name disputes and a canonical dataset of European Court of Human Rights violation cases. Topic models based on latent semantic analysis as well as language model embeddings are evaluated. We show that topics derived by the pipeline are consistent with legal doctrines in both areas and can be useful in other related legal analysis tasks.

In multivariate functional data analysis, different functional covariates can be homogeneous. The hidden homogeneity structure is informative about the connectivity or association of different covariates. The covariates with pronounced homogeneity can be analyzed jointly within the same group, which gives rise to a way of parsimoniously modeling multivariate functional data. In this paper, a novel grouped multivariate functional regression model with a new regularization approach termed "coefficient shape alignment" is developed to tackle the potential homogeneity of different functional covariates. The modeling procedure includes two main steps: first detect the unknown grouping structure with the new regularization approach to aggregate covariates into disjoint groups; and then the grouped multivariate functional regression model is established based on the detected grouping structure. In this new grouped model, the coefficient functions of covariates in the same homogeneous group share the same shape invariant to scaling. The new regularization approach builds on penalizing the discrepancy of coefficient shape. The consistency property of the detected grouping structure is thoroughly investigated, and the conditions that guarantee uncovering the underlying true grouping structure are developed. The asymptotic properties of the model estimates are also developed. Extensive simulation studies are conducted to investigate the finite-sample properties of the developed methods. The practical utility of the proposed methods is illustrated in the real data analysis on sugar quality evaluation. This work provides a novel means for analyzing the underlying homogeneity of functional covariates and developing parsimonious model structures for multivariate functional data.

Recent work has focused on the very common practice of prediction-based inference: that is, (i) using a pre-trained machine learning model to predict an unobserved response variable, and then (ii) conducting inference on the association between that predicted response and some covariates. As pointed out by Wang et al. (2020), applying a standard inferential approach in (ii) does not accurately quantify the association between the unobserved (as opposed to the predicted) response and the covariates. In recent work, Wang et al. (2020) and Angelopoulos et al. (2023) propose corrections to step (ii) in order to enable valid inference on the association between the unobserved response and the covariates. Here, we show that the method proposed by Angelopoulos et al. (2023) successfully controls the type 1 error rate and provides confidence intervals with correct nominal coverage, regardless of the quality of the pre-trained machine learning model used to predict the unobserved response. However, the method proposed by Wang et al. (2020) provides valid inference only under very strong conditions that rarely hold in practice: for instance, if the machine learning model perfectly estimates the true regression function in the study population of interest.