How fast does coordinate ascent variational inference actually converge?

Coordinate ascent variational inference is a workhorse algorithm for Bayesian computation, but its convergence behavior was poorly understood. This work proves the algorithm provably contracts toward the true posterior in Wasserstein distance under mild conditions on smoothness and information geometry. The results apply broadly—including Gaussian mixtures, Bayesian logistic regression, and probit models—and provide both global and local convergence guarantees on smooth and non-smooth spaces.