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# Improved Rate of First Order Algorithms for Entropic Optimal Transport

*Proceedings of The 26th International Conference on Artificial Intelligence and Statistics*, PMLR 206:2723-2750, 2023.

#### 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.