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We develop a method to combine Markov chain Monte Carlo (MCMC) andvariational inference (VI), leveraging the advantages of both inferenceapproaches. Specifically, we improve the variational distribution by running afew MCMC steps. To make inference tractable, we introduce the variationalcontrastive divergence (VCD), a new divergence that replaces the standardKullback-Leibler (KL) divergence used in VI. The VCD captures a notion ofdiscrepancy between the initial variational distribution and its improvedversion (obtained after running the MCMC steps), and it convergesasymptotically to the symmetrized KL divergence between the variationaldistribution and the posterior of interest. The VCD objective can be optimizedefficiently with respect to the variational parameters via stochasticoptimization. We show experimentally that optimizing the VCD leads to betterpredictive performance on two latent variable models: logistic matrixfactorization and variational autoencoders (VAEs).

A Contrastive Divergence for Combining Variational Inference and MCMC