Furthermore, we show that the WAE objective is related to other statistical quantities such as thef-divergence and in particular, upper bounded by the Wasserstein distance, which then allows us to tap into existing efficient(regularized)optimaltransportsolvers.

Variational Bayes (VB) is a scalable alternative to Markov chain Monte Carlo (MCMC) for Bayesian posterior inference. Though popular, VB comes with few theoretical guarantees, most of which focus on well-specified models. However, models are rarely well-specified in practice. In this work, we study VB under model misspecification. We prove the VB posterior is asymptotically normal and centers at the value that minimizes the Kullback-Leibler (KL) divergence to the true data-generating distribution. Moreover, the VB posterior mean centers at the same value and is also asymptotically normal.

We propose a robust and scalable variational Bayes (VB) framework designed to effectively handle contamination and outliers in dataset. Our approach partitions the data into $m$ disjoint subsets and formulates a joint optimization problem based on robust aggregation principles. A key insight is that the full posterior distribution is equivalent to the minimizer of the mean Kullback-Leibler (KL) divergence from the $m$-powered local posterior distributions. To enhance robustness, we replace the mean KL divergence with a min-max median formulation. The min-max formulation not only ensures consistency between the KL minimizer and the Evidence Lower Bound (ELBO) maximizer but also facilitates the establishment of improved statistical rates for the mean of variational posterior. We observe a notable discrepancy in the $m$-powered marginal log likelihood function contingent on the presence of local latent variables. To address this, we treat these two scenarios separately to guarantee the consistency of the aggregated variational posterior. Specifically, when local latent variables are present, we introduce an aggregate-and-rescale strategy. Theoretically, we provide a non-asymptotic analysis of our proposed posterior, incorporating a refined analysis of Bernstein-von Mises (BvM) theorem to accommodate a diverging number of subsets $m$. Our findings indicate that the two-stage approach yields a smaller approximation error compared to directly aggregating the $m$-powered local posteriors. Furthermore, we establish a nearly optimal statistical rate for the mean of the proposed posterior, advancing existing theories related to min-max median estimators. The efficacy of our method is demonstrated through extensive simulation studies.

Variational Bayes methods are popular due to their computational efficiency and adaptability to diverse applications. In specifying the variational family, mean-field classes are commonly used, which enables efficient algorithms such as coordinate ascent variational inference (CAVI) but fails to capture parameter dependence and typically underestimates uncertainty. In this work, we introduce a variational bagging approach that integrates a bagging procedure with variational Bayes, resulting in a bagged variational posterior for improved inference. We establish strong theoretical guarantees, including posterior contraction rates for general models and a Bernstein-von Mises (BVM) type theorem that ensures valid uncertainty quantification. Notably, our results show that even when using a mean-field variational family, our approach can recover off-diagonal elements of the limiting covariance structure and provide proper uncertainty quantification. In addition, variational bagging is robust to model misspecification, with covariance structures matching those of the target covariance. We illustrate our variational bagging method in numerical studies through applications to parametric models, finite mixture models, deep neural networks, and variational autoencoders (VAEs).

Variational approaches are often used to approximate intractable posteriors or normalization constants in hierarchical latent variable models. While often effective in practice, it is known that the approximation error can be arbitrarily large. We propose a new class of bounds on the marginal log-likelihood of directed latent variable models. Our approach relies on random projections to simplify the posterior. In contrast to standard variational methods, our bounds are guaranteed to be tight with high probability. We provide a new approach for learning latent variable models based on optimizing our new bounds on the log-likelihood. We demonstrate empirical improvements on benchmark datasets in vision and language for sigmoid belief networks, where a neural network is used to approximate the posterior.

Bayes type methods, which would otherwise be well suited to the i.i.d. Our focus in this paper is not to explore the generalities of finite-dimensional processes.

Variational inference plays a vital role in learning graphical models, especially on large-scale datasets. Much of its success depends on a proper choice of auxiliary distribution class for posterior approximation. However, how to pursue an auxiliary distribution class that achieves both good approximation ability and computation efficiency remains a core challenge. In this paper, we proposed coupled variational Bayes which exploits the primal-dual view of the ELBO with the variational distribution class generated by an optimization procedure, which is termed optimization embedding.

Theoretically, we establish an interesting connection to gradient flow and demonstrate the extreme flexibility of this implicit distribution family in the limit sense.