Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm

Abstract

We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and Hölder continuity of the optimal coupling in p-Wasserstein distance are obtained under general conditions including quadratic costs and unbounded marginals. The results for the value extend to regularization by an arbitrary divergence. Two techniques are presented: The first compares an optimal coupling with its so-called shadow, a coupling induced on other marginals by an explicit construction. The second transforms one set of marginals by a change of coordinates and thus reduces the comparison of differing marginals to the comparison of differing cost functions under the same marginals.

Computational methods for adapted optimal transport

“…The first property (i) is clear by definition of K. Property (ii) follows immediately from the data processing inequality (cf. [20,Lemma 4.1]).…”

“…Section 1.3 below), and in this case we even obtain stability results for the respective optimizing couplings. Our main technique is based on an adapted version of the concept of the shadow -a coupling which shadows the dependence structure of another coupling between differing marginals (see Definition 3.3) -which builds on the recent work in [20]. A necessary component of our stability results is that marginals are compared with respect to an adapted Wasserstein distance, and not just a Wasserstein distance.…”

Stability of Schrödinger Potentials and Convergence of Sinkhorn's Algorithm

“…Obtained by control-theoretic arguments through a transport inequality, the main strength of this result lies in being quantitative (which the present one is not). On the other hand, [17] is once again silent about the densities or potentials, and does not yield a convergence in total variation. Indeed, we are not aware of previous stability results in total variation beyond bounded settings.…”

“…In the present work, we effectively reduce the dimension by focusing on densities with a decomposition given by potentials and then obtain compactness through the potentials. A last related work is [17], which was conducted concurrently. Here stability of the coupling in Wasserstein distance W p is shown under certain growth and integrability conditions.…”

Numerical method for feasible and approximately optimal solutions of multi-marginal optimal transport beyond discrete measures

“…Most notably, Cuturi [22] proposed to use entropic regularization and the Sinkhorn algorithm for solving the optimal transport problem (i.e., when N = 2). See also [27,47] for the theoretical properties of entropic regularization and the Sinkhorn algorithm. While most regularizationbased approaches deal with discrete marginals, see, e.g., [9,49,56], there are also regularization-based approaches for solving MMOT problems with non-discrete marginals.…”