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

TL;DR

This paper introduces a new criterion based on the Brenier formulation of optimal transport to tune parameters and select models that best approximate the ground truth OT map, improving model selection in stochastic OT settings.

Contribution

It proposes a convex optimization-based criterion for parameter tuning and model selection in stochastic OT, validated across various models and questioning OT's role in domain adaptation.

Findings

The criterion guarantees the lowest quadratic error to the ground truth under certain conditions.

It effectively distinguishes the closest potential to the true OT map in domain adaptation.

Selected potentials may not always yield the best downstream classification performance.

Abstract

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.