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We study the variational inference problem of minimizing a regularizedR\'enyi divergence over an exponential family, and propose a relaxedmoment-matching algorithm, which includes a proximal-like step. Using theinformation-geometric link between Bregman divergences and the Kullback-Leiblerdivergence, this algorithm is shown to be equivalent to a Bregman proximalgradient algorithm. This novel perspective allows us to exploit the geometry ofour approximate model while using stochastic black-box updates. We use thispoint of view to prove strong convergence guarantees including monotonicdecrease of the objective, convergence to a stationary point or to theminimizer, and geometric convergence rates. These new theoretical insights leadto a versatile, robust, and competitive method, as illustrated by numericalexperiments.

Regularized R\'enyi divergence minimization through Bregman proximal gradient algorithms