Convergence rate of entropy-regularized multi-marginal optimal transport costs

TL;DR

This paper analyzes how quickly entropy-regularized multi-marginal optimal transport costs converge to the unregularized costs as the regularization parameter approaches zero, providing bounds and conditions for various cost types.

Contribution

It extends convergence rate results from two-marginal to multi-marginal cases, including degenerate costs, with explicit bounds depending on problem parameters.

Findings

Established bounds of order pspspspspspspsps for convergence rates.

Derived bounds for Lipschitz and semi-concave costs, and lower bounds for ^2 costs with signature conditions.

Provided matching bounds in cases with deterministic optimal plans.

Abstract

We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann-Shannon entropy, as the noise parameter $\varepsilon$ tends to $0$. We establish lower and upper bounds on the difference with the unregularized cost of the form $C\varepsilon\log(1/\varepsilon)+O(\varepsilon)$ for some explicit dimensional constants $C$ depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semi-concave costs for a finer estimate, and lower bounds for $\mathscr{C}^2$ costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for non-degenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.