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• W2947656817 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 $tilde{O}left(n^2/epsilonright)$ 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 $tilde{O}(min(n^2 / epsilon^2, n^{2.5} / epsilon))$ (Dvurechensky et. al. '18, Lin et. al. '19). 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 (Sherman '17)." @default.
• W2947656817 created "2019-06-07" @default.
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• W2947656817 date "2019-06-03" @default.
• W2947656817 modified "2023-09-27" @default.
• W2947656817 title "A Direct $tilde{O}(1/epsilon)$ Iteration Parallel Algorithm for Optimal Transport" @default.
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