Loss-Aware Distributionally Robust Optimization via Trainable Optimal Transport Ambiguity Sets

Optimal-transport distributionally robust optimization (OT-DRO) robustifies data-driven decision-making under uncertainty by capturing the sampling-induced statistical error via optimal transport ambiguity sets. The standard OT-DRO pipeline consists of a two-step procedure, where the ambiguity set is first designed and subsequently embedded into the downstream OT-DRO problem. However, this separation between uncertainty quantification and optimization may lead to excessive conservatism. We introduce an end-to-end pipeline to automatically learn decision-focused ambiguity sets for OT-DRO problems, where the loss function informs the shape of the ambiguity set, leading to less conservative decisions whose distributional robustness is enforced via data-driven bootstrapping. We formulate the learning problem as a bilevel optimization program and solve it via a hypergradient-based method. By leveraging the recently introduced nonsmooth conservative implicit function theorem, we establish convergence to a critical point of the bilevel problem. We present experiments validating our method on standard portfolio optimization and linear regression tasks.

When data is used to guide decisions by an algorithm, it may provide an incomplete picture because it is only a limited sample of what could happen. A common way to handle this is to choose an action that performs well not only for the observed data, but also for other plausible versions of the data. This can protect against unpleasant surprises. However, existing methods often define these plausible alternatives in a one-size-fits-all way, before considering what decision is actually being made. As a result, they can be overly cautious, much like buying every kind of insurance, even when only a few risks are relevant. We introduce a method that learns which changes in the data matter most for the decision at hand. For example, in portfolio selection, it focuses on changes that affect financial risk, while in prediction tasks, it focuses on changes that could reduce prediction accuracy. The method uses repeated resampling of the available data to help calibrate these possibilities and keep them tied to the observed data. The goal is to preserve the main benefit of robust decision-making, that is, protection against surprises, while avoiding unnecessary caution. In experiments on portfolio selection and prediction tasks, our approach reduced conservatism and performed better on new data than standard robust approaches. We also provide an efficient algorithm and open-source code so others can apply and build on the method.