Stability of Entropic Optimal Transport and Schrödinger Bridges

Abstract

We establish the stability of solutions to the entropically regularized optimal transport problem with respect to the marginals and the cost function. The result is based on the geometric notion of cyclical invariance and inspired by the use of c-cyclical monotonicity in classical optimal transport. As a consequence of stability, we obtain the wellposedness of the solution in this geometric sense, even when all transports have infinite cost. More generally, our results apply to a class of static Schrödinger bridge problems including entropic optimal transport.

Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure

“…For entropy-regularised OT and the static Schrödinger bridge problem, the first qualitative result appeared very recently in Ghosal et al (2021), based on a version of cyclical monotonicity for entropy-regularised OT introduced by Bernton et al (2021). We present here the first, to the best of our knowledge, quantitative stability result for entropy-regularised OT.…”

Stability of Schrödinger Potentials and Convergence of Sinkhorn's Algorithm

“…Theorem 2.1 (i). The present result also applies in an infinite-dimensional context; the more important difference, however, is that we achieve a strong form of convergence, whereas [20] is silent about any convergence of the densities or potentials. In particular, we can infer stability in the sense of total variation convergence (Corollary 2.6) and the corresponding convergence of Sinkhorn's algorithm (Corollary 3.2).…”

“…Still with bounded cost (and some other conditions), [15] establishes uniform continuity of the potentials relative to the marginals in Wasserstein distance W 1 ; this result is based on the Hilbert-Birkhoff projective metric. Closer to the present unbounded setting, [20] obtains stability of the optimal couplings in weak convergence for general continuous costs. Based on the geometric approach first proposed in [4], the main restriction of the technique is that the underlying spaces need to satisfy Lebesgue's theorem on differentiation of measures which generally holds only in finite-dimensional spaces.…”

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

“…This condition holds as soon as the marginals have a finite exponential moment; in particular, the result covers quadratic costs when marginals are σ 2 -subgaussian for some (arbitrarily small) σ. We remark that Theorem 3.6 is the first quantitative stability result for unbounded costs, and in settings without differentiation of measures as assumed in [31], even the qualitative result alone would be novel.…”

“…This result is obtained by a differential approach establishing invertibility of the Schrödinger system. More recently, [31] obtain the first result on stability in a general setting. Using a geometric approach called cyclical invariance, continuity of optimizers is established in the sense of weak convergence.…”