Sample complexity for divergence regularized optimal transport with radial cost

We prove a new sample complexity result for divergence regularized optimal transport. Our bound holds for probability measures on~${\mathbb{R}}^d$ with exponential tail decay and for radial cost functions that satisfy a local Lipschitz condition. It is sharp up to logarithmic factors, and captures the intrinsic dimension of the marginal distributions through a generalized covering number of their supports. Examples that fit into our framework include subexponential and subgaussian distributions and radial cost functions $c(x,y)=\mathrm{\mid }x-y{\mathrm{\mid }}^p$ for $p\ge 1$ with logarithmic entropy or polynomial $\alpha$-divergence.