Extending Mean-Field Variational Inference via Entropic Regularization: Theory and Computation
This work addresses the challenge of dependency recovery in variational inference for statisticians and machine learning practitioners, offering an incremental improvement over existing mean-field methods.
The paper tackles the problem of approximate inference in high-dimensional Bayesian models by proposing a novel variational inference method that extends mean-field via entropic regularization, showing it recovers posterior dependencies and offers computational efficiency with polynomial-time guarantees, demonstrated through improved performance on simulated and real data.
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.