Exponential Convergence of Sinkhorn Under Regularization Scheduling

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 η 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.