Quantitative contraction rates for Sinkhorn algorithm: beyond bounded costs and compact marginals
Résumé
We show non-asymptotic exponential convergence of Sinkhorn iterates to the Schr\"odinger potentials, solutions of the quadratic Entropic Optimal Transport problem on $\mathbb{R}^d$. Our results hold under mild assumptions on the marginal inputs: in particular, we only assume that they admit an asymptotically positive log-concavity profile, covering as special cases log-concave distributions and bounded smooth perturbations of quadratic potentials. More precisely, we provide exponential $\mathrm{L}^1$ and pointwise convergence of the iterates (resp. their gradient and Hessian) to Schr\"odinger potentials (resp. their gradient and Hessian) for large enough values of the regularization parameter. As a corollary, we establish exponential convergence of Sinkhorn plans with respect to a symmetric relative entropy and in Wasserstein distance of order one. Up to the authors' knowledge, these are the first results which establish exponential convergence of Sinkhorn's algorithm in a general setting without assuming bounded cost functions or compactly supported marginals.