Convergence rate of general entropic optimal transport costs

TL;DR

This paper analyzes how quickly the entropic optimal transport cost converges to the classical cost as the noise parameter approaches zero, providing precise asymptotic rates under broad conditions.

Contribution

It establishes the convergence rate of entropic optimal transport costs to the classical cost for a wide class of cost functions and marginals, including non-unique plans.

Findings

Convergence rate is approximately (d/2) * ε * log(1/ε) for small ε.

Upper bounds achieved via block approximation and Alexandrov's theorem.

Lower bounds established under an invertibility condition on the cost function.

Abstract

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 $\mathbb{R}^d\times \mathbb{R}^d$ (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and $L^{\infty}$ marginals, one has $v_\varepsilon-v_0= \frac{d}{2} \varepsilon \log(1/\varepsilon)+ O(\varepsilon)$. 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 $\nabla_{xy}^2 c(x,y)$, we get the lower bound by establishing a quadratic detachment of the duality gap in $d$ dimensions thanks to Minty's trick.