Generalized Guarantees for Variational Inference in the Presence of Even and Elliptical Symmetry

We extend several recent results providing symmetry-based guarantees for variational inference (VI) with location-scale families. VI approximates a target density $p$ by the best match $q^*$ in a family $Q$ of tractable distributions that in general does not contain $p$. It is known that VI can recover key properties of $p$, such as its mean and correlation matrix, when $p$ and $Q$ exhibit certain symmetries and $q^*$ is found by minimizing the reverse Kullback-Leibler divergence. We extend these guarantees in two important directions. First, we provide symmetry-based guarantees for $f$-divergences, a broad class that includes the reverse and forward Kullback-Leibler divergences and the $α$-divergences. We highlight properties specific to the reverse Kullback-Leibler divergence under which we obtain our strongest guarantees. Second, we obtain further guarantees for VI when the target density $p$ exhibits even and elliptical symmetries in some but not all of its coordinates. These partial symmetries arise naturally in Bayesian hierarchical models, where the prior induces a challenging geometry but still possesses axes of symmetry. We illustrate these theoretical results in a number of experimental settings.