  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.
