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This paper establishes Wasserstein contraction for coordinate ascent variational inference (CAVI) under transport-information inequalities and functional smoothness conditions. The analysis provides local convergence guarantees applicable to smooth manifolds and some non-smooth spaces, offering sharper results than existing theory. The framework is applied to Bayesian Gaussian Mixture Models, high-dimensional Bayesian Probit Regression, and Logistic Regression with P\'olya-Gamma random variables, demonstrating its broad applicability.
CAVI, a workhorse of approximate Bayesian inference, is now provably convergent in Wasserstein distance under mild conditions, offering a new theoretical foundation for its use.
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 P\'olya-Gamma random variables (i.e. Jaakkola-Jordan's algorithm).