Convex transport potential selection with semi-dual criterion

Over the past few years, numerous computational models have been developed to solve Optimal Transport (OT) in a stochastic setting, where distributions are represented by samples. In such situations, the goal is to find a transport map that has good generalization properties on unseen data, ideally the closest map to the ground truth, unknown in practical settings. However, in the absence of ground truth, no quantitative criterion has been put forward to measure its generalization performance although it is crucial for model selection. We propose to leverage the Brenier formulation of OT to perform this task. Theoretically, we show that this formulation guarantees that, up to a distortion parameter that depends on the smoothness/strong convexity and a statistical deviation term, the selected map achieves the lowest quadratic error to the ground truth. This criterion, estimated via convex optimization, enables parameter and model selection among entropic regularization of OT, input convex neural networks and smooth and strongly convex nearest-Brenier (SSNB) models. Last, we make an experiment questioning the use of OT in Domain-Adaptation. Thanks to the criterion, we can identify the potential that is closest to the true OT map between the source and the target and we observe that this selected potential is not the one that performs best for the downstream transfer classification task.