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输运距离，是赋能多种应用的算法子程序。本文提出一种可扩展的一阶优化方法，可在$\widetilde{O}( n^2/\varepsilon)$的运行时间内，以$\varepsilon$加性精度计算最优输运距离，其中$n$表示目标概率分布的维度。我们的算法在所有一阶方法中实现了最先进的计算保证，同时与Sinkhorn、Greenkhorn等经典算法相比展现出更优的数值性能。我们算法设计的两个核心要素是：(a) 将原问题转化为概率分布上的双线性极小极大问题；(b) 结合熵正则化与自适应学习率，利用外梯度思想加速收敛。