The optimal transport (OT) problem or earth mover's distance has wide applications to machine learning and statistical problems. 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, we introduce the exponentially converging Sinkhorn algorithm ExpSinkhorn, by modifying the Sinkhorn algorithm to adaptively double the regularization parameter $\eta$ periodically. The resulting method has the benefits of automatically finding the regularization parameter $\eta$ for a desired accuracy $\varepsilon$, and has iteration complexity depending on $\log(1/\varepsilon)$, as opposed to $\varepsilon^{-O(1)}$ as in previous analyses [Cut13] [ANWR17]. We also show empirically that our algorithm is efficient and robust.

翻译：最佳运输(OT)问题或地球移动器距离对于机器学习和统计问题有着广泛的应用。 2013年,Cuturi [Cut13] 引入了矩阵缩放Sinkhorn算法,作为计算正规化最佳运输问题解决方案的一种方法。在本文中,我们引入了Sinkhorn算法ExmSinkhorn指数,将Sinkhorn算法修改为适应性地使正规化参数定期翻一倍$\eta$(美元)。由此得出的方法的好处是自动找到正规化参数$\eta$(美元)以达到理想的精确度 $\varepsilon$,并且根据$\log(1/\varepsilon) 来进行循环复杂度,而不是像以前的分析[Cut13] [ANWR17] 那样,我们从经验上表明我们的算法是高效和稳健的。