Quantitative contraction rates for Sinkhorn algorithm: beyond bounded costs and compact marginals

Giovanni Conforti, Alain Durmus, Giacomo Greco

Published 2023-04-10Version 1

We show non-asymptotic geometric 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 and bridges w.r.t. a symmetric relative entropy. Up to the authors' knowledge, these are the first results which establish geometric convergence of Sinkhorn algorithm in a general setting without assuming bounded cost functions or compactly supported marginals. Our results are proven following a probabilistic approach that rests on integrated semiconvexity estimates for Sinkhorn iterates that are of independent interest.