  We develop a method to combine Markov chain Monte Carlo (MCMC) and
variational inference (VI), leveraging the advantages of both inference
approaches. Specifically, we improve the variational distribution by running a
few MCMC steps. To make inference tractable, we introduce the variational
contrastive divergence (VCD), a new divergence that replaces the standard
Kullback-Leibler (KL) divergence used in VI. The VCD captures a notion of
discrepancy between the initial variational distribution and its improved
version (obtained after running the MCMC steps), and it converges
asymptotically to the symmetrized KL divergence between the variational
distribution and the posterior of interest. The VCD objective can be optimized
efficiently with respect to the variational parameters via stochastic
optimization. We show experimentally that optimizing the VCD leads to better
predictive performance on two latent variable models: logistic matrix
factorization and variational autoencoders (VAEs).
