Variational Inference in Location-Scale Families: Exact Recovery of the Mean and Correlation Matrix

TL;DR

This paper demonstrates that variational inference with location-scale families can exactly recover the mean and correlation matrix of symmetric target distributions, even under misspecification.

Contribution

It provides theoretical guarantees for VI's robustness in recovering key parameters of symmetric distributions despite model misspecification.

Findings

VI recovers the mean for even symmetric distributions.

VI recovers the correlation matrix for elliptically symmetric distributions.

Guarantees hold even with factorized approximations and tail differences.

Abstract

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.