Linear convergence of proximal descent schemes on the Wasserstein space

Razvan-Andrei Lascu; Mateusz B. Majka; David Šiška; Łukasz Szpruch

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arXiv · 2411.15067

Linear convergence of proximal descent schemes on the Wasserstein space

Razvan-Andrei Lascu·Mateusz B. Majka·David Šiška·Łukasz Szpruch

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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 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 has a unique Wasserstein sub-gradient, and that the relative Fisher information is indeed finite.

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Razvan-Andrei Lascu, Mateusz B. Majka, David Šiška, Łukasz Szpruch. 2024-11-22. Linear convergence of proximal descent schemes on the Wasserstein space. https://arxiv.org/abs/2411.15067

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