Convergence rate of general entropic optimal transport costs

We investigate the convergence rate of the optimal entropic cost v_ε to the optimal transport cost as the noise parameter ε↓ 0 . We show that for a large class of cost functions c on ℝ^d×ℝ^d (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and L^∞ marginals, one has v_ε -v_0= d/2εlog (1/ε )+ O(ε ) . 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 ∇ _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.