# On the Efficiency of Entropic Regularized Algorithms for Optimal Transport

Tianyi Lin, Nhat Ho, Michael I. Jordan.

Year: 2022, Volume: 23, Issue: 137, Pages: 1−42

#### Abstract

We present several new complexity results for the entropic regularized algorithms that approximately solve the optimal transport (OT) problem between two discrete probability measures with at most $n$ atoms. First, we improve the complexity bound of a greedy variant of Sinkhorn, known as Greenkhorn, from $\tilde{O}(n^2\varepsilon^{-3})$ to $\tilde{O}(n^2\varepsilon^{-2})$. Notably, our result can match the best known complexity bound of Sinkhorn and help clarify why Greenkhorn significantly outperforms Sinkhorn in practice in terms of row/column updates as observed by Altschuler et al. (2017). Second, we propose a new algorithm, which we refer to as APDAMD and which generalizes an adaptive primal-dual accelerated gradient descent (APDAGD) algorithm (Dvurechensky et al., 2018) with a prespecified mirror mapping $\phi$. We prove that APDAMD achieves the complexity bound of $\tilde{O}(n^2\sqrt{\delta}\varepsilon^{-1})$ in which $\delta>0$ stands for the regularity of $\phi$. In addition, we show by a counterexample that the complexity bound of $\tilde{O}(\min\{n^{9/4}\varepsilon^{-1}, n^2\varepsilon^{-2}\})$ proved for APDAGD before is invalid and give a refined complexity bound of $\tilde{O}(n^{5/2}\varepsilon^{-1})$. Further, we develop a deterministic accelerated variant of Sinkhorn via appeal to estimated sequence and prove the complexity bound of $\tilde{O}(n^{7/3}\varepsilon^{-4/3})$. As such, we see that accelerated variant of Sinkhorn outperforms Sinkhorn and Greenkhorn in terms of $1/\varepsilon$ and APDAGD and accelerated alternating minimization (AAM) (Guminov et al., 2021) in terms of $n$. Finally, we conduct the experiments on synthetic and real data and the numerical results show the efficiency of Greenkhorn, APDAMD and accelerated Sinkhorn in practice.