  Mean field variational inference (VI) is the problem of finding the closest
product (factorized) measure, in the sense of relative entropy, to a given
high-dimensional probability measure $\rho$. The well known Coordinate Ascent
Variational Inference (CAVI) algorithm aims to approximate this product measure
by iteratively optimizing over one coordinate (factor) at a time, which can be
done explicitly. Despite its popularity, the convergence of CAVI remains poorly
understood. In this paper, we prove the convergence of CAVI for log-concave
densities $\rho$. If additionally $\log \rho$ has Lipschitz gradient, we find a
linear rate of convergence, and if also $\rho$ is strongly log-concave, we find
an exponential rate. Our analysis starts from the observation that mean field
VI, while notoriously non-convex in the usual sense, is in fact displacement
convex in the sense of optimal transport when $\rho$ is log-concave. This
allows us to adapt techniques from the optimization literature on coordinate
descent algorithms in Euclidean space.
