Convergence of coordinate ascent variational inference for log-concave measures via optimal transport

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 ρ. 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 ρ. If additionally logρ has Lipschitz gradient, we find a linear rate of convergence, and if also ρ is strongly log-concave, we find an exponential rate. Our analysis starts from the observation that mean field VI, while notoriously nonconvex in the usual sense, is in fact displacement convex in the sense of optimal transport when ρ is log-concave. This allows us to adapt techniques from the optimization literature on coordinate descent algorithms in Euclidean space.