This work studies the distributionally robust evaluation of expected values over temporal data. A set of alternative measures is characterized by the causal optimal transport. We prove the strong duality and recast the causality constraint as minimization over an infinite-dimensional test function space. We approximate test functions by neural networks and prove the sample complexity with Rademacher complexity. An example is given to validate the feasibility of technical assumptions. Moreover, when structural information is available to further restrict the ambiguity set, we prove the dual formulation and provide efficient optimization methods. Our framework outperforms the classic counterparts in the distributionally robust portfolio selection problem. The connection with the naive strategy is also investigated numerically.

翻译：本研究探讨了时间数据期望值的分布鲁棒评估问题。我们采用因果最优传输来刻画替代测度集合的特征。证明了强对偶性，并将因果约束重构为无限维检验函数空间上的最小化问题。通过神经网络逼近检验函数，并利用Rademacher复杂度证明了样本复杂性。给出示例验证了技术假设的可行性。此外，当可获得结构信息以进一步限制模糊集时，我们证明了其对偶形式并提供了高效优化方法。在分布鲁棒投资组合选择问题中，本框架性能优于经典方法。通过数值实验进一步探究了与朴素策略的关联性。