  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.
