Fast Computation of Optimal Transport via Entropy-Regularized Extragradient Methods

TL;DR

This paper introduces a scalable first-order method for computing optimal transport distances efficiently, achieving state-of-the-art guarantees and improved numerical performance over classical algorithms.

Contribution

It develops a novel entropy-regularized extragradient approach that converts the problem into a bilinear minimax form and accelerates convergence with adaptive learning rates.

Findings

Achieves $ ilde{O}(n^2/\varepsilon)$ runtime for $\\varepsilon$-approximate solutions.

Outperforms Sinkhorn and Greenkhorn algorithms in numerical experiments.

Provides the first first-order method with optimal theoretical guarantees for this problem.

Abstract

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.