Quantitative Stability of Many-Marginal Schrodinger Bridge

Abstract

In this paper, we explore quantitative stability of multi-marginal Schr\"odinger bridges with respect to the marginal constraints. We focus on the case where the number of marginal constraints is large (i.e. ``many-marginals"). When this number increases, we show that the Kullback--Leibler (KL) divergence between two multi-marginal Schr\"odinger bridges, as measures on the path space, can be asymptotically bounded by the terminal marginal KL divergence and a time-integrated squared discrepancy {that combines} Wasserstein-2 geodesic velocity fields with a log-density gradient term. Our stability upper bound is also asymptotically tight: it converges to zero as the number of marginal constraints increases with unperturbed marginal constraints. To the best of our knowledge, this is the first such stability result that addresses the many-marginal regime, giving error estimates that are asymptotically independent of the number of marginals. To achieve our result, the key step is to derive an asymptotic expansion (of order $k\ge 2$) of Schr\"odinger potentials with respect to a diminishing regularization coefficient. This result can also be applied to deriving asymptotic expansions of entropic Brenier maps in entropic optimal self-transport problems. As byproducts of our analyses, we also establish the asymptotic expansion of entropic optimal transport cost with respect to the diminishing regularization coefficient when two marginal constraints are sufficiently close. We also prove a stability property of the Schr\"odinger functional.