Coresets for Wasserstein Distributionally Robust Optimization Problems

Wasserstein distributionally robust optimization (textsfWDRO) is a popular model to enhance the robustness of machine learning with ambiguous data. However, the complexity of textsfWDRO can be prohibitive in practice since solving its ``minimax'' formulation requires a great amount of computation. Recently, several fast textsfWDRO training algorithms for some specific machine learning tasks (e.g., logistic regression) have been developed. However, the research on designing efficient algorithms for general large-scale textsfWDROs is still quite limited, to the best of our knowledge. textitCoreset is an important tool for compressing large dataset, and thus it has been widely applied to reduce the computational complexities for many optimization problems. In this paper, we introduce a unified framework to construct the ε-coreset for the general textsfWDRO problems. Though it is challenging to obtain a conventional coreset for textsfWDRO due to the uncertainty issue of ambiguous data, we show that we can compute a ``dual coreset'' by using the strong duality property of textsfWDRO. Also, the error introduced by the dual coreset can be theoretically guaranteed for the original textsfWDRO objective. To construct the dual coreset, we propose a novel grid sampling approach that is particularly suitable for the dual formulation of textsfWDRO. Finally, we implement our coreset approach and illustrate its effectiveness for several textsfWDRO problems in the experiments.