# Convergence rate of general entropic optimal transport costs

arXiv: https://arxiv.org/abs/2206.03347 · HTML: https://mlsanity.com/paper/2206.03347/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2022.

## Built on by (most cited first, 9 of 9)

- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm](https://mlsanity.com/paper/2307.16421/) (2023)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [On Rank Energy Statistics via Optimal Transport: Continuity, Convergence, and Change Point Detection](https://mlsanity.com/paper/2302.07964/) (2023)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Convergence rate of Tsallis entropic regularized optimal transport](https://mlsanity.com/paper/2304.06616/) (2023)
- [Linear Bandits beyond Inner Product Spaces, the case of Bandit Optimal Transport](https://mlsanity.com/paper/2502.07397/) (2025)
- [General and Estimable Learning Bound Unifying Covariate and Concept Shifts](https://mlsanity.com/paper/2506.12829/) (2025)
- [General Quantification of Covariate and Concept Shifts](https://mlsanity.com/paper/2609.11918/) (2026)

## Abstract (arXiv)

We investigate the convergence rate of the optimal entropic cost v_varepsilon to the optimal transport cost as the noise parameter varepsilon downarrow 0. We show that for a large class of cost functions c on ℝd× ℝd (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and L∞ marginals, one has v_varepsilon-v0= fracd2 varepsilon log(1/varepsilon)+ O(varepsilon). Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under an infinitesimal twist condition on c, i.e. invertibility of nablaxy2 c(x,y), we get the lower bound by establishing a quadratic detachment of the duality gap in d dimensions thanks to Minty's trick.