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 a broader family of divergences, highlighting the properties of variational objectives under which VI provably recovers the mean and correlation matrix. 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.

In this work, we present a sampling algorithm for single hidden layer neural networks. This algorithm is built upon a recursive series of Bayesian posteriors using a method we call Greedy Bayes. Sampling of the Bayesian posterior for neuron weight vectors $w$ of dimension $d$ is challenging because of its multimodality. Our algorithm to tackle this problem is based on a coupling of the posterior density for $w$ with an auxiliary random variable $\xi$. The resulting reverse conditional $w|\xi$ of neuron weights given auxiliary random variable is shown to be log concave. In the construction of the posterior distributions we provide some freedom in the choice of the prior. In particular, for Gaussian priors on $w$ with suitably small variance, the resulting marginal density of the auxiliary variable $\xi$ is proven to be strictly log concave for all dimensions $d$. For a uniform prior on the unit $\ell_1$ ball, evidence is given that the density of $\xi$ is again strictly log concave for sufficiently large $d$. The score of the marginal density of the auxiliary random variable $\xi$ is determined by an expectation over $w|\xi$ and thus can be computed by various rapidly mixing Markov Chain Monte Carlo methods. Moreover, the computation of the score of $\xi$ permits methods of sampling $\xi$ by a stochastic diffusion (Langevin dynamics) with drift function built from this score. With such dynamics, information-theoretic methods pioneered by Bakry and Emery show that accurate sampling of $\xi$ is obtained rapidly when its density is indeed strictly log-concave. After which, one more draw from $w|\xi$, produces neuron weights $w$ whose marginal distribution is from the desired posterior.