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1. Distributionally Robust Mean-Variance Portfolio Selection with Wasserstein Distances

   https://doi.org/10.1287/mnsc.2021.4155  

   Blanchet, Jose; Chen, Lin; Zhou, Xun Yu (December 2021, Management Science)

   We revisit Markowitz’s mean-variance portfolio selection model by considering a distributionally robust version, in which the region of distributional uncertainty is around the empirical measure and the discrepancy between probability measures is dictated by the Wasserstein distance. We reduce this problem into an empirical variance minimization problem with an additional regularization term. Moreover, we extend the recently developed inference methodology to our setting in order to select the size of the distributional uncertainty as well as the associated robust target return rate in a data-driven way. Finally, we report extensive back-testing results on S&P 500 that compare the performance of our model with those of several well-known models including the Fama–French and Black–Litterman models. This paper was accepted by David Simchi-Levi, finance. 

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2. Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach

   https://doi.org/10.1287/moor.2020.1085  

   Duchi, John C.; Glynn, Peter W.; Namkoong, Hongseok (August 2021, Mathematics of Operations Research)

   We study statistical inference and distributionally robust solution methods for stochastic optimization problems, focusing on confidence intervals for optimal values and solutions that achieve exact coverage asymptotically. We develop a generalized empirical likelihood framework—based on distributional uncertainty sets constructed from nonparametric f-divergence balls—for Hadamard differentiable functionals, and in particular, stochastic optimization problems. As consequences of this theory, we provide a principled method for choosing the size of distributional uncertainty regions to provide one- and two-sided confidence intervals that achieve exact coverage. We also give an asymptotic expansion for our distributionally robust formulation, showing how robustification regularizes problems by their variance. Finally, we show that optimizers of the distributionally robust formulations we study enjoy (essentially) the same consistency properties as those in classical sample average approximations. Our general approach applies to quickly mixing stationary sequences, including geometrically ergodic Harris recurrent Markov chains. 

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3. Distributionally Robust Post-hoc Classifiers under Prior Shifts

   Wei, Jiaheng; Narasimhan, Harikrishna; Amid, Ehsan; Chu, Wen-Sheng; Liu, Yang; Kumar, Abhishek (January 2023, International Conference on Learning Representations (ICLR))

   The generalization ability of machine learning models degrades significantly when the test distribution shifts away from the training distribution. We investigate the problem of training models that are robust to shifts caused by changes in the distribution of class-priors or group-priors. The presence of skewed training priors can often lead to the models overfitting to spurious features. Unlike existing methods, which optimize for either the worst or the average performance over classes or groups, our work is motivated by the need for finer control over the robustness properties of the model. We present an extremely lightweight post-hoc approach that performs scaling adjustments to predictions from a pre-trained model, with the goal of minimizing a distributionally robust loss around a chosen target distribution. These adjustments are computed by solving a constrained optimization problem on a validation set and applied to the model during test time. Our constrained optimization objective is inspired from a natural notion of robustness to controlled distribution shifts. Our method comes with provable guarantees and empirically makes a strong case for distributional robust post-hoc classifiers. An empirical implementation is available at https://github.com/weijiaheng/Drops. 

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4. Wasserstein Robust Classification with Fairness Constraints

   https://doi.org/10.1287/msom.2022.0230  

   Wang, Yijie; Nguyen, Viet Anh; Hanasusanto, Grani A (July 2024, Manufacturing & Service Operations Management)

   Problem definition: Data analytics models and machine learning algorithms are increasingly deployed to support consequential decision-making processes, from deciding which applicants will receive job offers and loans to university enrollments and medical interventions. However, recent studies show these models may unintentionally amplify human bias and yield significant unfavorable decisions to specific groups. Methodology/results: We propose a distributionally robust classification model with a fairness constraint that encourages the classifier to be fair in the equality of opportunity criterion. We use a type-[Formula: see text] Wasserstein ambiguity set centered at the empirical distribution to represent distributional uncertainty and derive a conservative reformulation for the worst-case equal opportunity unfairness measure. We show that the model is equivalent to a mixed binary conic optimization problem, which standard off-the-shelf solvers can solve. We propose a convex, hinge-loss-based model for large problem instances whose reformulation does not incur binary variables to improve scalability. Moreover, we also consider the distributionally robust learning problem with a generic ground transportation cost to hedge against the label and sensitive attribute uncertainties. We numerically examine the performance of our proposed models on five real-world data sets related to individual analysis. Compared with the state-of-the-art methods, our proposed approaches significantly improve fairness with negligible loss of predictive accuracy in the testing data set. Managerial implications: Our paper raises awareness that bias may arise when predictive models are used in service and operations. It generally comes from human bias, for example, imbalanced data collection or low sample sizes, and is further amplified by algorithms. Incorporating fairness constraints and the distributionally robust optimization (DRO) scheme is a powerful way to alleviate algorithmic biases. Funding: This work was supported by the National Science Foundation [Grants 2342505 and 2343869] and the Chinese University of Hong Kong [Grant 4055191]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2022.0230 . 

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5. UNDERSTANDING WHY GENERALIZED REWEIGHTING DOES NOT IMPROVE OVER ERM

   Zhai, Runtian; Dan, Chen; Kolter, Zico; Ravikumar, Pradeep (May 2023, International Conference on Learning Representations)

   Empirical risk minimization (ERM) is known to be non-robust in practice to distributional shift where the training and the test distributions are different. A suite of approaches, such as importance weighting, and variants of distributionally robust optimization (DRO), have been proposed to solve this problem. But a line of recent work has empirically shown that these approaches do not significantly improve over ERM in real applications with distribution shift. The goal of this work is to obtain a comprehensive theoretical understanding of this intriguing phenomenon. We first posit the class of Generalized Reweighting (GRW) algorithms, as a broad category of approaches that iteratively update model parameters based on iterative reweighting of the training samples. We show that when overparameterized models are trained under GRW, the resulting models are close to that obtained by ERM. We also show that adding small regularization which does not greatly affect the empirical training accuracy does not help. Together, our results show that a broad category of what we term GRW approaches are not able to achieve distributionally robust generalization. Our work thus has the following sobering takeaway: to make progress towards distributionally robust generalization, we either have to develop non-GRW approaches, or perhaps devise novel classification/regression loss functions that are adapted to GRW approaches. 

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