Variational Inference in Location-Scale Families: Exact Recovery of the Mean and Correlation Matrix
This provides theoretical guarantees for VI robustness in Bayesian inference, addressing a fundamental limitation for practitioners using approximate inference methods, though it is incremental as it builds on existing VI frameworks with specific symmetry assumptions.
The paper tackles the problem of variational inference (VI) being misspecified when approximating intractable target densities, and proves that VI can exactly recover the mean and correlation matrix of the target under certain symmetry conditions, even with severe misspecifications like factorized approximations or different tail behaviors.
Given an intractable target density $p$, variational inference (VI) attempts to find the best approximation $q$ from a tractable family $Q$. This is typically done by minimizing the exclusive Kullback-Leibler divergence, $\text{KL}(q||p)$. In practice, $Q$ is not rich enough to contain $p$, and the approximation is misspecified even when it is a unique global minimizer of $\text{KL}(q||p)$. In this paper, we analyze the robustness of VI to these misspecifications when $p$ exhibits certain symmetries and $Q$ is a location-scale family that shares these symmetries. We prove strong guarantees for VI not only under mild regularity conditions but also in the face of severe misspecifications. Namely, we show that (i) VI recovers the mean of $p$ when $p$ exhibits an \textit{even} symmetry, and (ii) it recovers the correlation matrix of $p$ when in addition~$p$ exhibits an \textit{elliptical} symmetry. These guarantees hold for the mean even when $q$ is factorized and $p$ is not, and for the correlation matrix even when~$q$ and~$p$ behave differently in their tails. We analyze various regimes of Bayesian inference where these symmetries are useful idealizations, and we also investigate experimentally how VI behaves in their absence.