Convergence rate of general entropic optimal transport costs

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.