Fairness is a central pillar of trustworthy machine learning, especially in domains where accuracy- or profit-driven optimization is insufficient. While most fairness research focuses on supervised learning, fairness in policy learning remains less explored. Because policy learning is interventional, it induces two distinct fairness targets: action fairness (equitable action assignments) and outcome fairness (equitable downstream consequences). Crucially, equalizing actions does not generally equalize outcomes when groups face different constraints or respond differently to the same action. We propose a novel double fairness learning (DFL) framework that explicitly manages the trade-off among three objectives: action fairness, outcome fairness, and value maximization. We integrate fairness directly into a multi-objective optimization problem for policy learning and employ a lexicographic weighted Tchebyshev method that recovers Pareto solutions beyond convex settings, with theoretical guarantees on the regret bounds. Our framework is flexible and accommodates various commonly used fairness notions. Extensive simulations demonstrate improved performance relative to competing methods. In applications to a motor third-party liability insurance dataset and an entrepreneurship training dataset, DFL substantially improves both action and outcome fairness while incurring only a modest reduction in overall value.

We develop a new framework for flexible, nonlinear, and interpretable off-policy evaluation for infinite-horizon reinforcement learning. To handle large state spaces and support transparent decision-making, we model the Q-function using a nonlinear function class with a sparse additive structure. We derive high-probability finite-sample error bounds for estimating the value function of a target policy and show that the bounds depend only logarithmically on the ambient dimension d, thereby alleviating the curse of dimensionality. In contrast to most existing theory for off-policy evaluation, which typically assumes access to many trajectories, our analysis guarantees accurate value estimation when either the number of trajectories or the time horizon is sufficiently large. In addition, we propose a group-sparsity-based feature screening procedure that identifies, with high probability, a reduced feature set containing all relevant covariates. Numerical experiments demonstrate the effectiveness of the proposed approach.

This work is concerned with the Backfitting (BF) algorithm for the nonparametric additive quantile regression (AQR) model. We establish a strong uniform consistency rate for the Bahadur representation of the BF estimators, which includes the two stage estimator of [11] as a special case. These results are fundamental for statistical inference in AQR models and for applications that involve plugging such estimators into other functionals where some control over higher order terms is required. Two examples of applications are provided: one is on the root-n consistency of the parameter estimates for the partially linear AQR model, the other concerns structure recovery for the nonparametric AQR model. MSC2020 subject classifications: Primary 62G08, 62H12; secondary 62G20.

Motivated by the human-machine interaction such as training chatbots for improving customer satisfaction, we study human-guided human-machine interaction involving private information. We model this interaction as a two-player turn-based game, where one player (Alice, a human) guides the other player (Bob, a machine) towards a common goal. Specifically, we focus on offline reinforcement learning (RL) in this game, where the goal is to find a policy pair for Alice and Bob that maximizes their expected total rewards based on an offline dataset collected a priori. The offline setting presents two challenges: (i) We cannot collect Bob's private information, leading to a confounding bias when using standard RL methods, and (ii) a distributional mismatch between the behavior policy used to collect data and the desired policy we aim to learn. To tackle the confounding bias, we treat Bob's previous action as an instrumental variable for Alice's current decision making so as to adjust for the unmeasured confounding. We develop a novel identification result and use it to propose a new off-policy evaluation (OPE) method for evaluating policy pairs in this two-player turn-based game. To tackle the distributional mismatch, we leverage the idea of pessimism and use our OPE method to develop an off-policy learning algorithm for finding a desirable policy pair for both Alice and Bob. Finally, we prove that under mild assumptions such as partial coverage of the offline data, the policy pair obtained through our method converges to the optimal one at a satisfactory rate.

In the literature of reinforcement learning (RL), off-policy evaluation is mainly focused on estimating a value of a target policy given the pre-collected data generated by some behavior policy. Motivated by the recent success of distributional RL in many practical applications, we study the distributional off-policy evaluation problem in the batch setting when the reward is multi-variate. We propose an offline Wasserstein-based approach to simultaneously estimate the joint distribution of a multivariate discounted cumulative reward given any initial state-action pair in the setting of an infinite-horizon Markov decision process. Finite sample error bound for the proposed estimator with respect to a modified Wasserstein metric is established in terms of both the number of trajectories and the number of decision points on each trajectory in the batch data. Extensive numerical studies are conducted to demonstrate the superior performance of our proposed method. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Regularized quantile regression (QR) is a useful technique for analyzing heterogeneous data under potentially heavy-tailed error contamination in high dimensions. This paper provides a new analysis of the estimation/prediction error bounds of the global solution of $L_1$ -regularized QR (QR-LASSO) and the local solutions of nonconvex regularized QR (QR-NCP) when the number of covariates is greater than the sample size. Our results build upon and significantly generalize the earlier work in the literature. For certain heavy-tailed error distributions and a general class of design matrices, the least-squares-based LASSO cannot achieve the near-oracle rate derived under the normality assumption no matter the choice of the tuning parameter. In contrast, we establish that QR-LASSO achieves the near-oracle estimation error rate for a broad class of models under conditions weaker than those in the literature. For QR-NCP, we establish the novel results that all local optima within a feasible region have desirable estimation accuracy. Our analysis applies to not just the hard sparsity setting commonly used in the literature, but also to the soft sparsity setting which permits many small coefficients. Our approach relies on a unified characterization of the global/local solutions of regularized QR via subgradients using a generalized Karush–Kuhn–Tucker condition. The theory of the paper establishes a key property of the subdifferential of the quantile loss function in high dimensions, which is of independent interest for analyzing other high-dimensional nonsmooth problems.

The data-driven newsvendor problem with features has recently emerged as a significant area of research, driven by the proliferation of data across various sectors such as retail, supply chains, e-commerce, and healthcare. Given the sensitive nature of customer or organizational data often used in feature-based analysis, it is crucial to ensure individual privacy to uphold trust and confidence. Despite its importance, privacy preservation in the context of inventory planning remains unexplored. A key challenge is the nonsmoothness of the newsvendor loss function, which sets it apart from existing work on privacy-preserving algorithms in other settings. This paper introduces a novel approach to estimating a privacy-preserving optimal inventory policy within the f-differential privacy framework, an extension of the classical (epsilon, delta)-differential privacy with several appealing properties. We develop a clipped noisy gradient descent algorithm based on convolution smoothing for optimal inventory estimation to simultaneously address three main challenges: (i) unknown demand distribution and nonsmooth loss function, (ii) provable privacy guarantees for individual-level data, and (iii) desirable statistical precision. We derive finite-sample high-probability bounds for optimal policy parameter estimation and regret analysis. By leveraging the structure of the newsvendor problem, we attain a faster excess population risk bound compared with that obtained from an indiscriminate application of existing results for general nonsmooth convex loss. Our bound aligns with that for strongly convex and smooth loss function. Our numerical experiments demonstrate that the proposed new method can achieve desirable privacy protection with a marginal increase in cost.

The vast majority of literature on evaluating the significance of a treatment effect based on observational data has been confined to discrete treatments. These methods are not applicable to drawing inference for a continuous treatment, which arises in many important applications. To adjust for confounders when evaluating a continuous treatment, existing inference methods often rely on discretizing the treatment or using (possibly misspecified) parametric models for the effect curve. Recently, Kennedy et al. (2017) proposed nonparametric doubly robust estimation for a continuous treatment effect in observational studies. However, inference for the continuous treatment effect is a harder problem. To the best of our knowledge, a completely nonparametric doubly robust approach for inference in this setting is not yet available. We develop such a nonparametric doubly robust procedure in this paper for making inference on the continuous treatment effect curve. Using empirical process techniques for local U- and V-processes, we establish the test statistic's asymptotic distribution. Furthermore, we propose a wild bootstrap procedure for implementing the test in practice. We illustrate the new method via simulations and a study of a constructed dataset relating the effect of nurse staffing hours on hospital performance. We implement our doubly robust dose response test in the R package DRDRtest on CRAN.

This paper introduces the partial Gini covariance, a novel dependence measure that addresses the challenges of high-dimensional inference with heavy-tailed errors, often encountered in fields like finance, insurance, climate, and biology. Conventional high-dimensional regression inference methods suffer from inaccurate type I errors and reduced power in heavy-tailed contexts, limiting their effectiveness. Our proposed approach leverages the partial Gini covariance to construct a robust statistical inference framework that requires minimal tuning and does not impose restrictive moment conditions on error distributions. Unlike traditional methods, it circumvents the need for estimating the density of random errors and enhances the computational feasibility and robustness. Extensive simulations demonstrate the proposed method's superior power and robustness over standard high-dimensional inference approaches, such as those based on the debiased Lasso. The asymptotic relative efficiency analysis provides additional theoretical insight on the improved efficiency of the new approach in the heavy-tailed setting. Additionally, the partial Gini covariance extends to the multivariate setting, enabling chi-square testing for a group of coefficients. We illustrate the method's practical application with a real-world data example.

This work aims to study off-policy evaluation (OPE) under scenarios where two key reinforcement learning (RL) assumptions -- temporal stationarity and individual homogeneity are both violated. To handle the ``double inhomogeneities", we propose a class of latent factor models for the reward and observation transition functions, under which we develop a general OPE framework that consists of both model-based and model-free approaches. To our knowledge, this is the first paper that develops statistically sound OPE methods in offline RL with double inhomogeneities. It contributes to a deeper understanding of OPE in environments, where standard RL assumptions are not met, and provides several practical approaches in these settings. We establish the theoretical properties of the proposed value estimators and empirically show that our approach outperforms competing methods that ignore either temporal nonstationarity or individual heterogeneity. Finally, we illustrate our method on a data set from the Medical Information Mart for Intensive Care.

We study offline dynamic pricing when historical data provide incomplete coverage of the price space such that some candidate prices, including the optimal one, may be entirely unobserved. This setting is common in practice and is especially difficult in dynamic environments. Existing offline reinforcement learning methods typically rely on full or partial coverage and can therefore perform poorly in such settings. We develop a nonparametric partial identification framework for offline dynamic pricing that exploits the monotonicity of demand in price to bound the value of unobserved prices. Within this framework, we formulate two dynamic decision rules: a pessimistic policy that maximizes worst-case revenue and an opportunistic policy that minimizes worst-case regret. These rules are tailored to a sequential no-coverage environment and are not direct extensions of existing pessimistic offline RL or static opportunistic approaches. We establish finite-sample regret bounds for both policies, recovering the standard rate when the optimal price is covered and quantifying the additional cost when it is not. We also develop efficient algorithms and show, through simulations and an airline ticket application, that our methods outperform standard offline RL baselines in no-coverage settings. Managerially, the framework provides a practical mapping from a firm's risk posture to its pricing policy: firms seeking revenue stability and downside protection should prefer the pessimistic policy, whereas firms willing to bear measured risk for potential gains from underexplored prices should prefer the opportunistic policy.

Abstract Mendelian randomization (MR) is a statistical approach to inferring the causal relationships from genome-wide association studies (GWAS) by using genetic variants as instrumental variables (IVs). As IVs, the selected genetic variants should be solely associated with the exposure of interest, and have no associations with confounders and the studied outcome except through the exposure. Sometimes the selected genetic variants have effects on the outcome through other pathways, a phenomenon known as horizontal pleiotropy, which makes the selected variants violate IV assumptions. Two different approaches have been proposed to address this issue: one is to improve robustness to pleiotropic effects, and the other one is to generalize univariate exposure to multivariate cases so that we can incorporate possible pathways into MR analysis. Compared to pleiotropy-robust methods, multivariable Mendelian randomization (MVMR) can uncover the exposures having direct effects on the outcome. However, measuring all possible pathways from genetic variants to the outcome in MVMR analysis is difficult. Although MVMR methods with robustness to unmeasured pleiotropy have been proposed recently, they are statistically inefficient as a result of ignoring correlations among exposures and distributions of effect sizes. Given the limitations from both directions, we propose a novel method named MVMR-PRESS that can infer causal relationships between multivariate exposures and the outcome with robustness to horizontal pleiotropic effects from unconsidered pathways. MVMR-PRESS estimates causal effects by using a Penalized Regression on Summary Statistics from GWAS considering both correlations among exposures and distributions of effect sizes. One merit of MVMR-PRESS is that samples in GWAS of different exposures can have overlaps, which allows us to include GWAS summary statistics from the same cohort or consortium. Simulation experiments showed that our method achieved the smallest bias and highest power compared to existing pleiotropy-robust MR and MVMR methods while the type 1 error rate was well-controlled. Applying MVMR-PRESS to publicly available GWAS summary statistics demonstrated that body mass index (BMI), height (HT), and low-density lipoprotein cholesterol (LDL) have significant causal effects on coronary artery disease (CAD), and BMI and high-density lipoprotein cholesterol (HDL) were causally related to type 2 diabetes (T2D). On the contrary, the causal estimates of triglycerides (TG), HDL, and total cholesterol (TC) on CAD, and the estimates of HT, LDL, TG, and TC on T2D were non-significant. In addition, we found no evidence suggesting BMI, HT, and lipid levels have causal effects on inflammatory bowel disease (IBD) and schizophrenia (SCZ).

This article develops new tools to quantify uncertainty in optimal decision making and to gain insight into which variables one should collect information about given the potential cost of measuring a large number of variables. We investigate simultaneous inference to determine if a group of variables is relevant for estimating an optimal decision rule in a high-dimensional semiparametric framework. The unknown link function permits flexible modeling of the interactions between the treatment and the covariates, but leads to nonconvex estimation in high dimension and imposes significant challenges for inference. We first establish that a local restricted strong convexity condition holds with high probability and that any feasible local sparse solution of the estimation problem can achieve the near-oracle estimation error bound. We further rigorously verify that a wild bootstrap procedure based on a debiased version of the local solution can provide asymptotically honest uniform inference for the effect of a group of variables on optimal decision making. The advantage of honest inference is that it does not require the initial estimator to achieve perfect model selection and does not require the zero and nonzero effects to be well-separated. We also propose an efficient algorithm for estimation. Our simulations suggest satisfactory performance. An example from a diabetes study illustrates the real application. Supplementary materials for this article are available online.

In many important applications of precision medicine, the outcome of interest is time to an event (e.g., death, relapse of disease) and the primary goal is to identify the optimal individualized decision rule (IDR) to prolong survival time. Existing work in this area have been mostly focused on estimating the optimal IDR to maximize the restricted mean survival time in the population. We propose a new robust framework for estimating an optimal static or dynamic IDR with time-to-event outcomes based on an easy-to-interpret quantile criterion. The new method does not need to specify an outcome regression model and is robust for heavy-tailed distribution. The estimation problem corresponds to a nonregular M-estimation problem with both finite and infinite-dimensional nuisance parameters. Employing advanced empirical process techniques, we establish the statistical theory of the estimated parameter indexing the optimal IDR. Furthermore, we prove a novel result that the proposed approach can consistently estimate the optimal value function under mild conditions even when the optimal IDR is nonunique, which happens in the challenging setting of exceptional laws. We also propose a smoothed resampling procedure for inference. The proposed methods are implemented in the R-package QTOCen. We demonstrate the performance of the proposed new methods via extensive Monte Carlo studies and a real data application. Supplementary materials for this article are available online.

The U.S. electrical grid has undergone substantial transformation with increased penetration of wind and solar -- forms of variable renewable energy (VRE). Despite the benefits of VRE for decarbonization, it has garnered some controversy for inducing unwanted effects in regional electricity markets. In this study, the role of VRE penetration is examined on the system electricity price and price volatility based on hourly, real-time, historical data from six Independent System Operators (ISOs) in the U.S. using quantile and skew t-distribution regressions. After correcting for temporal effects, we found an increase in VRE penetration is associated with decrease in system electricity price in all ISOs studied. The increase in VRE penetration is associated with decrease in temporal price volatility in five out of six ISOs studied. The relationships are non-linear. These results are consistent with the modern portfolio theory where diverse volatile assets may lead to more stable and less risky portfolios.

The National Alzheimer's Coordinating Center Uniform Data Set includes test results from a battery of cognitive exams. Motivated by the need to model the cognitive ability of low-performing patients we create a composite score from ten tests and propose to model this score using a partially linear quantile regression model for longitudinal studies with non-ignorable dropouts. Quantile regression allows for modeling non-central tendencies. The partially linear model accommodates nonlinear relationships between some of the covariates and cognitive ability. The data set includes patients that leave the study prior to the conclusion. Ignoring such dropouts will result in biased estimates if the probability of dropout depends on the response. To handle this challenge, we propose a weighted quantile regression estimator where the weights are inversely proportional to the estimated probability a subject remains in the study. We prove that this weighted estimator is a consistent and efficient estimator of both linear and nonlinear effects.

There has recently been a surge on the methodological development for optimal individualized treatment rule (ITR) estimation. The standard methods in the literature are designed to maximize the potential average performance (assuming larger outcomes are desirable). A notable drawback of the standard approach, due to heterogeneity in treatment response, is that the estimated optimal ITR may be suboptimal or even detrimental to certain disadvantaged subpopulations. Motivated by the importance of incorporating an appropriate fairness constraint in optimal decision making (e.g., assign treatment with protection to those with shorter survival time, or assign a job training program with protection to those with lower wages), we propose a new framework that aims to estimate an optimal ITR to maximize the average value with the guarantee that its tail performance exceeds a prespecified threshold. The optimal fairness-oriented ITR corresponds to a solution of a nonconvex optimization problem. To handle the computational challenge, we develop a new efficient first-order algorithm. We establish theoretical guarantees for the proposed estimator. Furthermore, we extend the proposed method to dynamic optimal ITRs. The advantages of the proposed approach over existing methods are demonstrated via extensive numerical studies and real data analysis.

l(1)-penalized quantile regression (QR) is widely used for analysing high-dimensional data with heterogeneity. It is now recognized that the l(1)-penalty introduces non-negligible estimation bias, while a proper use of concave regularization may lead to estimators with refined convergence rates and oracle properties as the signal strengthens. Although folded concave penalized M-estimation with strongly convex loss functions have been well studied, the extant literature on QR is relatively silent. The main difficulty is that the quantile loss is piecewise linear: it is non-smooth and has curvature concentrated at a single point. To overcome the lack of smoothness and strong convexity, we propose and study a convolution-type smoothed QR with iteratively reweighted l(1)-regularization. The resulting smoothed empirical loss is twice continuously differentiable and (provably) locally strongly convex with high probability. We show that the iteratively reweighted l(1)-penalized smoothed QR estimator, after a few iterations, achieves the optimal rate of convergence, and moreover, the oracle rate and the strong oracle property under an almost necessary and sufficient minimum signal strength condition. Extensive numerical studies corroborate our theoretical results.