A Contrastive Divergence for Combining Variational Inference and MCMC

TL;DR

This paper introduces a novel divergence, VCD, that combines MCMC and variational inference to improve posterior approximation, leading to better predictive performance in latent variable models.

Contribution

We propose the variational contrastive divergence (VCD), a new divergence that integrates MCMC steps into VI and is optimized efficiently for improved inference.

Findings

VCD converges to the symmetrized KL divergence.

Optimizing VCD improves predictive accuracy.

Method outperforms standard VI in experiments.

Abstract

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).