An Accelerated Stochastic Algorithm for Solving the Optimal Transport Problem

TL;DR

This paper introduces PDASGD, an accelerated stochastic gradient algorithm with variance reduction, achieving the best-known computational complexity for solving the discrete optimal transport problem, outperforming previous primal-dual methods.

Contribution

The paper proposes PDASGD, a novel accelerated stochastic algorithm with variance reduction, that improves the computational complexity for optimal transport problems compared to existing methods.

Findings

PDASGD achieves $ ilde{O}(n^2/)$ complexity for OT.

Numerical experiments show superior practical performance.

Theoretical analysis explains the complexity improvement over previous algorithms.

Abstract

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.