Distributionally Robust Optimization (DRO) offers a promising framework by optimizing worst-case performance over a set of candidate distributions, referred to as the uncertainty set. However, the efficacy of DRO heavily depends on the design of the uncertainty set, and existing methods often perform suboptimally due to an inappropriate or inflexible uncertainty set. In this work, we first propose a novel perspective that casts entropy-regularized Wasserstein DRO as a dynamic process of distributional exploration and semantic alignment, both driven by optimal transport (OT). This unified viewpoint yields two key new techniques: semantic calibration, which bootstraps semantically meaningful transport costs via inverse OT, and adaptive refinement, which adjusts uncertainty set using OT-driven feedback. Together, these components form an exploration-and-feedback system, where the transport costs and uncertainty set evolve jointly during training, enabling the model to better adapt to potential distribution shifts. Moreover, we provide an in-depth analysis of this adaptive process and prove theoretical guarantees of convergence. Finally, we present our experimental results across diverse distribution shift scenarios, which demonstrate that our approach significantly outperforms existing methods, achieving state-ofthe-art robustness.

A major challenge in aligning large language models (LLMs) with human preferences is the issue of distribution shift. LLM alignment algorithms rely on static preference datasets, assuming that they accurately represent real-world user preferences. However, user preferences vary significantly across geographical regions, demographics, linguistic patterns, and evolving cultural trends. This preference distribution shift leads to catastrophic alignment failures in many real-world applications. We address this problem using the principled framework of distributionally robust optimization, and develop two novel distributionally robust direct preference optimization (DPO) algorithms, namely, Wasserstein DPO (WDPO) and Kullback-Leibler DPO (KLDPO). We characterize the sample complexity of learning the optimal policy parameters for WDPO and KLDPO. Moreover, we propose scalable gradient descent-style learning algorithms by developing suitable approximations for the challenging minimax loss functions of WDPO and KLDPO. Our empirical experiments using benchmark data sets and LLMs demonstrate the superior performance of WDPO and KLDPO in substantially improving the alignment when there is a preference distribution shift.

Distributionally robust optimization has been shown to offer a principled way to regularize learning models. In this paper, we find that Tikhonov regularization is distributionally robust in an optimal transport sense (i.e., if an adversary chooses distributions in a suitable optimal transport neighborhood of the empirical measure), provided that suitable martingale constraints are also imposed. Further, we introduce a relaxation of the martingale constraints which not only provides a unified viewpoint to a class of existing robust methods but also leads to new regularization tools. To realize these novel tools, tractable computational algorithms are proposed. As a byproduct, the strong duality theorem proved in this paper can be potentially applied to other problems of independent interest.

We consider the task of training machine learning models with data-dependent constraints. Such constraints often arise as empirical versions of expected value constraints that enforce fairness or stability goals. We reformulate data-dependent constraints so that they are calibrated: enforcing the reformulated constraints guarantees that their expected value counterparts are satisfied with a user-prescribed probability. The resulting optimization problem is amendable to standard stochastic optimization algorithms, and we demonstrate the efficacy of our method on a fairness-sensitive classification task where we wish to guarantee the classifier's fairness (at test time).

We study finite-sample statistical performance guarantees for distributionally robust optimization (DRO) with optimal transport (OT) and OT-regularized divergence model neighborhoods. Specifically, we derive concentration inequalities for supervised learning via DRO-based adversarial training, as commonly employed to enhance the adversarial robustness of machine learning models. Our results apply to a wide range of OT cost functions, beyond the $p$-Wasserstein case studied by previous authors. In particular, our results are the first to: 1) cover soft-constraint norm-ball OT cost functions; soft-constraint costs have been shown empirically to enhance robustness when used in adversarial training, 2) apply to the combination of adversarial sample generation and adversarial reweighting that is induced by using OT-regularized $f$-divergence model neighborhoods; the added reweighting mechanism has also been shown empirically to further improve performance. In addition, even in the $p$-Wasserstein case, our bounds exhibit better behavior as a function of the DRO neighborhood size than previous results when applied to the adversarial setting.

We develop efficient solution methods for a robust empirical risk minimization problem designed to give calibrated confidence intervals on performance and provide optimal tradeoffs between bias and variance. Our methods apply to distributionally robust optimization problems proposed by Ben-Tal et al., which put more weight on observations inducing high loss via a worst-case approach over a non-parametric uncertainty set on the underlying data distribution. Our algorithm solves the resulting minimax problems with nearly the same computational cost of stochastic gradient descent through the use of several carefully designed data structures. For a sample of size n, the per-iteration cost of our method scales as O(logn), which allows us to give optimality certificates that distributionally robust optimization provides at little extra cost compared to empirical risk minimization and stochastic gradient methods.

Distributionally Robust Optimization (DRO) accounts for uncertainty in data distributions by optimizing the model performance against the worst possible distribution within an ambiguity set. In this paper, we propose a DRO framework that relies on a new distance inspired by Unbalanced Optimal Transport (UOT). The proposed UOT distance employs a soft penalization term instead of hard constraints, enabling the construction of an ambiguity set that is more resilient to outliers. Under smoothness conditions, we establish strong duality of the proposed DRO problem. Moreover, we introduce a computationally efficient Lagrangian penalty formulation for which we show that strong duality also holds. Finally, we provide empirical results that demonstrate that our method offers improved robustness to outliers and is computationally less demanding for regression and classification tasks.

We consider the penalized distributionally robust optimization (DRO) problem with a closed, convex uncertainty set, a setting that encompasses learning using f -DRO and spectral/ L-risk minimization.