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arXiv · 2207.00736

Exponential Convergence of Sinkhorn Under Regularization Scheduling

Jingbang Chen·Li Chen·Yang P. Liu·Richard Peng·Arvind Ramaswami
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Abstract

In 2013, Cuturi [Cut13] introduced the Sinkhorn algorithm for matrix scaling as a method to compute solutions to regularized optimal transport problems. In this paper, aiming at a better convergence rate for a high accuracy solution, we work on understanding the Sinkhorn algorithm under regularization scheduling, and thus modify it with a mechanism that adaptively doubles the regularization parameter $\eta$ periodically. We prove that such modified version of Sinkhorn has an exponential convergence rate as iteration complexity depending on $\log(1/\varepsilon)$ instead of $\varepsilon^{-O(1)}$ from previous analyses [Cut13][ANWR17] in the optimal transport problems with integral supply and demand. Furthermore, with cost and capacity scaling procedures, the general optimal transport problem can be solved with a logarithmic dependence on $1/\varepsilon$ as well.

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Jingbang Chen, Li Chen, Yang P. Liu, Richard Peng, Arvind Ramaswami. 2022-07-02. Exponential Convergence of Sinkhorn Under Regularization Scheduling. https://arxiv.org/abs/2207.00736

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