Exponential Convergence of Sinkhorn Under Regularization Scheduling

The optimal transport (OT) problem or earth mover's distance has wide applications to machine learning and statistical problems. 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, we introduce the exponentially converging Sinkhorn algorithm ExpSinkhorn, by modifying the Sinkhorn algorithm to adaptively double the regularization parameter $\eta$ periodically. The resulting method has the benefits of automatically finding the regularization parameter $\eta$ for a desired accuracy $\varepsilon$, and has iteration complexity depending on $\log(1/\varepsilon)$, as opposed to $\varepsilon^{-O(1)}$ as in previous analyses [Cut13] [ANWR17]. We also show empirically that our algorithm is efficient and robust.