Distributionally Robust Optimization

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DRO studies how to make predictions and decisions reliable when the distribution at deployment may differ from the observed data. Methods define ambiguity sets using tools such as Wasserstein distance, statistical divergences, or group-based reweighting, then solve the resulting worst-case optimization with dual formulations, gradient methods, or saddle-point algorithms. Applications include machine learning under subgroup and domain shifts, reinforcement learning, control, and decisions in areas such as healthcare and energy. A central challenge is balancing protection against distribution shift with computational cost and excessive conservatism, while choosing ambiguity sets that reflect real uncertainty and still support useful guarantees.

Latest papers

86 papers18 papers in the last four weeks, up 350% on the four weeks before. 0.1% of all new papers.

Jul 18, 2026

Relative Entropy-Bounded Ambiguous Chance Constraints for Robust Planning in Nonlinear Systems

We consider defining risk probability in stochastic control problems under distribution ambiguity. Current approaches for chance-constrained control typically assume that the true state distribution is known and Gaussian distributed. These assumptions are not amenable to many real-world engineering applications where system dynamics are nonlinear and only approximately modeled. In this work, we define a distribution ambiguity set and, with a variational expression for exponential integrals, bound the expected risk value under an unknown distribution that resides within a relative entropy distance of a nominal Gaussian reference distribution. Our bound recovers the reference risk value in the zero-divergence limit. A method is presented to determine the relative entropy distance defining the ambiguity set that is a function of the reference covariance evolution and second-order dynamical truncation errors. The resulting contributions provide a framework for handling distributional ambiguity in nonlinear covariance steering problems. A stochastic spacecraft guidance example is presented to demonstrate our contributions.

Trevor N. Wolf, Jay W. McMahon

Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO 80303, USA

Chance-Constrained Optimization Stochastic Optimal Control Distributionally Robust Optimization

Jul 18, 2026

Approximate Relative Entropy Constraints for Nonlinear Covariance Steering Under Distribution Ambiguity

Covariance steering provides an efficient framework for designing linear stochastic feedback policies, but its extension to nonlinear systems relies on a Gaussian surrogate obtained through local linearization. Because this surrogate may differ substantially from the true nonlinear state distribution, risk-sensitive quantities such as collision probability and mean-squared error may be inaccurately estimated. This work develops a distributionally robust covariance-steering framework based on the relative entropy, also known as the Kullback-Leibler divergence (KLD), to account for ambiguity in the propagated probability density function. Using a variational representation of exponential integrals, we derive computable upper bounds on risk-sensitive quantities over a KLD ambiguity set. We then formulate an upper bound on the time rate of change of the KLD between the true nonlinear distribution and a Gaussian reference surrogate. Under some assumptions, this bound is controlled by decision variables within a covariance-steering formulation. The resulting constraints are incorporated into a sequential convex programming algorithm to design stochastic guidance policies that keep the true distribution close to its Gaussian surrogate while enforcing bounds on risk-sensitive performance measures. The proposed approach is demonstrated on a challenging nonlinear spacecraft transfer between two near-rectilinear halo orbits.

Trevor N. Wolf, Jay W. McMahon

*Postdoctoral Associate, Department of Aerospace Engineering Sciences, University of Colorado Boulder, AAS Member. · †Associate Professor, University of Colorado Boulder, Department of Aerospace Engineering Sciences, and AAS Fellow.

Distributionally Robust Optimization

Jul 15, 2026

Distributionally Robust and Safe Imitation Learning

Imitation learning (IL) has achieved remarkable success in complex decision-making tasks. However, its performance is highly sensitive to distribution shifts, which can pose significant safety risks. We propose a distributionally robust and safe IL framework that explicitly addresses both policy-induced and uncertainty-induced distribution shifts. Our approach develops a unified framework leveraging Taylor Series Imitation Learning (TaSIL) to mitigate policy-induced shifts and distributionally robust adaptive control to handle uncertainty-induced shifts. This architecture enables the formulation of an IL problem that optimizes performance under distributional uncertainty while systematically accounting for safety constraints. We demonstrate the effectiveness of the proposed approach on an unmanned aerial vehicle (UAV) case study where the UAV performs a task in an uncertain environment while avoiding unsafe regions.

Ahmed Aboudonia, Naira Hovakimyan

University of California Berkeley, Berkeley, CA 94720, USA · University of Illinois Urbana-Champaign, Urbana, IL 61801, USA

Distribution Shift Robustness Distributionally Robust Optimization Imitation Learning

Jul 10, 2026

Learning Predictive Ambiguity Sets for Decision-Focused Distributionally Robust Optimization

Predict-then-optimize systems usually compress uncertainty into a point forecast and then solve a downstream optimization problem as if the forecast were reliable. Distributionally robust optimization (DRO) offers protection against misspecification, but the ambiguity set is often centered at historical samples and uses a fixed radius. We propose learned predictive ambiguity sets (LPAS): a deep contextual model outputs a finite nominal scenario distribution, a state-dependent Wasserstein radius, and optionally an anisotropic ground metric. These outputs define a contextual ambiguity set that feeds a DRO decision layer. The radius is trained by a combination of conditional quantile calibration, size regularization, and downstream decision loss, so that robustness is adaptive rather than globally fixed. We derive the finite dual form used by the decision layer, present a staged training algorithm, and evaluate the method on distributionally robust portfolio optimization with 20 S&P 500 constituents from 2018--2026. The proposed method substantially improves over equal-weight, predict-then-optimize, and historical Wasserstein DRO baselines, achieving 26.28% annualized return, Sharpe ratio 1.30, final wealth 1.61, and lower tail loss than a deep fixed-radius DRO baseline while using a smaller average radius. The results show that learned ambiguity radii can recover most of the performance of strong fixed-radius DRO while reducing unnecessary conservatism and improving regime adaptivity.

Junjie Guo

Rutgers University

Distributionally Robust Optimization Decision-Focused Learning Wasserstein Distributionally Robust Optimization Portfolio Optimization

Jul 7, 2026

Decision-Focused Scenario Generation and Selection for Efficient and Robust Grid Dispatch

The increasing uncertainty from flexible demand and renewable generation has made distributionally robust optimization (DRO) an important tool for robust power system dispatch. DRO relies on forecast scenarios to construct ambiguity sets, but conventional scenario generation pipelines are often trained in an accuracy-oriented manner and may neglect spatial correlations among uncertainties. This mismatch can produce ambiguity sets that are statistically plausible but suboptimal for downstream operation. This work proposes a decision-focused generative framework for correlated scenario generation in DRO-based dispatch. Instead of training generative models solely to fit the historical uncertainty distribution, the proposed framework optimizes generated scenarios according to their induced downstream operational cost. The proposed framework is tailored to mainstream generative models, including variational autoencoders, generative adversarial networks, and diffusion models, while capturing the joint distribution of uncertainties across buses. To improve computational tractability, we further develop a differentiable scenario selector that selects decision-relevant scenarios from a generated pool and can be trained within the same decision-focused pipeline. Case studies demonstrate that the proposed framework effectively reduces 0.80%-2.02% operational cost across different generative models compared to accuracy-oriented methods.

Yangze Zhou, Yihong Zhou, Thomas Morstyn +1

Distributionally Robust Optimization Decision-Focused Learning Generative Modeling Decision-Making under Uncertainty Power Systems

Jul 2, 2026

Beyond the Performance Illusion: Structure-Aware Stratified Partitioning and Curriculum Distributionally Robust Optimization for Spatially Correlated Domains

Performance evaluation in AI systems commonly assumes that random dataset splits produce independent and identically distributed (i.i.d.) subsets. We show that this assumption often breaks down in spatiotemporally correlated domains such as aerial surveillance, precision agriculture, and medical imaging, leading to two systematic failures: data leakage, where correlated samples span training and validation splits and inflate performance estimates, and hidden stratification, where errors on minority subpopulations are obscured by aggregate metrics. To address these issues, we propose a unified evaluation and training framework for spatially correlated data. We introduce Structure-Aware Stratified Partitioning (SASP), which constructs validation splits that reduce spatiotemporal leakage while preserving meaningful class balance, and Curriculum Distributionally Robust Optimization (CDRO), a curriculum-based relaxation of distributionally robust training that stabilizes optimization under these stricter splits. Across multiple benchmarks, this combination yields consistently improved generalization, more reliable confidence calibration, and exposes failure modes that remain hidden under conventional random-split evaluation.

Prathamesh Patil, Arpit Jain, Aswanth Krishnan

1QpiAI India Pvt. Ltd.

Data Leakage Distributionally Robust Optimization

Jul 2, 2026

Distributionally Robust Listwise Preference Optimization

Existing robust preference optimization for language-model alignment mainly studies pairwise supervision and places robustness at the dataset, prompt, or preference-pair level. We instead study listwise preference optimization under ranking-label uncertainty: given a prompt and a candidate list, the observed ranking over that list may be ambiguous due to annotator inconsistency, near-ties, lossy rankwise feedback, or reward-model noise. We propose a pointwise total-variation robust Plackett--Luce objective that directly robustifies the ranking label conditional on the candidate list. The robust loss admits an exact decomposition into the nominal PL loss plus a worst-case PL correction, and the worst-case ranking is obtained by sorting current implicit scores in ascending order, reducing the inner maximization from

K!

enumeration to

O(K\log K)

. This tractable structure yields strong offline and online optimization guarantees. In the offline fixed-list setting, the robust objective is convex and projected stochastic subgradient reaches global

ε

-suboptimality with

O(ε^{-2})

sample complexity. In the online policy-induced setting, where candidate lists are generated by the current policy, we establish weak convexity and

\widetilde O(ε^{-2})

Moreau-envelope stationarity. Experiments in offline LLM alignment show that the proposed robust correction largely preserves performance under clean labels and improves robustness under noise. In online alignment, it makes reward-model-ranked candidate expansion more reliable and improves both reward-model and external GPT-4 judge metrics.

Xudong Wu, Jian Qian, Pangpang Liu +2

The University of Hong Kong · Hong Kong SAR · Yale University +3

LLM Alignment Distributionally Robust Optimization Robust Optimization Robust Loss Functions Preference Optimization

Jun 30, 2026

Distributionally Robust Linear Regression With Block Lewis Weights

We present an algorithm for the group distributionally robust (GDR) least squares problem. Given

m

groups, a parameter vector in

\mathbb{R}^d

, and stacked design matrices and responses

\mathbf{A}

and

\mathbf{b}

, our algorithm obtains a

(1+\varepsilon)

-multiplicative optimal solution using

\widetilde{O}(\min\{\mathsf{rank}(\mathbf{A}),m\}^{1/3}\varepsilon^{-2/3})

linear-system-solves of matrices of the form

\mathbf{A}^{\top}\mathbf{B}\mathbf{A}

for block-diagonal

\mathbf{B}

. Our technical methods follow from a recent geometric construction, block Lewis weights, that relates the empirical GDR problem to a carefully chosen least squares problem and an application of accelerated proximal methods. Our algorithm improves over known interior point methods for moderate accuracy regimes and matches the state-of-the-art guarantees for the special case of

\ell_{\infty}

regression. We also give algorithms that smoothly interpolate between minimizing the average least squares loss and the distributionally robust loss.

Naren Sarayu Manoj, Kumar Kshitij Patel

TTIC · Institute for Foundations of Data Science, Yale University

Robust Regression Group Distributionally Robust Optimization Distributionally Robust Optimization Linear Regression Robust Estimation

Jun 29, 2026

A Distributionally Robust Framework for Learned Reconstructions in Inverse Problems

Learned reconstruction operators for inverse problems are typically trained under a fixed noise model, and generalize poorly when the distribution during testing differs from the one assumed during training. Distributionally robust optimization (DRO) addresses this by optimizing against the worst-case distribution within a prescribed ambiguity set, but standard Wasserstein DRO perturbs the full joint distribution uniformly, which can be overly conservative and ignores the physics of the measurement process. We develop a structured DRO framework in which the ambiguity set is restricted to structured perturbations aligned with the data-acquisition process. This allows us to learn data-driven reconstruction operators that remain robust to distributional shifts. By constraining perturbations to subsets such as

P(Y|X)

, our framework models uncertainty in the forward operator and noise model more faithfully, accommodating any noise model expressible as a stochastic forward operator. We establish strong duality for this general formulation and derive explicit finite-dimensional dual representations for perturbations in the joint, marginal, and conditional distributions. A central result is an explicit worst-case risk bound that induces Tikhonov regularization on the Lipschitz constant of the reconstruction operator, and is less conservative relative to standard DRO for well-posed problems. Numerical experiments on deblurring and sinogram-to-CT reconstruction demonstrate improved robustness, stability, and interpretability over standard DRO and MSE baselines. In the linear setting, the learned operator becomes effectively low-rank, truncating at the intrinsic dimension of the data and recovering a data-driven analogue of truncated-SVD regularization.

Floor van Maarschalkerwaart, Subhadip Mukherjee, Christoph Brune +1

University of Twente, Enschede, the Netherlands · IIT Kharagpur, Kharagpur, India

Distribution Shift Robustness Distributionally Robust Optimization Inverse Problems

May 28, 2026

Distributionally Robust Set Representation Learning Under Inference-Time Element Corruption

Standard Set Representation Learning methods typically excel on curated data but often overlook the challenge of inference-time element corruption. This refers to scenarios where deployed models encounter element-level degradations, such as outliers or missing components, that may distort set representation and degrade performance. We propose SW-DRSO, a distributionally robust optimization framework tailored for sets. Rather than minimizing loss solely on observed training data, SW-DRSO optimizes a tractable surrogate of the worst-case expected loss over a family of plausible inference-time variations. We introduce a barycentric adversary that approximates the intractable search over corrupted sets by a differentiable training-time optimization over simplex weights. Extensive experiments across four tasks demonstrate that SW-DRSO effectively enhances robustness against corruption while maintaining high overall performance.

Yankai Chen, Hanrong Zhang, Bowei He +2

MBZUAI · McGill University · University of Illinois Chicago

Distribution Shift Robustness Representation Learning Distributionally Robust Optimization

May 27, 2026

Conservative neural posterior estimation via distributionally robust training

Simulation-based inference with neural posterior estimation (NPE) often yields overconfident and unreliable posteriors under limited simulation budgets. To address this, we propose DRO-NPE, a distributionally robust approach that replaces the standard NPE objective with a worst-case loss over a Wasserstein ambiguity set. We introduce KL-based metrics for miscoverage and miscalibration, and use these to show that the DRO-NPE objective controls overfitting and reduces posterior overconfidence. Our method is tractable, parallelisable, and readily integrates with standard normalising flows. Across benchmark SBI tasks, DRO-NPE consistently improves coverage and calibration, while narrowing the gap between empirical and population NPE loss, leading to more reliable inference in low-simulation regimes.

William Laplante, Yuga Hikida, Charita Dellaporta +2

Department of Statistical Science, University College London, UK · The Alan Turing Institute, UK · Department of Computer Science, Aalto University, Finland

Simulation-Based Inference Neural Posterior Estimation Probability Calibration Distributionally Robust Optimization Wasserstein Distributionally Robust Optimization Robust Estimation

May 24, 2026

Choosing Online Experiment Designs under Interference in Ads, Recommendations, and Member-Experience Systems

Online experiments in ads, recommendation, and member-experience systems are often planned before the dominant interference mechanism is known. A treatment may propagate through budgets, inventory, producer exposure, graph spillovers, or temporal carryover, making the randomization design itself a statistical decision. We formulate this problem as robust design selection over uncertain exposure mechanisms. Given a finite catalog of six implementable designs, the selector compares each design by worst-case planning risk over an ambiguity set. The risk combines exposure bias, assignment-unit variance, minimum detectable effect, contamination or carryover, operational cost, and estimand mismatch. For theoretical justification, the paper develops a geometry-aware guarantee, stating that design bias is bounded by Wasserstein distance to the launch exposure distribution, and this penalty is minimax tight under Lipschitz exposure response. We also prove finite-catalog approximation and a robust selector theorem with excess-risk control, exact recovery under separation, and certified shortlists when the risk surface is flat. Empirically, the same selector gives different recommendations across samples from public datasets. It selects user-randomization on Criteo ads with dimensionless robust risk 1.295, switchbacks on Open Bandit-bts/men with risk 2.105, and cluster-randomization on KuaiRand with risk 2.240. The Open Bandit case stresses known but uneven logging support, with propensities from 0.00006 to 0.594 and a 5.17% IPS effective-sample share. Overall, the paper contributes an interference-aware experiment design framework based on mechanism-robust design decisions, where the output is either a justified design choice or an uncertainty shortlist.

Prashant Shekhar, Caroline Howard

Department of Mathematics Embry-Riddle Aeronautical University Daytona Beach, FL, USA

Distributionally Robust Optimization

May 18, 2026

Distributionally Robust Control via Stein Variational Inference for Contact-Rich Manipulation

Reliable robotic manipulation requires control policies that can accurately represent and adapt to uncertainty arising from contact-rich interactions. Modern data-driven methods mitigate uncertainty through large-scale training and computation, and degrade significantly in performance with limited number of training samples. By contrast, classical model-based controllers are computationally efficient and reliable, but their limited ability to represent task-relevant uncertainty can hinder performance in contact-rich interactions. In this work, we propose to expand the capabilities of model-based manipulation control through more flexible uncertainty modeling that retains performance while exactly adapting to uncertainty. Our approach casts the manipulation problem as a distributionally robust control optimization and proposes a novel deterministic formulation based on Stein variational inference that preserves performance while explicitly modeling task-sensitive parameter uncertainty. As a result, the derived controllers are more aware of task sensitivities to uncertainty, yielding high reliability without compromising performance. Experimental results demonstrate up to 3

\times

improved robustness across a range of contact-rich manipulation tasks under broad parametric uncertainty, outperforming existing model-based control methods.

Hrishikesh Sathyanarayan, Victor Vantilborgh, Harish Ravichandar +2

Yale University, New Haven, CT, USA · Ghent University, Belgium · Georgia Institute of Technology, Atlanta, GA, USA +1

Contact-Rich Robotic Manipulation Robust Control Distributionally Robust Optimization Variational Inference

May 13, 2026

Distributionally Robust Safety Under Arbitrary Uncertainties: A Safety Filtering Approach

In this work, we study how to ensure probabilistic safety for nonlinear systems under distributional ambiguity. Our approach builds on a backup-based safety filtering framework that switches between a high-performance nominal policy and a certified backup policy to ensure safety. To handle arbitrary uncertainties from ambiguous distributions, i.e., where the distribution is not of specific structure and the true distribution is unknown, we adopt a distributionally robust (DR) formulation using Wasserstein ambiguity sets. Rather than solving a high-dimensional DR trajectory optimization problem online, we exploit the structure of backup-based safety filtering to reduce safety certification to a one-dimensional search over the switching time between nominal and backup policies. We then develop a sampling-based certification procedure with finite-sample guarantees, where empirical failure probabilities are compared against a Wasserstein-inflated threshold. We validate our method through simulations across three systems, from a Dubins vehicle to a high-speed racing car and a fighter jet, demonstrating the broad applicability and computational efficiency.

Daniel M. Cherenson, Haejoon Lee, Taekyung Kim +1

Department of Robotics, University of Michigan, Ann Arbor, MI, USA

Safety-Critical Control Distributionally Robust Optimization Safety Filtering Wasserstein Distributionally Robust Optimization

May 11, 2026

DuetFair: Coupling Inter- and Intra-Subgroup Robustness for Fair Medical Image Segmentation

Medical image segmentation models can perform unevenly across subgroups. Most existing fairness methods focus on improving average subgroup performance, implicitly treating each subgroup as internally homogeneous. However, this can hide difficult cases within a subgroup, where high-loss samples are obscured by the subgroup mean. We call this problem intra-group hidden failure. To solve this, we propose DuetFair mechanism, a dual-axis fairness framework that jointly considers inter-subgroup adaptation and intra-subgroup robustness. Based on DuetFair, we introduce FairDRO, which combines distribution-aware mixture-of-experts (dMoE) with subgroup-conditioned distributionally robust optimization (DRO) loss aggregation. This design allows the model to adapt across subgroups while also reducing hidden failures within each subgroup. We evaluate FairDRO on three medical image segmentation benchmarks with varying degrees of within-group heterogeneity. FairDRO achieves the best equity-scaled performance on Harvard-FairSeg and improves worst-case subgroup performance on HAM10000 under both age- and race-based grouping schemes. On the 3D radiotherapy target cohort, FairDRO further improves worst-group Dice by 3.5 points (

\uparrow 6.0\%

) under the tumor-stage grouping and by 4.1 points (

\uparrow 7.4\%

) under the institution grouping over the strongest baseline.

Yiqi Tian, Sangjoon Park, Bo Zeng +3

Center for Advanced Medical Computing and Analysis, Massachusetts General Hospital and Harvard Medical School, Boston, MA 02114 · Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA 15261 · Department of Radiation Oncology, College of Medicine, Yonsei University, Seoul, South Korea +2

Subgroup Robustness Healthcare Distributionally Robust Optimization Group Fairness Medical Image Segmentation

May 8, 2026

Interactive Trajectory Planning with Learning-based Distributionally Robust Model Predictive Control and Markov Systems

We investigate interactive trajectory planning subject to uncertainty in the decisions of surrounding agents. To control the ego-agent, we aim to first learn the decision distribution and solve a Stochastic Model Predictive Control (SMPC) problem. To account for errors in the learned distribution, we show that it is possible to utilize Probably Approximately Correct (PAC) learning in combination with Distributionally Robust (DR) optimization to obtain a solution which accounts for the errors induced by the learning model. The results indicate that our PAC learning-based DR-MPC framework provides a method to interpolate between a robust MPC and an omnipotent SMPC, based on the available number of samples.

Erik Börve, Nikolce Murgovski, Morteza Haghir Chehreghani +1

Chalmers University of Technology, Chalmersgatan 4, 412 96 Göteborg, Sweden · Volvo Group Trucks Technology, Gropegårdsgatan 2, 417 15 Göteborg, Sweden

Distributionally Robust Optimization Model Predictive Control Motion Planning Optimal Control

May 8, 2026

Ensemble Distributionally Robust Bayesian Optimisation with Continuous Context

We study Bayesian Optimisation (BO) in settings where the objective function is influenced by uncontrollable environmental contexts governed by an unknown probability distribution. In practice, the contextual distribution must be estimated from empirical data, a process that inherently introduces distributional mismatch, producing sub-optimal results. While Distributionally Robust Optimisation (DRO) provides a framework to mitigate these risks, existing robust BO methods frequently suffer from high computational complexity, rely on discretisation of continuous context spaces, or impose restrictive assumptions on the structure of the ambiguity set. To overcome these limitations, we propose Ensemble Distributionally Robust Bayesian Optimisation (EDRBO). Our framework leverages the expressive power of ensemble surrogate models to approximate the black-box function while simultaneously accounting for contextual uncertainty. By utilising Wasserstein ball as ambiguity sets, EDRBO provides a robustified acquisition function that remains computationally tractable and natively handles continuous context spaces. We establish a rigorous theoretical foundation for our approach by proving sublinear cumulative regret guarantees of order

\mathcal{O}(γ_T \sqrt{T})

, where

γ_T

represents the maximum information gain within the ensemble. Finally, we provide extensive empirical evaluations that corroborate our theory and demonstrate the state-of-the-art performance of EDRBO.

Tigran Ramazyan, Denis Derkach

Faculty of Computer Science, HSE University, Pokrovsky Boulevard 11, Moscow, 109028, Moscow, Russia.

Bayesian Optimization Distributionally Robust Optimization Wasserstein Distributionally Robust Optimization

May 7, 2026

Distributionally-Robust Learning to Optimize

We propose a distributionally robust approach to learning hyperparameters for first-order methods in convex optimization. Given a dataset of problem instances, we minimize a Wasserstein distributionally robust version of the performance estimation problem (PEP) over algorithm parameters such as step sizes. Our framework unifies two extremes: as the robustness radius vanishes, we recover classical learning to optimize (L2O); as it grows, we recover worst-case optimal algorithm design via PEP. We solve the resulting problem with stochastic gradient descent, differentiating through the solution of an inner semidefinite program at each step. We prove high-probability bounds showing that the true risk of the learned algorithm is at most the in-sample L2O optimum plus a slack that shrinks with the sample size, and is no worse than the worst-case PEP bound. On unconstrained quadratic minimization, LASSO, and linear programming benchmarks, our learned algorithms achieve strong out-of-sample performance with certifiable robustness, outperforming both worst-case optimal and vanilla L2O baselines.

Vinit Ranjan, Jisun Park, Bartolomeo Stellato

Department of Operations Research and Financial Engineering, Princeton University. · Department of Operations Research and Financial Engineering, Princeton University, and Research Institute of Mathematics, Seoul National University. · Research Institute of Mathematics, Seoul National University.

Learning to Optimize Distributionally Robust Optimization Robust Optimization Wasserstein Distributionally Robust Optimization

May 7, 2026

From Dual Tracking to Clipping: Provably Faster Distributionally Robust Multi-Objective Optimization

Multi-objective optimization (MOO) has received growing attention in applications that require learning under multiple criteria. However, most existing MOO formulations do not explicitly account for distributional shifts in the data. We introduce distributionally robust multi-objective optimization (DR-MOO), which minimizes multiple objectives under their respective worst-case distributions. We propose Pareto-type solution concepts for DR-MOO and develop multi-gradient descent algorithms (MGDA) with provable guarantees. Leveraging a Lagrangian dual reformulation, we first design a double-loop MGDA that uses an inner loop to estimate dual variables and achieves a total sample complexity

\mathcal{O}(ε^{-8})

for reaching an

ε

-Pareto-stationary point. To further improve convergence, we combine large-batch sampling with gradient clipping to accommodate generalized smoothness and control bias in stochastic preference updates, eliminating the need for double sampling. This yields a single-loop double-clip MGDA with substantially improved sample complexity

\mathcal{O}(ε^{-4})

. Our theory applies to nonconvex problems without requiring uniformly bounded gradients of the dual objectives. Experiments demonstrate that our methods are competitive with state-of-the-art MGDA baselines.

Yufeng Yang, Fangning Zhuo, Ziyi Chen +2

Texas A&M University · A*STAR · University of Maryland

Gradient Clipping Gradient Descent Nonconvex Stochastic Optimization Distributionally Robust Optimization Robust Optimization Multi-Objective Optimization

May 6, 2026

Information Theoretic Adversarial Training of Large Language Models

Large language models (LLMs) remain vulnerable to adversarial prompting despite advances in alignment and safety, often exhibiting harmful behaviors under novel attack strategies. While adversarial training can improve robustness, existing approaches are computationally expensive and difficult to scale. Recent continuous adversarial training methods, such as Continuous adversarial training (CAT) and Continuous Adversarial Preference Optimization (CAPO), address this challenge by leveraging gradient-based perturbations in the embedding space, enabling more efficient and expressive attacks. Building on this paradigm, we propose WARDEN, a distributionally robust adversarial training framework for LLMs that dynamically reweights adversarial examples through an f -divergence ambiguity set around the empirical training distribution. Our method optimizes the worst-case adversarial loss within a divergence ball around the empirical data distribution, automatically emphasizing harder adversarial examples. Using the convex dual formulation, the objective reduces to a log-sum-exp form under the KL divergence, with a dynamical parameter controlling the strength of reweighting. This study leads to a new class of information-theoretic objectives that significantly reduce attack success rates while maintaining model utility. Across multiple LLMs and attack settings, WARDEN substantially reduces attack success rates with computational and utility costs comparable to CAT-, CAPO-, and MixAT-based baselines, making it a practical approach for scalable robust alignment.

Yiwei Zhang, Jeremiah Birrell, Reza Ebrahimi +3

Purdue University · West Lafayette, IN 47907, USA · Texas State University +5

Adversarial Training LLM Alignment Adversarial Attacks on LLMs Distributionally Robust Optimization Language Model Robustness

May 5, 2026

Quantile Geometry Regularization for Distributional Reinforcement Learning

Quantile-based distributional reinforcement learning methods learn return distributions through sampled quantile regression, but their bootstrapped target quantiles may induce distorted or degenerate distribution estimates. We propose Robust Quantile-based Implicit Quantile Networks (RQIQN), a lightweight Wasserstein distributionally robust enhancement boosted from a quantile estimation perspective. We first reinterpret a snapshot of IQN loss as a collection of local empirical quantile estimation problems over sampled current fractions. We then robustify each local slot with a Wasserstein distributionally robust quantile estimation formulation, yielding a closed-form, fraction-dependent correction to the Bellman target. This correction directly addresses distributional degeneration: its median antisymmetry preserves the risk-neutral quantile average, while its monotonicity enlarges upper-lower quantile gaps and counteracts collapsed distributional spread. RQIQN thus regularizes quantile geometry without changing the underlying value objective or requiring additional sample set reconstruction. Finally, we empirically show that the proposed RQIQN outperforms other existing quantile-based distributional reinforcement learning algorithms in risk-sensitive navigation and Atari games.

Zhaofan Zhang, Minghao Yang, Rufeng Chen +2

Information Hub, AI Thrust Hong Kong University of Science and Technology (Guangzhou) Guangzhou, Guangdong, China

Distributional RL Quantile Regression Risk-Sensitive RL Distributionally Robust Optimization Distributionally Robust RL

Apr 30, 2026

Wasserstein Distributionally Robust Regret Optimization for Reinforcement Learning from Human Feedback

Reinforcement learning from human feedback (RLHF) is a central post-training tool for aligning large language models, but its training reward is only a learned proxy for true human utility. This creates a decision problem under objective misspecification: the policy is optimized against an estimated reward, while deployment performance is governed by an unobserved population preference. The resulting gap leads to reward over-optimization, where proxy reward keeps improving after true quality deteriorates. We propose distributionally robust regret optimization (DRRO) for RLHF with a Wasserstein ambiguity set over reward laws, using promptwise

\ell_p

distances between reward vectors as transport costs. Unlike standard distributionally robust optimization, which pessimizes worst-case value, DRRO pessimizes worst-case regret relative to the best policy under the same plausible reward perturbation. We show that the expressive-policy problem decomposes into promptwise regret problems. For each prompt, the inner adversary has a dual-norm closed form; under the

\ell_1

transport cost used by our algorithm, the optimizer has a water-filling structure. These results lead to a practical policy-gradient algorithm that adds a simple sampled bonus to GRPO-style training. Theory and experiments both show that DRRO is less over-pessimistic than standard DRO and mitigates over-optimization more effectively than existing baselines.

Yikai Wang, Shang Liu, Jose Blanchet

Department of Statistics and Operations Research, University of North Carolina · Imperial Business School, Imperial College London · Department of Management Science and Engineering, Stanford University

Reward Over-Optimization Regret Minimization in RL Reinforcement Learning Distributionally Robust Optimization Wasserstein Distributionally Robust Optimization Distributionally Robust RL+2

Apr 30, 2026

Sampler-Robust Optimization under Generative Models

Modern stochastic optimization pipelines increasingly rely on learned generative models to represent uncertainty, while downstream decisions are evaluated almost entirely through Monte Carlo scenarios. This shifts the operational object of uncertainty from an explicit probability law to the sampler induced by the learned generator. Reliability therefore depends on two errors: sampler misspecification and finite-simulation error. We propose Sampler-Robust Optimization (SRO), which optimizes decisions against the worst-case sampler induced by perturbing the learned generator. This sampler-first formulation aligns with simulation-based decision pipelines and admits a sharpness-aware interpretation: it favors decisions whose performance is stable under generator perturbations, rather than merely under the nominal sampler. Under a coverage assumption, we show that the empirical worst-case objective provides a high-probability upper certificate for the true population objective, with finite-simulation error partially absorbed by the robustification used to guard against sampler misspecification. The framework accommodates generative models with or without explicit densities and admits efficient minimax procedures. Portfolio-optimization experiments show that SRO produces more stable decisions and improves out-of-sample performance under distribution shift.

Ziwei Zhang, Jonathan Yu-Meng Li

Telfer School of Management University of Ottawa

Distribution Shift Robustness Distributionally Robust Optimization Robust Optimization

Apr 28, 2026

A Distributionally Robust Reinforcement Learning Framework for Constrained Urban EV Dispatch

We study city-scale control of electric-vehicle (EV) ride-hailing fleets where dispatch, repositioning, and charging decisions must respect charger and feeder limits under uncertain, spatially correlated demand and travel times. We formulate the problem as a hex-grid semi-Markov decision process (semi-MDP) with mixed actions -- discrete actions for serving, repositioning, and charging, together with continuous charging power -- and variable action durations. To guarantee physical feasibility during both training and deployment, the policy learns over high-level intentions produced by a masked, temperature-annealed actor. These intentions are projected at every decision step through a time-limited rolling mixed-integer linear program (MILP) that strictly enforces state-of-charge, port, and feeder constraints. To mitigate distributional shifts, we optimize a Soft Actor-Critic (SAC) agent against a Wasserstein-1 ambiguity set with a graph-aligned Mahalanobis ground metric that captures spatial correlations. The robust backup uses the Kantorovich-Rubinstein dual, a projected subgradient inner loop, and a primal-dual risk-budget update. Our architecture combines a two-layer Graph Convolutional Network (GCN) encoder, twin critics, and a value network that drives the adversary. Experiments on a large-scale EV fleet simulator built from NYC taxi data show that PD-RSAC achieves the highest net profit, reaching $1.22M, compared with $0.58M-$0.70M for strong heuristic, single-agent RL, and multi-agent RL baselines, including Greedy, SAC, MAPPO, and MADDPG, while maintaining zero feeder-limit violations.

An Nguyen, Hoang Nguyen, Phuong Le +3

College of Engineering and Computer Science, VinUniversity, Hanoi, Vietnam · Center for Environmental Intelligence, VinUniversity, Hanoi, Vietnam

Electric Vehicle Charging Constrained RL Distributionally Robust Optimization Distributionally Robust RL Robust RL

Apr 22, 2026

Robust Out-of-Distribution Stochastic Optimization

Data-driven decision-making under uncertainty typically presumes the collection of historical data from an unknown target probability distribution. However, one may have no access to any data from the target distribution prior to decision-making. To address this challenge, we propose robust out-of-distribution stochastic optimization, a novel data-driven framework that effectively utilizes relevant data distributions for robust decision-making under unseen distributions. A key feature of our framework is that all data distributions are assumed to be randomly generated from a meta-distribution over distributions. To describe uncertainty in distribution generation, we propose to learn a data-driven uncertainty set in a reproducing kernel Hilbert space (RKHS) from relevant data distributions, with adjustable conservatism. We then incorporate this set into a min-max stochastic program to derive robust decisions. Notably, under randomness of distribution generation, we establish rigorous out-of-distribution generalization guarantees for the uncertainty set as well as the solution. To ease problem-solving in RKHS, an approximate parametrization with a provably bounded suboptimality and a row generation strategy are presented. Extensive numerical experiments on multi-item newsvendor and portfolio optimization demonstrate the superior out-of-distribution performance of our decision-making framework under unseen data distribution, even when only a small or moderate number of relevant sources are available.

Xianyu Li, Huan Xu, Xiaolin Huang +1

Department of Automation, Tsinghua University · Antai College of Economics & Management, Shanghai Jiao Tong University · Institute of Image Processing and Pattern Recognition, Shanghai Jiao Tong University

Stochastic Optimization OOD Generalization Distributionally Robust Optimization Decision-Making under Uncertainty

Apr 20, 2026

Wasserstein Distributionally Robust Risk-Sensitive Estimation via Conditional Value-at-Risk

We propose a distributionally robust approach to risk-sensitive estimation of an unknown signal x from an observed signal y. The observation and unknown signal are modeled as random vectors whose joint probability distribution is unknown, but assumed to belong to a given type-2 Wasserstein ball of distributions, termed the ambiguity set. The performance of an estimator is measured according to the conditional value-at-risk (CVaR) of the squared estimation error. Within this framework, we study the problem of computing affine estimators that minimize the worst-case CVaR over all distributions in the given ambiguity set. As our main result, we show that, when the nominal distribution at the center of the Wasserstein ball is finitely supported, such estimators can be exactly computed by solving a tractable semidefinite program. We evaluate the proposed estimators on a wholesale electricity price forecasting task using real market data and show that they deliver lower out-of-sample CVaR of squared error compared to existing methods.

Feras Al Taha, Eilyan Bitar

School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853 USA

Distributionally Robust Optimization Wasserstein Distributionally Robust Optimization Conditional Value-at-Risk Robust Estimation

Apr 9, 2026

STaR-DRO: Stateful Tsallis Reweighting for Group-Robust Structured Prediction

Structured prediction with large language models requires outputs that are label-accurate, ontology-constrained, structurally valid, and evidence-grounded under label imbalance and heterogeneous group difficulty. We present a unified framework for ontology-constrained generation. First, we introduce a modular prompt-engineering architecture combining XML-style structure, expert disambiguation rules, chain-of-thought reasoning, metadata-aware decision logic, schema contracts, and a self-validation gate. It targets recurrent in-context failures, including format drift, label ambiguity, evidence hallucination, and metadata-conditioned confusion. Second, we propose STaR-DRO, combining Tsallis mirror ascent, sparse entmax-style primal mapback, EMA-smoothed group-loss tracking, rescaled ascent signals, and bounded excess-only multipliers. Unlike conventional DRO, which relies on dense Shannon-entropy exponentiated-gradient updates, can introduce high-variance stochastic reweighting, assigns positive adversarial mass to groups that are not persistently hard, and incurs costs through simplex competition, STaR-DRO upweights only persistently hard groups without suppressing easier ones. We evaluate the framework on EPPC Miner, a clinically grounded high-stakes structured-prediction task requiring hierarchical label prediction and evidence-span extraction from patient-provider secure messages. Across 1B-70B Llama models, prompt engineering improves zero-shot extraction, yielding an average label F1 gain of +14.46 and a Span F1 gain of +17.40. Building on supervised fine-tuning, STaR-DRO further improves accuracy and robustness, increasing average label F1 by +1.08 and +2.20 while reducing mean groupwise validation cross-entropy by 21.3% and 14.8% relative to SFT and standard DRO, respectively. These results advance reliable automated communication mining for patient-centered clinical care analysis.

Samah Fodeh, Ganesh Puthiaraju, Elyas Irankhah +5

Yale University, New Haven, CT, USA. · Medical Oncology, Yale School of Medicine, New Haven, CT, USA.

Structured Prediction Distributionally Robust Optimization LLM Prompting Clinical Information Extraction Medical LLMs

Apr 7, 2026

Robust Learning of Heterogeneous Dynamic Systems

Ordinary differential equations (ODEs) provide a powerful framework for modeling dynamic systems arising in a wide range of scientific domains. However, most existing ODE methods focus on a single system, and do not adequately address the problem of learning shared patterns from multiple heterogeneous dynamic systems. In this article, we propose a novel distributionally robust learning approach for modeling heterogeneous ODE systems. Specifically, we construct a robust dynamic system by maximizing a worst-case reward over an uncertainty class formed by convex combinations of the derivatives of trajectories. We show the resulting estimator admits an explicit weighted average representation, where the weights are obtained from a quadratic optimization that balances information across multiple data sources. We further develop a bi-level stabilization procedure to address potential instability in estimation. We establish rigorous theoretical guarantees for the proposed method, including consistency of the stabilized weights, error bound for robust trajectory estimation, and asymptotical validity of pointwise confidence interval. We demonstrate that the proposed method considerably improves the generalization performance compared to the alternative solutions through both extensive simulations and the analysis of an intracranial electroencephalogram data.

Shuoxun Xu, Zijian Guo, Brooke R. Staveland +2

University of California at Berkeley · Zhejiang University · University of California at San Francisco

Dynamical Systems Robust Estimation System Identification Distributionally Robust Optimization

Apr 6, 2026

Reproducibility study on how to find Spurious Correlations, Shortcut Learning, Clever Hans or Group-Distributional non-robustness and how to fix them

Deep Neural Networks (DNNs) are increasingly utilized in high-stakes domains like medical diagnostics and autonomous driving where model reliability is critical. However, the research landscape for ensuring this reliability is terminologically fractured across communities that pursue the same goal of ensuring models rely on causally relevant features rather than confounding signals. While frameworks such as distributionally robust optimization (DRO), invariant risk minimization (IRM), shortcut learning, simplicity bias, and the Clever Hans effect all address model failure due to spurious correlations, researchers typically only reference work within their own domains. This reproducibility study unifies these perspectives through a comparative analysis of correction methods under challenging constraints like limited data availability and severe subgroup imbalance. We evaluate recently proposed correction methods based on explainable artificial intelligence (XAI) techniques alongside popular non-XAI baselines using both synthetic and real-world datasets. Findings show that XAI-based methods generally outperform non-XAI approaches, with Counterfactual Knowledge Distillation (CFKD) proving most consistently effective at improving generalization. Our experiments also reveal that the practical application of many methods is hindered by a dependency on group labels, as manual annotation is often infeasible and automated tools like Spectral Relevance Analysis (SpRAy) struggle with complex features and severe imbalance. Furthermore, the scarcity of minority group samples in validation sets renders model selection and hyperparameter tuning unreliable, posing a significant obstacle to the deployment of robust and trustworthy models in safety-critical areas.

Ole Delzer, Sidney Bender

Machine Learning Group Technische Universität Berlin

Group Distributionally Robust Optimization Spurious Correlation Robustness Shortcut Learning Distributionally Robust Optimization ML Reproducibility

Apr 6, 2026

Generative models for decision-making under distributional shift

Many data-driven decision problems are formulated using a nominal distribution estimated from historical data, while performance is ultimately determined by a deployment distribution that may be shifted, context-dependent, partially observed, or stress-induced. This tutorial presents modern generative models, particularly flow- and score-based methods, as mathematical tools for constructing decision-relevant distributions. From an operations research perspective, their primary value lies not in unconstrained sample synthesis but in representing and transforming distributions through transport maps, velocity fields, score fields, and guided stochastic dynamics. We present a unified framework based on pushforward maps, continuity, Fokker-Planck equations, Wasserstein geometry, and optimization in probability space. Within this framework, generative models can be used to learn nominal uncertainty, construct stressed or least-favorable distributions for robustness, and produce conditional or posterior distributions under side information and partial observation. We also highlight representative theoretical guarantees, including forward-reverse convergence for iterative flow models, first-order minimax analysis in transport-map space, and error-transfer bounds for posterior sampling with generative priors. The tutorial provides a principled introduction to using generative models for scenario generation, robust decision-making, uncertainty quantification, and related problems under distributional shift.

Xiuyuan Cheng, Yunqin Zhu, Yao Xie

Department of Mathematics, Duke University, Durham, NC, USA · H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

Decision-Making under Uncertainty Distribution Shift Robustness Distributionally Robust Optimization