  We study the variational inference problem of minimizing a regularized
R\'enyi divergence over an exponential family, and propose a relaxed
moment-matching algorithm, which includes a proximal-like step. Using the
information-geometric link between Bregman divergences and the Kullback-Leibler
divergence, this algorithm is shown to be equivalent to a Bregman proximal
gradient algorithm. This novel perspective allows us to exploit the geometry of
our approximate model while using stochastic black-box updates. We use this
point of view to prove strong convergence guarantees including monotonic
decrease of the objective, convergence to a stationary point or to the
minimizer, and geometric convergence rates. These new theoretical insights lead
to a versatile, robust, and competitive method, as illustrated by numerical
experiments.
