Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm

We analyze two algorithms for approximating the general optimal transport (OT) distance between two discrete distributions of size $n$, up to accuracy $\varepsilon$. For the first algorithm, which is based on the celebrated Sinkhorn's algorithm, we prove the complexity bound $\widetilde{O}\left({n^2/\varepsilon^2}\right)$ arithmetic operations. For the second one, which is based on our novel Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD) algorithm, we prove the complexity bound $\widetilde{O}\left(\min\left\{n^{9/4}/\varepsilon, n^{2}/\varepsilon^2 \right\}\right)$ arithmetic operations. Both bounds have better dependence on $\varepsilon$ than the state-of-the-art result given by $\widetilde{O}\left({n^2/\varepsilon^3}\right)$. Our second algorithm not only has better dependence on $\varepsilon$ in the complexity bound, but also is not specific to entropic regularization and can solve the OT problem with different regularizers.