A Sinkhorn-type Algorithm for Constrained Optimal Transport

TL;DR

This paper introduces a Sinkhorn-type algorithm for solving constrained optimal transport problems with theoretical guarantees, including error bounds and convergence rates, enabling practical computation of transport plans under complex constraints.

Contribution

The work extends entropic optimal transport and Sinkhorn algorithms to handle combined equality and inequality constraints with proven convergence and error bounds.

Findings

Approximation error reduces exponentially with regularization parameter.

The proposed algorithm achieves sublinear convergence in the dual space.

Dynamic regularization and second-order acceleration improve convergence speed.

Abstract

Entropic optimal transport (OT) and the Sinkhorn algorithm have made it practical for machine learning practitioners to perform the fundamental task of calculating transport distance between statistical distributions. In this work, we focus on a general class of OT problems under a combination of equality and inequality constraints. We derive the corresponding entropy regularization formulation and introduce a Sinkhorn-type algorithm for such constrained OT problems supported by theoretical guarantees. We first bound the approximation error when solving the problem through entropic regularization, which reduces exponentially with the increase of the regularization parameter. Furthermore, we prove a sublinear first-order convergence rate of the proposed Sinkhorn-type algorithm in the dual space by characterizing the optimization procedure with a Lyapunov function. To achieve fast and higher-order convergence under weak entropy regularization, we augment the Sinkhorn-type algorithm with dynamic regularization scheduling and second-order acceleration. Overall, this work systematically combines recent theoretical and numerical advances in entropic optimal transport with the constrained case, allowing practitioners to derive approximate transport plans in complex scenarios.