arXiv subjects

Wen-Xin Zhou

Publications and source records attributed to Wen-Xin Zhou.

At least 19 recordsLinked to original sources

arXiv2026

Generalized Rank Regression

Rank regression offers robustness to outliers and heavy-tailed response distributions, invariance to monotonic transformations, and improved efficiency under non-Gaussian errors, making it a versatile tool for analyzing complex data. This paper introduces Generalized Rank Regression (GRR), an extension of classical rank-based methods that accommodates non-monotonic score functions. While aimed at enhancing the statistical efficiency of robust estimators, this generalization results in a potentially non-convex and non-smooth objective function, presenting challenges for both theoretical analysis and algorithmic implementation. We derive a non-asymptotic Bahadur representation of the proposed estimator and establish its asymptotic normality under mild conditions. To address the optimization challenges, we propose a new two-stage sub-gradient descent algorithm that enables efficient computation of GRR estimators with desirable statistical properties. Furthermore, we develop a multiplier bootstrap procedure for conducting statistical inference. A close connection between GRR and variants of quantile regression is uncovered, which demonstrates that GRR and composite quantile regression share asymptotically equivalent variances. The advantages of GRR are illustrated through extensive simulation studies and a real data application.

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arXiv2026

Adapting to noise tails in private linear regression

While the traditional goal of statistics is to infer population parameters, modern practice increasingly demands protection of individual privacy. One way to address this need is to adapt classical statistical procedures into privacy-preserving algorithms. In this paper, we develop differentially private tail-robust methods for linear regression. The trade-off among bias, privacy, and robustness is controlled by a tunable robustification parameter in the Huber loss. We implement noisy clipped gradient descent for low-dimensional settings and noisy iterative hard thresholding for high-dimensional sparse models. Under sub-Gaussian errors, our method achieves near-optimal convergence rates while relaxing several assumptions required in earlier work. For heavy-tailed errors, we explicitly characterize how the non-asymptotic convergence rate depends on the moment index, privacy parameters, sample size, and intrinsic dimension. Our analysis shows how the moment index influences the choice of robustification parameters and, in turn, the resulting statistical error and privacy cost. By quantifying the interplay among bias, privacy, and robustness, we extend classical perspectives on privacy-preserving robust regression. The proposed methods are evaluated through simulations and two real datasets.

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arXiv2026

Connection Probabilities Estimation in Multi-layer Networks via Iterative Neighborhood Smoothing

Understanding the structural mechanisms of multi-layer networks is essential for analyzing complex systems characterized by multiple interacting layers. This work studies the problem of estimating connection probabilities in multi-layer networks and introduces a new Multi-layer Iterative Connection Probability Estimation (MICE) method. The proposed approach employs an iterative framework that jointly refines inter-layer and intra-layer similarity sets by dynamically updating distance metrics derived from current probability estimates. By leveraging both layer-level and node-level neighborhood information, MICE improves estimation accuracy while preserving computational efficiency. Theoretical analysis establishes the consistency of the estimator and shows that, under mild regularity conditions, the proposed method achieves an optimal convergence rate comparable to that of an oracle estimator. Extensive simulation studies across diverse graphon structures demonstrate the superior performance of MICE relative to existing methods. Empirical evaluations using brain network data from patients with Attention-Deficit/Hyperactivity Disorder (ADHD) and global food and agricultural trade network data further illustrate the robustness and effectiveness of the method in link prediction tasks. Overall, this work provides a theoretically grounded and practically scalable framework for probabilistic modeling and inference in multi-layer network systems.

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arXiv2026

SGD with Dependent Data: Optimal Estimation, Regret, and Inference

This work investigates the performance of the final iterate produced by stochastic gradient descent (SGD) under temporally dependent data. We consider two complementary sources of dependence: $(i)$ martingale-type dependence in both the covariate and noise processes, which accommodates non-stationary and non-mixing time series data, and $(ii)$ dependence induced by sequential decision making. Our formulation runs in parallel with classical notions of (local) stationarity and strong mixing, while neither framework fully subsumes the other. Remarkably, SGD is shown to automatically accommodate both independent and dependent information under a broad class of stepsize schedules and exploration rate schemes. Non-asymptotically, we show that SGD simultaneously achieves statistically optimal estimation error and regret, extending and improving existing results. In particular, our tail bounds remain sharp even for potentially infinite horizon $T=+\infty$. Asymptotically, the SGD iterates converge to a Gaussian distribution with only an $O_{\PP}(1/\sqrt{t})$ remainder, demonstrating that the supposed estimation-regret trade-off claimed in prior work can in fact be avoided. We further propose a new ``conic'' approximation of the decision region that allows the covariates to have unbounded support. For online sparse regression, we develop a new SGD-based algorithm that uses only $d$ units of storage and requires $O(d)$ flops per iteration, achieving the long term statistical optimality. Intuitively, each incoming observation contributes to estimation accuracy, while aggregated summary statistics guide support recovery.

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arXiv2025

A new non-parametric Kendall's tau for matrix-valued elliptical observations

In this article, we first propose generalized row/column matrix Kendall's tau for matrix-variate observations that are ubiquitous in areas such as finance and medical imaging. For a random matrix following a matrix-variate elliptically contoured distribution, we show that the eigenspaces of the proposed row/column matrix Kendall's tau coincide with those of the row/column scatter matrix respectively, with the same descending order of the eigenvalues. We perform eigenvalue decomposition to the generalized row/column matrix Kendall's tau for recovering the loading spaces of the matrix factor model. We also propose to estimate the pair of the factor numbers by exploiting the eigenvalue-ratios of the row/column matrix Kendall's tau. Theoretically, we derive the convergence rates of the estimators for loading spaces, factor scores and common components, and prove the consistency of the estimators for the factor numbers without any moment constraints on the idiosyncratic errors. Thorough simulation studies are conducted to show the higher degree of robustness of the proposed estimators over the existing ones. Analysis of a financial dataset of asset returns and a medical imaging dataset associated with COVID-19 illustrate the empirical usefulness of the proposed method.

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arXiv2025

Deep neural expected shortfall regression with tail-robustness

Expected shortfall (ES), also known as conditional value-at-risk, is a widely recognized risk measure that complements value-at-risk by capturing tail-related risks more effectively. Compared with quantile regression, which has been extensively developed and applied across disciplines, ES regression remains in its early stage, partly because the traditional empirical risk minimization framework is not directly applicable. In this paper, we develop a nonparametric framework for expected shortfall regression based on a two-step approach that treats the conditional quantile function as a nuisance parameter. Leveraging the representational power of deep neural networks, we construct a two-step ES estimator using feedforward ReLU networks, which can alleviate the curse of dimensionality when the underlying functions possess hierarchical composition structures. However, ES estimation is inherently sensitive to heavy-tailed response or error distributions. To address this challenge, we integrate a properly tuned Huber loss into the neural network training, yielding a robust deep ES estimator that is provably resistant to heavy-tailedness in a non-asymptotic sense and first-order insensitive to quantile estimation errors in the first stage. Comprehensive simulation studies and an empirical analysis of the effect of El Niño on extreme precipitation illustrate the accuracy and robustness of the proposed method.

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arXiv2025

Online Quantile Regression

This paper addresses the challenge of integrating sequentially arriving data within the quantile regression framework, where the number of features is allowed to grow with the number of observations, the horizon is unknown, and memory is limited. We employ stochastic sub-gradient descent to minimize the empirical check loss and study its statistical properties and regret performance. In our analysis, we unveil the delicate interplay between updating iterates based on individual observations versus batches of observations, revealing distinct regularity properties in each scenario. Our method ensures long-term optimal estimation irrespective of the chosen update strategy. Importantly, our contributions go beyond prior works by achieving exponential-type concentration inequalities and attaining optimal regret and error rates that exhibit only \textsf{ short-term} sensitivity to initial errors. A key insight from our study is the delicate statistical analyses and the revelation that appropriate stepsize schemes significantly mitigate the impact of initial errors on subsequent errors and regrets. This underscores the robustness of stochastic sub-gradient descent in handling initial uncertainties, emphasizing its efficacy in scenarios where the sequential arrival of data introduces uncertainties regarding both the horizon and the total number of observations. Additionally, when the initial error rate is well-controlled, there is a trade-off between short-term error rate and long-term optimality. Due to the lack of delicate statistical analysis for squared loss, we also briefly discuss its properties and proper schemes. Extensive simulations support our theoretical findings.

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arXiv2024

High-Dimensional Expected Shortfall Regression

Expected shortfall is defined as the average over the tail below (or above) a certain quantile of a probability distribution. Expected shortfall regression provides powerful tools for learning the relationship between a response variable and a set of covariates while exploring the heterogeneous effects of the covariates. In the health disparity research, for example, the lower/upper tail of the conditional distribution of a health-related outcome, given high-dimensional covariates, is often of importance. Under sparse models, we propose the lasso-penalized expected shortfall regression and establish non-asymptotic error bounds, depending explicitly on the sample size, dimension, and sparsity, for the proposed estimator. To perform statistical inference on a covariate of interest, we propose a debiased estimator and establish its asymptotic normality, from which asymptotically valid tests can be constructed. We illustrate the finite sample performance of the proposed method through numerical studies and a data application on health disparity.

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arXiv2024

Subgroup Identification with Latent Factor Structure

Subgroup analysis has garnered increasing attention for its ability to identify meaningful subgroups within heterogeneous populations, thereby enhancing predictive power. However, in many fields such as social science and biology, covariates are often highly correlated due to common factors. This correlation poses significant challenges for subgroup identification, an issue that is often overlooked in existing literature. In this paper, we aim to address this gap in the ``diverging dimension" regime by proposing a center-augmented subgroup identification method within the Factor Augmented (sparse) Linear Model framework. This method bridges dimension reduction and sparse regression. Our proposed approach is adaptable to the high cross-sectional dependence among covariates and offers computational advantages with a complexity of $O(nK)$, compared to the $O(n^2)$ complexity of the conventional pairwise fusion penalty method in the literature, where $n$ is the sample size and $K$ is the number of subgroups. We also investigate the asymptotic properties of the oracle estimators under conditions on the minimal distance between group centroids. To implement the proposed approach, we introduce a Difference of Convex functions-based Alternating Direction Method of Multipliers (DC-ADMM) algorithm and demonstrate its convergence to a local minimizer in a finite number of steps. We illustrate the superiority of the proposed method through extensive numerical experiments and a real macroeconomic data example. An \texttt{R} package, \texttt{SILFS}, implementing the method is also available on CRAN.

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arXiv2023

High-Dimensional Composite Quantile Regression: Optimal Statistical Guarantees and Fast Algorithms

The composite quantile regression (CQR) was introduced by Zou and Yuan [Ann. Statist. 36 (2008) 1108--1126] as a robust regression method for linear models with heavy-tailed errors while achieving high efficiency. Its penalized counterpart for high-dimensional sparse models was recently studied in Gu and Zou [IEEE Trans. Inf. Theory 66 (2020) 7132--7154], along with a specialized optimization algorithm based on the alternating direct method of multipliers (ADMM). Compared to the various first-order algorithms for penalized least squares, ADMM-based algorithms are not well-adapted to large-scale problems. To overcome this computational hardness, in this paper we employ a convolution-smoothed technique to CQR, complemented with iteratively reweighted $\ell_1$-regularization. The smoothed composite loss function is convex, twice continuously differentiable, and locally strong convex with high probability. We propose a gradient-based algorithm for penalized smoothed CQR via a variant of the majorize-minimization principal, which gains substantial computational efficiency over ADMM. Theoretically, we show that the iteratively reweighted $\ell_1$-penalized smoothed CQR estimator achieves near-minimax optimal convergence rate under heavy-tailed errors without any moment constraint, and further achieves near-oracle convergence rate under a weaker minimum signal strength condition than needed in Gu and Zou (2020). Numerical studies demonstrate that the proposed method exhibits significant computational advantages without compromising statistical performance compared to two state-of-the-art methods that achieve robustness and high efficiency simultaneously.

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arXiv2023

An Efficient Iterative Least Squares Algorithm for Large-dimensional Matrix Factor Model via Random Projection

The matrix factor model has drawn growing attention for its advantage in achieving two-directional dimension reduction simultaneously for matrix-structured observations. In this paper, we propose a simple iterative least squares algorithm for matrix factor models, in contrast to the Principal Component Analysis (PCA)-based methods in the literature. In detail, we first propose to estimate the latent factor matrices by projecting the observations with two deterministic weight matrices, which are chosen to diversify away the idiosyncratic components. We show that the inferences on factors are still asymptotically valid even if we overestimate both the row/column factor numbers. We then estimate the row/column loading matrices by minimizing the squared loss function under certain identifiability conditions. The resultant estimators of the loading matrices are treated as the new weight/projection matrices and thus the above update procedure can be iteratively performed until convergence. Theoretically, given the true dimensions of the factor matrices, we derive the convergence rates of the estimators for loading matrices and common components at any $s$-th step iteration. Additionally, we propose an eigenvalue-ratio method to estimate the pair of factor numbers consistently. Thorough numerical simulations are conducted to investigate the finite-sample performance of the proposed methods and two real datasets associated with financial portfolios and multinational macroeconomic indices are used to illustrate our algorithm's practical usefulness.

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arXiv2023

Huber Principal Component Analysis for Large-dimensional Factor Models

Factor models have been widely used in economics and finance. However, the heavy-tailed nature of macroeconomic and financial data is often neglected in the existing literature. To address this issue and achieve robustness, we propose an approach to estimate factor loadings and scores by minimizing the Huber loss function, which is motivated by the equivalence of conventional Principal Component Analysis (PCA) and the constrained least squares method in the factor model. We provide two algorithms that use different penalty forms. The first algorithm, which we refer to as Huber PCA, minimizes the $\ell_2$-norm-type Huber loss and performs PCA on the weighted sample covariance matrix. The second algorithm involves an element-wise type Huber loss minimization, which can be solved by an iterative Huber regression algorithm. Our study examines the theoretical minimizer of the element-wise Huber loss function and demonstrates that it has the same convergence rate as conventional PCA when the idiosyncratic errors have bounded second moments. We also derive their asymptotic distributions under mild conditions. Moreover, we suggest a consistent model selection criterion that relies on rank minimization to estimate the number of factors robustly. We showcase the benefits of Huber PCA through extensive numerical experiments and a real financial portfolio selection example. An R package named ``HDRFA" has been developed to implement the proposed robust factor analysis.

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arXiv2023

Retire: Robust Expectile Regression in High Dimensions

High-dimensional data can often display heterogeneity due to heteroscedastic variance or inhomogeneous covariate effects. Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity in high-dimensional data. The former is computationally challenging due to the non-smooth nature of the check loss, and the latter is sensitive to heavy-tailed error distributions. In this paper, we propose and study (penalized) robust expectile regression (retire), with a focus on iteratively reweighted $\ell_1$-penalization which reduces the estimation bias from $\ell_1$-penalization and leads to oracle properties. Theoretically, we establish the statistical properties of the retire estimator under two regimes: (i) low-dimensional regime in which $d \ll n$; (ii) high-dimensional regime in which $s\ll n\ll d$ with $s$ denoting the number of significant predictors. In the high-dimensional setting, we carefully characterize the solution path of the iteratively reweighted $\ell_1$-penalized retire estimation, adapted from the local linear approximation algorithm for folded-concave regularization. Under a mild minimum signal strength condition, we show that after as many as $\log(\log d)$ iterations the final iterate enjoys the oracle convergence rate. At each iteration, the weighted $\ell_1$-penalized convex program can be efficiently solved by a semismooth Newton coordinate descent algorithm. Numerical studies demonstrate the competitive performance of the proposed procedure compared with either non-robust or quantile regression based alternatives.

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arXiv2022

How do noise tails impact on deep ReLU networks?

This paper investigates the stability of deep ReLU neural networks for nonparametric regression under the assumption that the noise has only a finite p-th moment. We unveil how the optimal rate of convergence depends on p, the degree of smoothness and the intrinsic dimension in a class of nonparametric regression functions with hierarchical composition structure when both the adaptive Huber loss and deep ReLU neural networks are used. This optimal rate of convergence cannot be obtained by the ordinary least squares but can be achieved by the Huber loss with a properly chosen parameter that adapts to the sample size, smoothness, and moment parameters. A concentration inequality for the adaptive Huber ReLU neural network estimators with allowable optimization errors is also derived. To establish a matching lower bound within the class of neural network estimators using the Huber loss, we employ a different strategy from the traditional route: constructing a deep ReLU network estimator that has a better empirical loss than the true function and the difference between these two functions furnishes a low bound. This step is related to the Huberization bias, yet more critically to the approximability of deep ReLU networks. As a result, we also contribute some new results on the approximation theory of deep ReLU neural networks.

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arXiv2022

Robust Estimation and Inference for Expected Shortfall Regression with Many Regressors

Expected Shortfall (ES), also known as superquantile or Conditional Value-at-Risk, has been recognized as an important measure in risk analysis and stochastic optimization, and is also finding applications beyond these areas. In finance, it refers to the conditional expected return of an asset given that the return is below some quantile of its distribution. In this paper, we consider a recently proposed joint regression framework that simultaneously models the quantile and the ES of a response variable given a set of covariates, for which the state-of-the-art approach is based on minimizing a joint loss function that is non-differentiable and non-convex. This inevitably raises numerical challenges and limits its applicability for analyzing large-scale data. Motivated by the idea of using Neyman-orthogonal scores to reduce sensitivity with respect to nuisance parameters, we propose a statistically robust (to highly skewed and heavy-tailed data) and computationally efficient two-step procedure for fitting joint quantile and ES regression models. With increasing covariate dimensions, we establish explicit non-asymptotic bounds on estimation and Gaussian approximation errors, which lay the foundation for statistical inference. Finally, we demonstrate through numerical experiments and two data applications that our approach well balances robustness, statistical, and numerical efficiencies for expected shortfall regression.

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arXiv2022

Scalable estimation and inference for censored quantile regression process

Censored quantile regression (CQR) has become a valuable tool to study the heterogeneous association between a possibly censored outcome and a set of covariates, yet computation and statistical inference for CQR have remained a challenge for large-scale data with many covariates. In this paper, we focus on a smoothed martingale-based sequential estimating equations approach, to which scalable gradient-based algorithms can be applied. Theoretically, we provide a unified analysis of the smoothed sequential estimator and its penalized counterpart in increasing dimensions. When the covariate dimension grows with the sample size at a sublinear rate, we establish the uniform convergence rate (over a range of quantile indexes) and provide a rigorous justification for the validity of a multiplier bootstrap procedure for inference. In high-dimensional sparse settings, our results considerably improve the existing work on CQR by relaxing an exponential term of sparsity. We also demonstrate the advantage of the smoothed CQR over existing methods with both simulated experiments and data applications.

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arXiv2022

Communication-Constrained Distributed Quantile Regression with Optimal Statistical Guarantees

We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. This is challenging due to the non-smooth nature of the quantile regression (QR) loss function, which invalidates the use of existing methodology. The difficulties are resolved through a double-smoothing approach that is applied to the local (at each data source) and global objective functions. Despite the reliance on a delicate combination of local and global smoothing parameters, the quantile regression model is fully parametric, thereby facilitating interpretation. In the low-dimensional regime, we establish a finite-sample theoretical framework for the sequentially defined distributed QR estimators. This reveals a trade-off between the communication cost and statistical error. We further discuss and compare several alternative confidence set constructions, based on inversion of Wald and score-type tests and resampling techniques, detailing an improvement that is effective for more extreme quantile coefficients. In high dimensions, a sparse framework is adopted, where the proposed doubly-smoothed objective function is complemented with an $\ell_1$-penalty. We show that the corresponding distributed penalized QR estimator achieves the global convergence rate after a near-constant number of communication rounds. A thorough simulation study further elucidates our findings.

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arXiv2022

A Unified Algorithm for Penalized Convolution Smoothed Quantile Regression

Penalized quantile regression (QR) is widely used for studying the relationship between a response variable and a set of predictors under data heterogeneity in high-dimensional settings. Compared to penalized least squares, scalable algorithms for fitting penalized QR are lacking due to the non-differentiable piecewise linear loss function. To overcome the lack of smoothness, a recently proposed convolution-type smoothed method brings an interesting tradeoff between statistical accuracy and computational efficiency for both standard and penalized quantile regressions. In this paper, we propose a unified algorithm for fitting penalized convolution smoothed quantile regression with various commonly used convex penalties, accompanied by an R-language package conquer available from the Comprehensive R Archive Network. We perform extensive numerical studies to demonstrate the superior performance of the proposed algorithm over existing methods in both statistical and computational aspects. We further exemplify the proposed algorithm by fitting a fused lasso additive QR model on the world happiness data.

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