Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm

TL;DR

This paper compares the computational complexity of Sinkhorn's algorithm and a new adaptive primal-dual accelerated gradient descent method for approximating optimal transport distances, showing the new method's superior efficiency and flexibility.

Contribution

Introduces a novel Adaptive Primal-Dual Accelerated Gradient Descent algorithm with improved complexity bounds for optimal transport approximation, applicable to various regularizers.

Findings

APDAGD has complexity $ ilde{O}( ext{min}igrace n^{9/4}/ ext{ε}, n^{2}/ ext{ε}^2 ig)$

Both algorithms outperform the previous $ ilde{O}(n^2/ ext{ε}^3)$ bound

APDAGD is versatile for different regularizers beyond entropic regularization

Abstract

We analyze two algorithms for approximating the general optimal transport (OT) distance between two discrete distributions of size $n$, up to accuracy $\varepsilon$. For the first algorithm, which is based on the celebrated Sinkhorn's algorithm, we prove the complexity bound $\widetilde{O}\left({n^2/\varepsilon^2}\right)$ arithmetic operations. For the second one, which is based on our novel Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD) algorithm, we prove the complexity bound $\widetilde{O}\left(\min\left\{n^{9/4}/\varepsilon, n^{2}/\varepsilon^2 \right\}\right)$ arithmetic operations. Both bounds have better dependence on $\varepsilon$ than the state-of-the-art result given by $\widetilde{O}\left({n^2/\varepsilon^3}\right)$. Our second algorithm not only has better dependence on $\varepsilon$ in the complexity bound, but also is not specific to entropic regularization and can solve the OT problem with different regularizers.