An Accelerated Stochastic Algorithm for Solving the Optimal Transport Problem

A primal-dual accelerated stochastic gradient descent with variance reduction algorithm (PDASGD) is proposed to solve linear-constrained optimization problems. PDASGD could be applied to solve the discrete optimal transport (OT) problem and enjoys the best-known computational complexity -- $\widetilde{\mathcal{O}}(n^2/\epsilon)$, where $n$ is the number of atoms, and $\epsilon>0$ is the accuracy. In the literature, some primal-dual accelerated first-order algorithms, e.g., APDAGD, have been proposed and have the order of $\widetilde{\mathcal{O}}(n^{2.5}/\epsilon)$ for solving the OT problem. To understand why our proposed algorithm could improve the rate by a factor of $\widetilde{\mathcal{O}}(\sqrt{n})$, the conditions under which our stochastic algorithm has a lower order of computational complexity for solving linear-constrained optimization problems are discussed. It is demonstrated that the OT problem could satisfy the aforementioned conditions. Numerical experiments demonstrate superior practical performances of the proposed PDASGD algorithm for solving the OT problem.