We provide theoretical analyses for two algorithms that solve the regularized optimal transport (OT) problem between two discrete probability measures with at most $n$ atoms. We show that a greedy variant of the classical Sinkhorn algorithm, known as the \emph{Greenkhorn algorithm}, can be improved to $\widetilde{\mathcal{O}}(n^2\varepsilon^{-2})$, improving on the best known complexity bound of $\widetilde{\mathcal{O}}(n^2\varepsilon^{-3})$. Notably, this matches the best known complexity bound for the Sinkhorn algorithm and helps explain why the Greenkhorn algorithm can outperform the Sinkhorn algorithm in practice. Our proof technique, which is based on a primal-dual formulation and a novel upper bound for the dual solution, also leads to a new class of algorithms that we refer to as \emph{adaptive primal-dual accelerated mirror descent} (APDAMD) algorithms. We prove that the complexity of these algorithms is $\widetilde{\mathcal{O}}(n^2\sqrt{\delta}\varepsilon^{-1})$, where $\delta > 0$ refers to the inverse of the strong convexity module of Bregman divergence with respect to $\|\cdot\|_\infty$. This implies that the APDAMD algorithm is faster than the Sinkhorn and Greenkhorn algorithms in terms of $\varepsilon$. Experimental results on synthetic and real datasets demonstrate the favorable performance of the Greenkhorn and APDAMD algorithms in practice.
翻译:我们为两种算法提供理论分析,这些算法可以解决两种离散的概率计量方法之间的正常最佳运输(OT)问题(OT ) 。 值得注意的是, 这符合Sinkhorn算法中已知的最佳复杂程度, 有助于解释为何古典Sinkhorn算法(称为\ emph{ Greenkhorn算法} ) 的贪婪变方, 可以改进成$宽的公式{O} (n2\ varepsilón}-2} 美元, 改进了已知最复杂程度, 约束于$宽度/ smart{ O} (n2\\ vareal- directrial_ lax) 。 我们证明这些算法的复杂性是 $link_ sink_ liver2 dalxx 的硬性能和 $leval_ deal_ daldal_ daldal_ dismodal_ dal_ discoal_Bral_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ dad_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ dal_ lax_ lax_ lad_ dal_ dal_ dal_ dal_ dal_ dal_d_ dal_ lad_ lad_ lad_ lad_ lad_ lad_ lad_ lad_ lad_ lad_ lad_d_d_d_d_d_d_d_d_d_d_d_d_d_d_ ladal_d_dal_ lad_ ladal_ lad_ lad_d___ lad_ lax_ lad) lad_ lad_ lad_ lad_ lad_ lad_ lad) lad_ lad_ ladal_ lad_ lad_ lax_ lad_