aiqbench analysis
This research advances the theoretical foundations of optimization in Wasserstein space by proving linear convergence for proximal descent methods under flat convexity, removing a restrictive assumption common in optimal transport literature. By utilizing uniform logarithmic Sobolev inequalities instead of discrete-time Evolution Variational Inequalities, the work provides more efficient analytical tools relevant to modern machine learning algorithms operating on probability distributions.
Linear convergence of proximal descent schemes on the Wasserstein space
arXiv:2411.15067v2 Announce Type: replace-cross 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.
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.
Optimization and Control (math.OC); Machine Learning (cs.LG); Probability (math.PR)
arXiv:2411.15067 [math.OC]
(or arXiv:2411.15067v2 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2411.15067 Focus to learn more arXiv-issued DOI via DataCite
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