Exponential Convergence of Sinkhorn Under Regularization Scheduling

TL;DR

This paper demonstrates that an adaptively scheduled regularization in the Sinkhorn algorithm achieves exponential convergence, significantly improving the iteration complexity for solving optimal transport problems.

Contribution

It introduces a modified Sinkhorn algorithm with regularization scheduling that guarantees exponential convergence rate in high-accuracy solutions.

Findings

Convergence rate depends on log(1/ε) instead of polynomial in 1/ε.

Adaptive doubling of regularization parameter improves convergence.

Applicable to general optimal transport problems with cost and capacity scaling.

Abstract

In 2013, Cuturi [Cut13] introduced the Sinkhorn algorithm for matrix scaling as a method to compute solutions to regularized optimal transport problems. In this paper, aiming at a better convergence rate for a high accuracy solution, we work on understanding the Sinkhorn algorithm under regularization scheduling, and thus modify it with a mechanism that adaptively doubles the regularization parameter $\eta$ periodically. We prove that such modified version of Sinkhorn has an exponential convergence rate as iteration complexity depending on $\log(1/\varepsilon)$ instead of $\varepsilon^{-O(1)}$ from previous analyses [Cut13][ANWR17] in the optimal transport problems with integral supply and demand. Furthermore, with cost and capacity scaling procedures, the general optimal transport problem can be solved with a logarithmic dependence on $1/\varepsilon$ as well.