A Direct tilde{O}(1/epsilon) Iteration Parallel Algorithm for Optimal Transport

Arun Jambulapati, Aaron Sidford, Kevin Tian

Abstract: Optimal transportation, or computing the Wasserstein or ``earth mover's'' distance between two distributions, is a fundamental primitive which arises in many learning and statistical settings. We give an algorithm which solves this problem to additive epsilon with tilde{O}(1/epsilon) parallel depth, and tilde{O}(n^2/epsilon) work. Barring a breakthrough on a long-standing algorithmic open problem, this is optimal for first-order methods. BlanchetJKS18, Quanrud19 obtained similar runtimes through reductions to positive linear programming and matrix scaling. However, these reduction-based algorithms use complicated subroutines which may be deemed impractical due to requiring solvers for second-order iterations (matrix scaling) or non-parallelizability (positive LP). The fastest practical algorithms run in time tilde{O}(min(n^2/epsilon^2, n^{2.5}/epsilon)) (DvurechenskyGK18, LinHJ19). We bridge this gap by providing a parallel, first-order, tilde{O}(1/epsilon) iteration algorithm without worse dependence on dimension, and provide preliminary experimental evidence that our algorithm may enjoy improved practical performance. We obtain this runtime via a primal-dual extragradient method, motivated by recent theoretical improvements to maximum flow (Sherman17).

CMT Num: 6067