A Direct tildeO(1/ε) Iteration Parallel Algorithm for Optimal Transport

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 ε with tildeO(1/ε) parallel depth, and tildeOleft(n2/εright) work. Barring a breakthrough on a long-standing algorithmic open problem, this is optimal for first-order methods. Blanchet et. al. '18, Quanrud '19 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 tildeO(min(n2 / ε2, n2.5 / ε)) (Dvurechensky et. al. '18, Lin et. al. '19). We bridge this gap by providing a parallel, first-order, tildeO(1/ε) 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 (Sherman '17).