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.

翻译：本文提出了一种带方差缩减的原对偶加速随机梯度下降算法（PDASGD），用于求解线性约束优化问题。PDASGD可应用于求解离散最优传输（OT）问题，并具有目前最优的计算复杂度——$\widetilde{\mathcal{O}}(n^2/\epsilon)$，其中$n$为原子数，$\epsilon>0$为精度。现有文献中已提出一些原对偶加速一阶算法（如APDAGD），其求解OT问题的复杂度为$\widetilde{\mathcal{O}}(n^{2.5}/\epsilon)$。为阐明所提算法为何能将计算速率提升$\widetilde{\mathcal{O}}(\sqrt{n})$倍，本文讨论了随机算法在满足何种条件下能够降低线性约束优化问题的计算复杂度阶数。理论证明最优传输问题恰好满足上述条件。数值实验表明，所提PDASGD算法在求解OT问题时具有优异的实际性能。