Convergence Rates for Regularized Optimal Transport via Quantization

TL;DR

This paper establishes precise convergence rates for divergence-regularized optimal transport problems as the regularization diminishes, using a novel quantization approach suitable for various divergences and marginals.

Contribution

Introduces a new quantization-based methodology to derive sharp convergence rates for regularized optimal transport with general divergences and non-compact marginals.

Findings

Sharp convergence rates for divergence-regularized optimal transport.

Method applicable to general divergences and transport costs.

Achieves the leading-order term for entropic regularization in 2-Wasserstein distance.

Abstract

We study the convergence of divergence-regularized optimal transport as the regularization parameter vanishes. Sharp rates for general divergences including relative entropy or $L^{p}$ regularization, general transport costs and multi-marginal problems are obtained. A novel methodology using quantization and martingale couplings is suitable for non-compact marginals and achieves, in particular, the sharp leading-order term of entropically regularized 2-Wasserstein distance for all marginals with finite $(2+\delta)$-moment.