Wasserstein Contraction of Coordinate Ascent Variational Inference

2026-05-28

AI Summary

This work addresses the lack of local convergence guarantees for coordinate ascent variational inference (CAVI) algorithms under the Wasserstein metric, particularly in non-smooth spaces where theoretical results have been absent. By introducing transport-information inequalities and appropriate smoothness conditions on the objective functional, the paper establishes Wasserstein contraction properties of CAVI for the first time—not only on general smooth manifolds but also in certain non-smooth settings—yielding a sharp local convergence theory. The resulting framework is both unified and broadly applicable, successfully encompassing Bayesian Gaussian mixture models, high-dimensional probit regression, and Pólya–Gamma-augmented logistic regression, thereby demonstrating the theoretical validity and practical versatility of the proposed approach.

Abstract

We study the contraction in Wasserstein distance of the coordinate ascent variational inference algorithm. This is shown to hold under a transport-information inequality at the fixed points and a functional smoothness condition. The results are general and sharp, allow for local convergence guarantees, hold for general smooth manifolds, and also in some non-smooth spaces. We consider applications to Bayesian Gaussian Mixture Models, and high-dimensional Bayesian Probit Regression, and Logistic Regression with Pólya-Gamma random variables (i.e. Jaakkola-Jordan's algorithm).

Problem

• Wasserstein contraction
• coordinate ascent variational inference
• transport-information inequality
• local convergence
• Bayesian inference

Innovation

• Wasserstein contraction
• coordinate ascent variational inference
• transport-information inequality
• local convergence
• non-smooth spaces