Decreasing Entropic Regularization Averaged Gradient for Semi-Discrete Optimal Transport

TL;DR

This paper introduces DRAG, a stochastic gradient descent algorithm that adaptively decreases entropic regularization in semi-discrete optimal transport, leading to faster convergence and reduced bias.

Contribution

The paper provides the first theoretical analysis showing that decreasing regularization accelerates convergence in semi-discrete OT and introduces DRAG, a novel algorithm implementing this idea.

Findings

DRAG achieves $ ext{O}(1/t)$ complexity for OT cost and potential estimation.

Decreasing regularization reduces bias and accelerates convergence.

Numerical experiments confirm the practical benefits of DRAG.

Abstract

Adding entropic regularization to Optimal Transport (OT) problems has become a standard approach for designing efficient and scalable solvers. However, regularization introduces a bias from the true solution. To mitigate this bias while still benefiting from the acceleration provided by regularization, a natural solver would adaptively decrease the regularization as it approaches the solution. Although some algorithms heuristically implement this idea, their theoretical guarantees and the extent of their acceleration compared to using a fixed regularization remain largely open. In the setting of semi-discrete OT, where the source measure is continuous and the target is discrete, we prove that decreasing the regularization can indeed accelerate convergence. To this end, we introduce DRAG: Decreasing (entropic) Regularization Averaged Gradient, a stochastic gradient descent algorithm where the regularization decreases with the number of optimization steps. We provide a theoretical analysis showing that DRAG benefits from decreasing regularization compared to a fixed scheme, achieving an unbiased $\mathcal{O}(1/t)$ sample and iteration complexity for both the OT cost and the potential estimation, and a $\mathcal{O}(1/\sqrt{t})$ rate for the OT map. Our theoretical findings are supported by numerical experiments that validate the effectiveness of DRAG and highlight its practical advantages.