Parameter tuning and model selection in Optimal Transport with semi-dual Brenier formulation

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 and where the goal is to find the closest map to the ground truth OT map, unknown in practical settings. So far, no quantitative criterion has yet been put forward to tune the parameters of these models and select maps that best approximate the ground truth. To perform this task, we propose to leverage the Brenier formulation of OT. Theoretically, we show that this formulation guarantees that, up to sharp a distortion parameter depending 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 tuning and model selection among entropic regularization of OT, input convex neural networks and smooth and strongly convex nearest-Brenier (SSNB) models. We also use this criterion to question the use of OT in Domain-Adaptation (DA). In a standard DA experiment, it enables us to identify the potential that is closest to the true OT map between the source and the target. Yet, we observe that this selected potential is far from being the one that performs best for the downstream transfer classification task.