On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithms

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 $\bigOtil\left(n^2/\varepsilon^2\right)$, improving on the best known complexity bound of $\bigOtil\left(n^2/\varepsilon^3\right)$. This matches the best known complexity bound for the Sinkhorn algorithm and helps explain why the Greenkhorn algorithm outperforms the Sinkhorn algorithm in practice. Our proof technique is based on a primal-dual formulation and provide a \textit{tight} upper bound for the dual solution, leading to a class of \emph{adaptive primal-dual accelerated mirror descent} (APDAMD) algorithms. We prove that the complexity of these algorithms is $\bigOtil\left(n^2\sqrt{\gamma}/\varepsilon\right)$ in which $\gamma \in (0, n]$ refers to some constants in the Bregman divergence. Experimental results on synthetic and real datasets demonstrate the favorable performance of the Greenkhorn and APDAMD algorithms in practice.