Convergence rate of general entropic optimal transport costs

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.