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1. Wasserstein Distributionally Robust Inverse Multiobjective Optimization

   Dong, Chaosheng ; Zeng, Bo ( February 2021 , the AAAI Conference on Artificial Intelligence)

   null (Ed.)

   Inverse multiobjective optimization provides a general framework for the unsupervised learning task of inferring parameters of a multiobjective decision making problem (DMP), based on a set of observed decisions from the human expert. However, the performance of this framework relies critically on the availability of an accurate DMP, sufficient decisions of high quality, and a parameter space that contains enough information about the DMP. To hedge against the uncertainties in the hypothetical DMP, the data, and the parameter space, we investigate in this paper the distributionally robust approach for inverse multiobjective optimization. Specifically, we leverage the Wasserstein metric to construct a ball centered at the empirical distribution of these decisions. We then formulate a Wasserstein distributionally robust inverse multiobjective optimization problem (WRO-IMOP) that minimizes a worst-case expected loss function, where the worst case is taken over all distributions in the Wasserstein ball. We show that the excess risk of the WRO-IMOP estimator has a sub-linear convergence rate. Furthermore, we propose the semi-infinite reformulations of the WRO-IMOP and develop a cutting-plane algorithm that converges to an approximate solution in finite iterations. Finally, we demonstrate the effectiveness of our method on both a synthetic multiobjective quadratic program and a real world portfolio optimization problem. 

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2. Confidence regions in Wasserstein distributionally robust estimation

   https://doi.org/10.1093/biomet/asab026  

   Blanchet, Jose ; Murthy, Karthyek ; Si, Nian ( April 2021 , Biometrika)

   Summary Estimators based on Wasserstein distributionally robust optimization are obtained as solutions of min-max problems in which the statistician selects a parameter minimizing the worst-case loss among all probability models within a certain distance from the underlying empirical measure in a Wasserstein sense. While motivated by the need to identify optimal model parameters or decision choices that are robust to model misspecification, these distributionally robust estimators recover a wide range of regularized estimators, including square-root lasso and support vector machines, among others. This paper studies the asymptotic normality of these distributionally robust estimators as well as the properties of an optimal confidence region induced by the Wasserstein distributionally robust optimization formulation. In addition, key properties of min-max distributionally robust optimization problems are also studied; for example, we show that distributionally robust estimators regularize the loss based on its derivative, and we also derive general sufficient conditions which show the equivalence between the min-max distributionally robust optimization problem and the corresponding max-min formulation. 

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3. Distributionally Robust Decision Making Leveraging Conditional Distributions

   Chen, Yuxiao ; Kim, Jip ; Anderson, James ( December 2022 , Proceedings of the IEEE 61st Conference on Decision and Control (CDC))

   Distributionally robust optimization (DRO) is a powerful tool for decision making under uncertainty. It is particularly appealing because of its ability to leverage existing data. However, many practical problems call for decision- making with some auxiliary information, and DRO in the context of conditional distributions is not straightforward. We propose a conditional kernel distributionally robust optimiza- tion (CKDRO) method that enables robust decision making under conditional distributions through kernel DRO and the conditional mean operator in the reproducing kernel Hilbert space (RKHS). In particular, we consider problems where there is a correlation between the unknown variable y and an auxiliary observable variable x. Given past data of the two variables and a queried auxiliary variable, CKDRO represents the conditional distribution P(y|x) as the conditional mean operator in the RKHS space and quantifies the ambiguity set in the RKHS as well, which depends on the size of the dataset as well as the query point. To justify the use of RKHS, we demonstrate that the ambiguity set defined in RKHS can be viewed as a ball under a metric that is similar to the Wasserstein metric. The DRO is then dualized and solved via a finite dimensional convex program. The proposed CKDRO approach is applied to a generation scheduling problem and shows that the result of CKDRO is superior to common benchmarks in terms of quality and robustness. 

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4. Learning while Respecting Privacy and Robustness to Adversarial Distributed Datasets

   Alireza Sadeghi and Georgios B. Giannakis ( August 2022 , EUSIPCO)

   Massive datasets are typically distributed geographically across multiple sites, where scalability, data privacy and integrity, as well as bandwidth scarcity typically discourage uploading these data to a central server. This has propelled the so-called federated learning framework where multiple workers exchange information with a server to learn a “centralized” model using data locally generated and/or stored across workers. This learning framework necessitates workers to communicate iteratively with the server. Although appealing for its scalability, one needs to carefully address the various data distribution shifts across workers, which degrades the performance of the learnt model. In this context, the distributionally robust optimization framework is considered here. The objective is to endow the trained model with robustness against adversarially manipulated input data, or, distributional uncertainties, such as mismatches between training and testing data distributions, or among datasets stored at different workers. To this aim, the data distribution is assumed unknown, and to land within a Wasserstein ball centered around the empirical data distribution. This robust learning task entails an infinite-dimensional optimization problem, which is challenging. Leveraging a strong duality result, a surrogate is obtained, for which a primal-dual algorithm is developed. Compared to classical methods, the proposed algorithm offers robustness with little computational overhead. Numerical tests using image datasets showcase the merits of the proposed algorithm under several existing adversarial attacks and distributional uncertainties. 

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5. Learning while Respecting Privacy and Robustness to Adversarial Distributed Datasets

   A. Sadeghi ; G. B. Giannakis ( October 2022 , Proceedings of European Signal Processing Conference)

   Massive datasets are typically distributed geographically across multiple sites, where scalability, data privacy and integrity, as well as bandwidth scarcity typically discourage uploading these data to a central server. This has propelled the so-called federated learning framework where multiple workers exchange information with a server to learn a “centralized” model using data locally generated and/or stored across workers. This learning framework necessitates workers to communicate iteratively with the server. Although appealing for its scalability one needs to carefully address the various data distribution shifts across workers, which degrades the performance of the learnt model. In this context, the distributionally robust optimization framework is considered here. The objective is to endow the trained model with robustness against adversarially manipulated input data, or, distributional uncertainties, such as mismatches between training and testing data distributions, or among datasets stored at different workers. To this aim, the data distribution is assumed unknown, and to land within a Wasserstein ball centered around the empirical data distribution. This robust learning task entails an infinite-dimensional optimization problem, which is challenging. Leveraging a strong duality result, a surrogate is obtained, for which a primal-dual algorithm is developed. Compared to classical methods, the proposed algorithm offers robustness with little computational overhead. Numerical tests using image datasets showcase the merits of the proposed algorithm under several existing adversarial attacks and distributional uncertainties. 

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