Efficient computation of the optimal transport distance between two distributions serves as an algorithm subroutine that empowers various applications. This paper develops a scalable first-order optimization-based method that computes optimal transport to within $\varepsilon$ additive accuracy with runtime $\widetilde{O}( n^2/\varepsilon)$, where $n$ denotes the dimension of the probability distributions of interest. Our algorithm achieves the state-of-the-art computational guarantees among all first-order methods, while exhibiting favorable numerical performance compared to classical algorithms like Sinkhorn and Greenkhorn. Underlying our algorithm designs are two key elements: (a) converting the original problem into a bilinear minimax problem over probability distributions; (b) exploiting the extragradient idea -- in conjunction with entropy regularization and adaptive learning rates -- to accelerate convergence.

翻译：高效计算两个分布之间的最优传输距离作为算法子程序，赋能了众多应用。本文提出了一种可扩展的基于一阶优化的方法，能够在$\widetilde{O}( n^2/\varepsilon)$的运行时间内，以$\varepsilon$的加性精度计算最优传输，其中$n$表示所关注概率分布的维度。我们的算法在所有一阶方法中达到了最先进的计算保证，同时在与Sinkhorn和Greenkhorn等经典算法的数值对比中展现出优越性能。算法设计的关键要素包括：(a) 将原始问题转化为关于概率分布的双线性极小极大问题；(b) 利用外梯度思想——结合熵正则化与自适应学习率——以加速收敛。