Bo-Han Wu

Shunan Sheng, Bohan Wu, Bennett Zhu et al.

Structured variational inference constitutes a core methodology in modern statistical applications. Unlike mean-field variational inference, the approximate posterior is assumed to have interdependent structure. We consider the natural setting of star-structured variational inference, where a root variable impacts all the other ones. We prove the first results for existence, uniqueness, and self-consistency of the variational approximation. In turn, we derive quantitative approximation error bounds for the variational approximation to the posterior, extending prior work from the mean-field setting to the star-structured setting. We also develop a gradient-based algorithm with provable guarantees for computing the variational approximation using ideas from optimal transport theory. We explore the implications of our results for Gaussian measures and hierarchical Bayesian models, including generalized linear models with location family priors and spike-and-slab priors with one-dimensional debiasing. As a by-product of our analysis, we develop new stability results for star-separable transport maps which might be of independent interest.

Bohan Wu, David Blei

Variational inference (VI) has emerged as a popular method for approximate inference for high-dimensional Bayesian models. In this paper, we propose a novel VI method that extends the naive mean field via entropic regularization, referred to as $Ξ$-variational inference ($Ξ$-VI). $Ξ$-VI has a close connection to the entropic optimal transport problem and benefits from the computationally efficient Sinkhorn algorithm. We show that $Ξ$-variational posteriors effectively recover the true posterior dependency, where the dependence is downweighted by the regularization parameter. We analyze the role of dimensionality of the parameter space on the accuracy of $Ξ$-variational approximation and how it affects computational considerations, providing a rough characterization of the statistical-computational trade-off in $Ξ$-VI. We also investigate the frequentist properties of $Ξ$-VI and establish results on consistency, asymptotic normality, high-dimensional asymptotics, and algorithmic stability. We provide sufficient criteria for achieving polynomial-time approximate inference using the method. Finally, we demonstrate the practical advantage of $Ξ$-VI over mean-field variational inference on simulated and real data.

Shunan Sheng, Bohan Wu, Alberto González-Sanz

Mean-field variational inference (MFVI) is a widely used method for approximating high-dimensional probability distributions by product measures. It has been empirically observed that MFVI optimizers often suffer from mode collapse. Specifically, when the target measure $π$ is a mixture $π= w P_0 + (1 - w) P_1$, the MFVI optimizer tends to place most of its mass near a single component of the mixture. This work provides the first theoretical explanation of mode collapse in MFVI. We introduce the notion to capture the separatedness of the two mixture components -- called $\varepsilon$-separateness -- and derive explicit bounds on the fraction of mass that any MFVI optimizer assigns to each component when $P_0$ and $P_1$ are $\varepsilon$-separated for sufficiently small $\varepsilon$. Our results suggest that the occurrence of mode collapse crucially depends on the relative position of the components. To address this issue, we propose the rotational variational inference (RoVI), which augments MFVI with a rotation matrix. The numerical studies support our theoretical findings and demonstrate the benefits of RoVI.