Provable Smoothness Guarantees for Black-Box Variational Inference

Black-box variational inference tries to approximate a complex target distribution though a gradient-based optimization of the parameters of a simpler distribution. Provable convergence guarantees require structural properties of the objective. This paper shows that for location-scale family approximations, if the target is M-Lipschitz smooth, then so is the objective, if the entropy is excluded. The key proof idea is to describe gradients in a certain inner-product space, thus permitting use of Bessel's inequality. This result gives insight into how to parameterize distributions, gives bounds the location of the optimal parameters, and is a key ingredient for convergence guarantees.