Entropic Optimal Transport: Geometry and Large Deviations

Abstract

We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the local exponential convergence rate as the regularization parameter vanishes. The exact rate function is determined in a general setting and linked to the Kantorovich potential of optimal transport. Our arguments are based on the geometry of the optimizers and inspired by the use of c-cyclical monotonicity in classical transport theory. The results can also be phrased in terms of Schrödinger bridges.

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.…”

Entropic estimation of optimal transport maps

“…Several recent works have bridged the regularized and unregularized optimal transport regimes, with particular interest in the setting where ε → 0. Convergence of π ε to π 0 was studied by Carlier et al (2017) and Léonard (2012), and recent work has quantified the convergence of the plans (Bernton et al, 2021;Ghosal et al, 2021) and the potentials (Altschuler et al, 2021;Nutz and Wiesel, 2021) in certain settings. Convergence of S ε (P, Q) to 1 2 W 2 2 (P, Q) has attracted significant research interest: under mild conditions, Pal (2019) proves a first-order convergence result for general convex costs (replacing 1 2 • 2 ), and a second order expansion was subsequently obtained by Chizat et al (2020) and Conforti and Tamanini (2021).…”

Estimation of Stationary Optimal Transport Plans

“…As an object of estimation, the entropic optimal transport cost also exhibits better sample complexity and faster rates of convergence [39,31,48]. Furthermore, solutions of the entropic optimal transport problem have been shown to converge to solutions of the unregularized problem as the regularization coefficient η converges to zero [60,5,52]. For additional details on entropic optimal transport, we refer the reader to [60].…”