In recent years, Wasserstein Distributionally Robust Optimization (DRO) has garnered substantial interest for its efficacy in data-driven decision-making under distributional uncertainty. However, limited research has explored the application of DRO to address individual fairness concerns, particularly when considering causal structures and discrete sensitive attributes in learning problems.To address this gap, we first formulate the DRO problem from the perspectives of causality and individual fairness. We then present the DRO dual formulation as an efficient tool to convert the main problem into a more tractable and computationally efficient form. Next, we characterize the closed form of the approximate worst-case loss quantity as a regularizer, eliminating the max-step in the Min-Max DRO problem. We further estimate the regularizer in more general cases and explore the relationship between DRO and classical robust optimization. Finally, by removing the assumption of a known structural causal model, we provide finite sample error bounds when designing DRO with empirical distributions and estimated causal structures to ensure efficiency and robust learning.

Distributionally robust optimization offers a compelling framework for model fitting in machine learning, as it systematically accounts for data uncertainty. Focusing on Wasserstein distributionally robust optimization, we investigate the regularized problem where entropic smoothing yields a sampling-based approximation of the original objective. We establish the convergence of the approximate gradient over a compact set, leading to the concentration of the regularized problem critical points onto the original problem critical set as regularization diminishes and the number of approximation samples increases. Finally, we deduce convergence guarantees for a projected stochastic gradient method. Our analysis covers a general machine learning situation with an unbounded sample space and mixed continuous-discrete data.

In recent years, Wasserstein Distributionally Robust Optimization (DRO) has garnered substantial interest for its efficacy in data-driven decision-making under distributional uncertainty. However, limited research has explored the application of DRO to address individual fairness concerns, particularly when considering causal structures and discrete sensitive attributes in learning problems.To address this gap, we first formulate the DRO problem from the perspectives of causality and individual fairness. We then present the DRO dual formulation as an efficient tool to convert the main problem into a more tractable and computationally efficient form. Next, we characterize the closed form of the approximate worst-case loss quantity as a regularizer, eliminating the max-step in the Min-Max DRO problem. We further estimate the regularizer in more general cases and explore the relationship between DRO and classical robust optimization. Finally, by removing the assumption of a known structural causal model, we provide finite sample error bounds when designing DRO with empirical distributions and estimated causal structures to ensure efficiency and robust learning.

Applications such as adversarially robust training and Wasserstein Distributionally Robust Optimization (WDRO) can be naturally formulated as min-sum-max optimization problems. While this formulation can be rewritten as an equivalent min-max problem, the summation of max terms introduces computational challenges, including increased complexity and memory demands, which must be addressed. These challenges are particularly evident in WDRO, where existing tractable algorithms often rely on restrictive assumptions on the objective function, limiting their applicability to state-of-the-art machine learning problems such as the training of deep neural networks. This study introduces a novel stochastic smoothing framework based on the \mbox{log-sum-exp} function, efficiently approximating the max operator in min-sum-max problems. By leveraging the Clarke regularity of the max operator, we develop an iterative smoothing algorithm that addresses these computational difficulties and guarantees almost surely convergence to a Clarke/directional stationary point. We further prove that the proposed algorithm finds an $\epsilon$-scaled Clarke stationary point of the original problem, with a worst-case iteration complexity of $\widetilde{O}(\epsilon^{-3})$. Our numerical experiments demonstrate that our approach outperforms or is competitive with state-of-the-art methods in solving the newsvendor problem, deep learning regression, and adversarially robust deep learning. The results highlight that our method yields more accurate and robust solutions in these challenging problem settings.

The library is based on distributionally robust optimization using optimal transport distances. For ease of use, it features both scikit-learn compatible estimators for popular objectives, as well as a wrapper for PyTorch modules, enabling researchers and practitioners to use it in a wide range of models with minimal code changes. Its implementation relies on an entropic smoothing of the original robust objective in order to ensure maximal model flexibility. The library is available at https://github.com/iutzeler/skwdro. Keywords: Distributionally robust optim., distribution shifts, entropic regularization

In many applications in statistics and machine learning, the availability of data samples from multiple possibly heterogeneous sources has become increasingly prevalent. On the other hand, in distributionally robust optimization, we seek data-driven decisions which perform well under the most adverse distribution from a nominal distribution constructed from data samples within a certain discrepancy of probability distributions. However, it remains unclear how to achieve such distributional robustness in model learning and estimation when data samples from multiple sources are available. In this work, we propose constructing the nominal distribution in optimal transport-based distributionally robust optimization problems through the notion of Wasserstein barycenter as an aggregation of data samples from multiple sources. Under specific choices of the loss function, the proposed formulation admits a tractable reformulation as a finite convex program, with powerful finite-sample and asymptotic guarantees. As an illustrative example, we demonstrate with the problem of distributionally robust sparse inverse covariance matrix estimation for zero-mean Gaussian random vectors that our proposed scheme outperforms other widely used estimators in both the low- and high-dimensional regimes.