Linear convergence of proximal descent schemes on the Wasserstein space

TL;DR

This paper proves linear convergence of proximal descent methods on the Wasserstein space for entropy-regularized functionals, using a novel approach that relaxes geodesic convexity assumptions and employs a uniform logarithmic Sobolev inequality.

Contribution

It introduces a new convergence analysis for proximal schemes on Wasserstein space that does not rely on geodesic convexity, using LSI and entropy sandwich lemma.

Findings

Establishes linear convergence under flat convexity assumptions.

Shows the iterates belong to a Sobolev class ensuring Fisher information finiteness.

Extends previous analyses by avoiding discrete EVI adaptations.

Abstract

We investigate proximal descent methods, inspired by the minimizing movement scheme introduced by Jordan, Kinderlehrer and Otto, for optimizing entropy-regularized functionals on the Wasserstein space. We establish linear convergence under flat convexity assumptions, thereby relaxing the common reliance on geodesic convexity. Our analysis circumvents the need for discrete-time adaptations of the Evolution Variational Inequality (EVI). Instead, we leverage a uniform logarithmic Sobolev inequality (LSI) and the entropy "sandwich" lemma, extending the analysis from arXiv:2201.10469 and arXiv:2202.01009. The major challenge in the proof via LSI is to show that the relative Fisher information $I(\cdot|\pi)$ is well-defined at every step of the scheme. Since the relative entropy is not Wasserstein differentiable, we prove that along the scheme the iterates belong to a certain class of Sobolev regularity, and hence the relative entropy $\operatorname{KL}(\cdot|\pi)$ has a unique Wasserstein sub-gradient, and that the relative Fisher information is indeed finite.