Improved Rate of First Order Algorithms for Entropic Optimal Transport

• URL: http://arxiv.org/abs/2301.09675v1
• Date: Mon, 23 Jan 2023 19:13:25 GMT
• Title: Improved Rate of First Order Algorithms for Entropic Optimal Transport
• Authors: Yiling Luo, Yiling Xie, Xiaoming Huo
• Abstract summary: This paper improves the state-of-the-art rate of a first-order algorithm for solving entropy regularized optimal transport. We propose an accelerated primal-dual mirror descent algorithm with variance reduction. Our algorithm may inspire more research to develop accelerated primal-dual algorithms that have rate $widetildeO(n2/epsilon)$ for solving OT.
• Score: 2.1485350418225244
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\widetilde{{O}}({n^{2.5}}/{\epsilon})$ to $\widetilde{{O}}({n^2}/{\epsilon})$, 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 $\widetilde{{O}}({n^2}/{\epsilon^2})$, 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 $\widetilde{{O}}({n^2}/{\epsilon})$ for solving OT.

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