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Over the past few years, numerous computational models have been developed tosolve Optimal Transport (OT) in a stochastic setting, where distributions arerepresented by samples and where the goal is to find the closest map to theground truth OT map, unknown in practical settings. So far, no quantitativecriterion has yet been put forward to tune the parameters of these models andselect maps that best approximate the ground truth. To perform this task, wepropose to leverage the Brenier formulation of OT.Theoretically, we show thatthis formulation guarantees that, up to sharp a distortion parameter dependingon the smoothness/strong convexity and a statistical deviation term, theselected map achieves the lowest quadratic error to the ground truth. Thiscriterion, estimated via convex optimization, enables parameter tuning andmodel selection among entropic regularization of OT, input convex neuralnetworks and smooth and strongly convex nearest-Brenier (SSNB) models.We alsouse this criterion to question the use of OT in Domain-Adaptation (DA). In astandard DA experiment, it enables us to identify the potential that is closestto the true OT map between the source and the target. Yet, we observe that thisselected potential is far from being the one that performs best for thedownstream transfer classification task.

Parameter tuning and model selection in optimal transport with semi-dual Brenier formulation