Forward-Backward Splitting for Optimal Transport based Problems

TL;DR

This paper introduces a forward-backward splitting algorithm using Bregman distances to efficiently solve optimal transport problems with additional constraints, demonstrating significant improvements in domain adaptation tasks.

Contribution

It presents a novel optimization algorithm tailored for extended optimal transport problems, enhancing computational efficiency and applicability in machine learning.

Findings

Significant speed-up over existing methods.

Improved performance in domain adaptation tasks.

Effective handling of complex constraints in optimal transport.

Abstract

Optimal transport aims to estimate a transportation plan that minimizes a displacement cost. This is realized by optimizing the scalar product between the sought plan and the given cost, over the space of doubly stochastic matrices. When the entropy regularization is added to the problem, the transportation plan can be efficiently computed with the Sinkhorn algorithm. Thanks to this breakthrough, optimal transport has been progressively extended to machine learning and statistical inference by introducing additional application-specific terms in the problem formulation. It is however challenging to design efficient optimization algorithms for optimal transport based extensions. To overcome this limitation, we devise a general forward-backward splitting algorithm based on Bregman distances for solving a wide range of optimization problems involving a differentiable function with Lipschitz-continuous gradient and a doubly stochastic constraint. We illustrate the efficiency of our approach in the context of continuous domain adaptation. Experiments show that the proposed method leads to a significant improvement in terms of speed and performance with respect to the state of the art for domain adaptation on a continually rotating distribution coming from the standard two moon dataset.