Improved Rate of First Order Algorithms for Entropic Optimal Transport
Yiling Luo, Yiling Xie, Xiaoming Huo
Abstract
This paper improves the state-of-the-art rate of a first-order algorithm for solving entropy regularized optimal transport. The resulting rate for approximating the optimal transport (OT) has been improved from $\widetilde{\mathcal{O}}({n^{2.5}}/{\epsilon})$ to $\widetilde{\mathcal{O}}({n^2}/{\epsilon})$, where $n$ is the problem size and $\epsilon$ is the accuracy level. In particular, we propose an accelerated primal-dual stochastic mirror descent algorithm with variance reduction. Such special design helps us improve the rate compared to other accelerated primal-dual algorithms. We further propose a batch version of our stochastic algorithm, which improves the computational performance through parallel computing. To compare, we prove that the computational complexity of the Stochastic Sinkhorn algorithm is $\widetilde{\mathcal{O}}({n^2}/{\epsilon^2})$, which is slower than our accelerated primal-dual stochastic mirror algorithm. Experiments are done using synthetic and real data, and the results match our theoretical rates. Our algorithm may inspire more research to develop accelerated primal-dual algorithms that have rate $\widetilde{\mathcal{O}}({n^2}/{\epsilon})$ for solving OT.
BibTeX
@InProceedings{pmlr-v206-luo23a,
title = {Improved Rate of First Order Algorithms for Entropic Optimal Transport},
author = {Luo, Yiling and Xie, Yiling and Huo, Xiaoming},
booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics},
pages = {2723--2750},
year = {2023},
editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem},
volume = {206},
series = {Proceedings of Machine Learning Research},
month = {25--27 Apr},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v206/luo23a/luo23a.pdf},
url = {https://proceedings.mlr.press/v206/luo23a.html},
abstract = {This paper improves the state-of-the-art rate of a first-order algorithm for solving entropy regularized optimal transport. The resulting rate for approximating the optimal transport (OT) has been improved from $\widetilde{\mathcal{O}}({n^{2.5}}/{\epsilon})$ to $\widetilde{\mathcal{O}}({n^2}/{\epsilon})$, where $n$ is the problem size and $\epsilon$ is the accuracy level. In particular, we propose an accelerated primal-dual stochastic mirror descent algorithm with variance reduction. Such special design helps us improve the rate compared to other accelerated primal-dual algorithms. We further propose a batch version of our stochastic algorithm, which improves the computational performance through parallel computing. To compare, we prove that the computational complexity of the Stochastic Sinkhorn algorithm is $\widetilde{\mathcal{O}}({n^2}/{\epsilon^2})$, which is slower than our accelerated primal-dual stochastic mirror algorithm. Experiments are done using synthetic and real data, and the results match our theoretical rates. Our algorithm may inspire more research to develop accelerated primal-dual algorithms that have rate $\widetilde{\mathcal{O}}({n^2}/{\epsilon})$ for solving OT.}
}