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Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm
Pavel Dvurechenskii · Alexander Gasnikov · Alexey Kroshnin

Wed Jul 11 07:40 AM -- 07:50 AM (PDT) @ A9
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 $\widetilde{O}\left(\frac{n^2}{\varepsilon^2}\right)$ arithmetic operations ($\widetilde{O}$ hides polylogarithmic factors $(\ln n)^c$, $c>0$). For the second one, which is based on our novel Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD) algorithm, we prove the complexity bound $\widetilde{O}\left(\min\left\{\frac{n^{9/4}}{\varepsilon}, \frac{n^{2}}{\varepsilon^2} \right\}\right)$ arithmetic operations. Both bounds have better dependence on $\varepsilon$ than the state-of-the-art result given by $\widetilde{O}\left(\frac{n^2}{\varepsilon^3}\right)$. 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.

#### Author Information

##### Pavel Dvurechenskii (Weierstrass Institute for Applied Analysis and Stochastics)

Graduated with honors from Moscow Institute of Physics and Technology. PhD on differential games in the same university. At the moment research associate in the area of optimization under inexact information in Berlin. Research interest include - algorithms for convex and non-convex large-scale optimization problems; - optimization under deterministic and stochastic inexact information; - randomized algorithms: random coordinate descent, random (derivative-free) directional search; - numerical aspects of Optimal Transport - Algorithms for saddle-point problems and variational inequalities