  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.
