A Sinkhorn-type Algorithm for Constrained Optimal Transport
Xun Tang Holakou Rahmanian Michael Shavlovsky Kiran Koshy Thekumparampil Tesi Xiao Lexing Ying
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.