Computational and statistical guarantees for star-structured variational inference

Music City Center 

We study star-structured variational inference (SVI), an extension of mean-field variational inference that approximates a target distribution $\pi$ over $\mathbb{R}^d$ with a star graphical model $\pi^*$, where a central latent variable is connected to all other variables. We establish the existence, uniqueness, and self-consistency of the star variational solution, derive quantitative approximation error bounds, and provide computational guarantees via projected gradient descent under curvature assumptions on $\pi$. We explore the implications of our results in Gaussian measures and hierarchical Bayesian models, including generalized linear models with location family priors and spike-and-slab priors with one-dimensional debiasing. Our analysis and algorithms rely on functional inequalities and displacement convexity from optimal transport theory.

structured variational inference

log-concavity

Bayesian regression

approximate Bayesian inference

Knothe–Rosenblatt (KR) maps 

IMS