# Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy

arXiv: https://arxiv.org/abs/2306.06940 · HTML: https://mlsanity.com/paper/2306.06940/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2023.

## Built on by (most cited first, 5 of 5)

- [Annealed Sinkhorn for Optimal Transport: convergence, regularization path and debiasing](https://mlsanity.com/paper/2408.11620/) (2024)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity](https://mlsanity.com/paper/2504.11133/) (2025)
- [E-ROBOT: a dimension-free method for robust statistics and machine learning via Schr"odinger bridge](https://mlsanity.com/paper/2509.11532/) (2025)
- [Scaling Online Optimal Transport Minibatches for Flow Matching](https://mlsanity.com/paper/2610.18838/) (2026)

## Abstract (arXiv)

We study the convergence of the transport plans γε towards γ0 as well as the cost of the entropy-regularized optimal transport (c,γε) towards (c,γ0) as the regularization parameter ε vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance W2(γε,γ0) is asymptotically greater than C√ε and the suboptimality (c,γε)-(c,γ0) is of order ε. In the quadratic cost case the compactness assumption is relaxed into a moment of order 2+δ assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance W2(γε,γ0) converges to 0 at rate √ε. Finally, if in addition the marginals have finite Fisher information, we prove (c,γε)-(c,γ0) ~ dε/2 and we provide a companion expansion of H(γε). These results are achieved by disentangling the role of the cost and the entropy in the regularized problem.