Improved Rate of First Order Algorithms for Entropic Optimal Transport

TL;DR

This paper introduces an accelerated primal-dual stochastic mirror descent algorithm with variance reduction for entropy-regularized optimal transport, achieving faster convergence rates than previous methods and demonstrating improved computational efficiency in experiments.

Contribution

It presents a novel accelerated primal-dual stochastic mirror descent algorithm with variance reduction for entropy-regularized OT, improving convergence rates from previous methods.

Findings

Achieves a convergence rate of O(n^2/) for OT approximation.

Outperforms the stochastic Sinkhorn algorithm in computational complexity.

Experimental results confirm theoretical rate improvements.

Abstract

This paper improves the state-of-the-art rate of a first-order algorithm for solving entropy regularized optimal transport. The resulting rate for approximating the optimal transport (OT) has been improved from $\widetilde{{O}}({n^{2.5}}/{\epsilon})$ to $\widetilde{{O}}({n^2}/{\epsilon})$, where $n$ is the problem size and $\epsilon$ is the accuracy level. In particular, we propose an accelerated primal-dual stochastic mirror descent algorithm with variance reduction. Such special design helps us improve the rate compared to other accelerated primal-dual algorithms. We further propose a batch version of our stochastic algorithm, which improves the computational performance through parallel computing. To compare, we prove that the computational complexity of the Stochastic Sinkhorn algorithm is $\widetilde{{O}}({n^2}/{\epsilon^2})$, which is slower than our accelerated primal-dual stochastic mirror algorithm. Experiments are done using synthetic and real data, and the results match our theoretical rates. Our algorithm may inspire more research to develop accelerated primal-dual algorithms that have rate $\widetilde{{O}}({n^2}/{\epsilon})$ for solving OT.