We study various types of consistency of honest decision trees and random forests in the regression setting. In contrast to related literature, our proofs are elementary and follow the classical arguments used for smoothing methods. Under mild regularity conditions on the regression function and data distribution, we establish weak and almost sure convergence of honest trees and honest forest averages to the true regression function, and moreover we obtain uniform convergence over compact covariate domains. The framework naturally accommodates ensemble variants based on subsampling and also a two-stage bootstrap sampling scheme. Our treatment synthesizes and simplifies existing analyses, in particular recovering several results as special cases. The elementary nature of the arguments clarifies the close relationship between data-adaptive partitioning and kernel-type methods, providing an accessible approach to understanding the asymptotic behavior of tree-based methods.

Differential Privacy (DP) provides a rigorous framework for releasing statistics while protecting individual information present in a dataset. Although substantial progress has been made on differentially private linear regression, existing methods almost exclusively address the item-level DP setting, where each user contributes a single observation. Many scientific and economic applications instead involve longitudinal or panel data, in which each user contributes multiple dependent observations. In these settings, item-level DP offers inadequate protection, and user-level DP - shielding an individual's entire trajectory - is the appropriate privacy notion. We develop a comprehensive framework for estimation and inference in longitudinal linear regression under user-level DP. We propose a user-level private regression estimator based on aggregating local regressions, and we establish finite-sample guarantees and asymptotic normality under short-range dependence. For inference, we develop a privatized, bias-corrected covariance estimator that is automatically heteroskedasticity- and autocorrelation-consistent. These results provide the first unified framework for practical user-level DP estimation and inference in longitudinal linear regression under dependence, with strong theoretical guarantees and promising empirical performance.

We propose a unified framework for fair regression tasks formulated as risk minimization problems subject to a demographic parity constraint. Unlike many existing approaches that are limited to specific loss functions or rely on challenging non-convex optimization, our framework is applicable to a broad spectrum of regression tasks. Examples include linear regression with squared loss, binary classification with cross-entropy loss, quantile regression with pinball loss, and robust regression with Huber loss. We derive a novel characterization of the fair risk minimizer, which yields a computationally efficient estimation procedure for general loss functions. Theoretically, we establish the asymptotic consistency of the proposed estimator and derive its convergence rates under mild assumptions. We illustrate the method's versatility through detailed discussions of several common loss functions. Numerical results demonstrate that our approach effectively minimizes risk while satisfying fairness constraints across various regression settings.

Estimating the Riesz representer is central to debiased machine learning for causal and structural parameter estimation. We propose generalized Riesz regression, a unified framework that estimates the Riesz representer by fitting a representer model via Bregman divergence minimization. This framework includes the squared loss and the Kullback--Leibler (KL) divergence as special cases: the former recovers Riesz regression, while the latter recovers tailored loss minimization. Under suitable model specifications, the dual problems correspond to covariate balancing, which we call automatic covariate balancing. Moreover, under the same specifications, outcome averages weighted by the estimated Riesz representer satisfy Neyman orthogonality even without estimating the regression function, a property we call automatic Neyman orthogonalization. This property not only reduces the estimation error of Neyman orthogonal scores but also clarifies a key distinction between debiased machine learning and targeted maximum likelihood estimation. Our framework can also be viewed as a generalization of density ratio fitting under Bregman divergences to Riesz representer estimation, and it applies beyond density ratio estimation. We provide convergence analyses for both reproducing kernel Hilbert space (RKHS) and neural network model classes. A Python package for generalized Riesz regression is available at https://github.com/MasaKat0/grr.

The graphical Lasso (GLASSO) is a widely used algorithm for learning high-dimensional undirected Gaussian graphical models (GGM). Given i.i.d. observations from a multivariate normal distribution, GLASSO estimates the precision matrix by maximizing the log-likelihood with an \ell_1-penalty on the off-diagonal entries. However, selecting an optimal regularization parameter λin this unsupervised setting remains a significant challenge. A well-known issue is that existing methods, such as out-of-sample likelihood maximization, select a single global λand do not account for heterogeneity in variable scaling or partial variances. Standardizing the data to unit variances, although a common workaround, has been shown to negatively affect graph recovery. Addressing the problem of nodewise adaptive tuning in graph estimation is crucial for applications like computational neuroscience, where brain networks are constructed from highly heterogeneous, region-specific fMRI data. In this work, we develop Locally Adaptive Regularization for Graph Estimation (LARGE), an approach to adaptively learn nodewise tuning parameters to improve graph estimation and selection. In each block coordinate descent step of GLASSO, we augment the nodewise Lasso regression to jointly estimate the regression coefficients and error variance, which in turn guides the adaptive learning of nodewise penalties. In simulations, LARGE consistently outperforms benchmark methods in graph recovery, demonstrates greater stability across replications, and achieves the best estimation accuracy in the most difficult simulation settings. We demonstrate the practical utility of our method by estimating brain functional connectivity from a real fMRI data set.

Multilabel Classification (MLC) deals with the simultaneous classification of multiple binary labels. The task is challenging because, not only may there be arbitrarily different and complex relationships between predictor variables and each label, but associations among labels may exist even after accounting for effects of predictor variables. In this paper, we present a Bayesian additive regression tree (BART) framework to model the problem. BART is a nonparametric and flexible model structure capable of uncovering complex relationships within the data. Our adaptation, MLCBART, assumes that labels arise from thresholding an underlying numeric scale, where a multivariate normal model allows explicit estimation of the correlation structure among labels. This enables the discovery of complicated relationships in various forms and improves MLC predictive performance. Our Bayesian framework not only enables uncertainty quantification for each predicted label, but our MCMC draws produce an estimated conditional probability distribution of label combinations for any predictor values. Simulation experiments demonstrate the effectiveness of the proposed model by comparing its performance with a set of models, including the oracle model with the correct functional form. Results show that our model predicts vectors of labels more accurately than other contenders and its performance is close to the oracle model. An example highlights how the method's ability to produce measures of uncertainty on predictions provides nuanced understanding of classification results.

Multiple binary responses arise in many modern data-analytic problems. Although fitting separate logistic regressions for each response is computationally attractive, it ignores shared structure and can be statistically inefficient, especially in high-dimensional and class-imbalanced regimes. Low-rank models offer a natural way to encode latent dependence across tasks, but existing methods for binary data are largely likelihood-based and focus on pointwise classification rather than ranking performance. In this work, we propose a unified framework for learning with multiple binary responses that directly targets discrimination by minimizing a surrogate loss for the area under the ROC curve (AUC). The method aggregates pairwise AUC surrogate losses across responses while imposing a low-rank constraint on the coefficient matrix to exploit shared structure. We develop a scalable projected gradient descent algorithm based on truncated singular value decomposition. Exploiting the fact that the pairwise loss depends only on differences of linear predictors, we simplify computation and analysis. We establish non-asymptotic convergence guarantees, showing that under suitable regularity conditions, leading to linear convergence up to the minimax-optimal statistical precision. Extensive simulation studies demonstrate that the proposed method is robust in challenging settings such as label switching and data contamination and consistently outperforms likelihood-based approaches.

We study differentially private ordinary least squares (DP-OLS) with bounded data. The dominant approach, adaptive sufficient-statistics perturbation (AdaSSP), adds an adaptively chosen perturbation to the sufficient statistics, namely, the matrix $X^{\top}X$ and the vector $X^{\top}Y$, and is known to achieve near-optimal accuracy and to have strong empirical performance. In contrast, methods that rely on Gaussian-sketching, which ensure differential privacy by pre-multiplying the data with a random Gaussian matrix, are widely used in federated and distributed regression, yet remain relatively uncommon for DP-OLS. In this work, we introduce the iterative Hessian mixing, a novel DP-OLS algorithm that relies on Gaussian sketches and is inspired by the iterative Hessian sketch algorithm. We provide utility analysis for the iterative Hessian mixing as well as a new analysis for the previous methods that rely on Gaussian sketches. Then, we show that our new approach circumvents the intrinsic limitations of the prior methods and provides non-trivial improvements over AdaSSP. We conclude by running an extensive set of experiments across standard benchmarks to demonstrate further that our approach consistently outperforms these prior baselines.

Covariate shift arises when covariate distributions differ between source and target populations while the conditional distribution of the response remains invariant, and it underlies problems in missing data and causal inference. We propose a minimum Wasserstein distance estimation framework for inference under covariate shift that avoids explicit modeling of outcome regressions or importance weights. The resulting W-estimator admits a closed-form expression and is numerically equivalent to the classical 1-nearest neighbor estimator, yielding a new optimal transport interpretation of nearest neighbor methods. We establish root-$n$ asymptotic normality and show that the estimator is not asymptotically linear, leading to super-efficiency relative to the semiparametric efficient estimator under covariate shift in certain regimes, and uniformly in missing data problems. Numerical simulations, along with an analysis of a rainfall dataset, underscore the exceptional performance of our W-estimator.

Decision trees are powerful machine learning algorithms, widely used in fields such as economics and medicine for their simplicity and interpretability. However, decision trees such as CART are prone to overfitting, especially when grown deep or the sample size is small. Conventional methods to reduce overfitting include pre-pruning and post-pruning, which constrain the growth of uninformative branches. In this paper, we propose a complementary approach by introducing a covariance-driven splitting criterion for regression trees (CovRT). This method is more robust to overfitting than the empirical risk minimization criterion used in CART, as it produces more balanced and stable splits and more effectively identifies covariates with true signals. We establish an oracle inequality of CovRT and prove that its predictive accuracy is comparable to that of CART in high-dimensional settings. We find that CovRT achieves superior prediction accuracy compared to CART in both simulations and real-world tasks.