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.

翻译：2013年，Cuturi [Cut13] 引入Sinkhorn算法用于矩阵缩放，作为求解正则化最优传输问题的一种方法。本文旨在提升高精度解的收敛速度，通过分析正则调度下的Sinkhorn算法，对其加以修改，引入自适应周期倍增正则化参数η的机制。我们证明，在供需为整数的最优传输问题中，该改进版Sinkhorn算法具有指数收敛速率，其迭代复杂度仅依赖于log(1/ε)，而非以往分析[Cut13][ANWR17]中的ε^{-O(1)}。此外，结合成本与容量缩放过程，一般最优传输问题的求解也可实现关于1/ε的对数依赖关系。