Adaptive variational Bayes: Optimality, computation and applications

In this paper, we explore adaptive inference based on variational Bayes. Although several studies have been conducted to analyze the contraction properties of variational posteriors, there is still a lack of a general and computationally tractable variational Bayes method that performs adaptive inference. To fill this gap, we propose a novel adaptive variational Bayes framework, which can operate on a collection of models. The proposed framework first computes a variational posterior over each individual model separately and then combines them with certain weights to produce a variational posterior over the entire model. It turns out that this combined variational posterior is the closest member to the posterior over the entire model in a predefined family of approximating distributions. We show that the adaptive variational Bayes attains optimal contraction rates adaptively under very general conditions. In addition, we provide a methodology to maintain the tractability and adaptive optimality of the adaptive variational Bayes even in the presence of an enormous number of individual models, such as sparse models. We apply the general results to several examples, including deep learning and sparse factor models, and derive new and adaptive inference results. Moreover, we consider the use of quasi-likelihoods in our framework. We formulate theoretical conditions on quasi-likelihoods to ensure adaptive concentration and discuss specific applications to stochastic block models and nonparametric regression with sub-Gaussian errors.