Black-Box Variational Inference as a Parametric Approximation to Langevin Dynamics

Variational inference (VI) and Markov chain Monte Carlo (MCMC) are approximate posterior inference algorithms that are often said to have complementary strengths, with VI being fast but biased and MCMC being slower but asymptotically unbiased. In this paper, we analyze gradientbased MCMC and VI procedures and find theoretical and empirical evidence that these procedures are not as different as one might think. In particular, a close examination of the Fokker-Planck equation that governs the Langevin dynamics (LD) MCMC procedure reveals that LD implicitly follows a gradient flow that corresponds to a variational inference procedure based on optimizing a nonparametric normalizing flow. This result suggests that the transient bias of LD (due to the Markov chain not having burned in) may track that of VI (due to the optimizer not having converged), up to differences due to VI's asymptotic bias and parameterization. Empirically, we find that the transient biases of these algorithms (and their momentum-accelerated counterparts) do evolve similarly. This suggests that practitioners with a limited time budget may get more accurate results by running an MCMC procedure (even if it's far from burned in) than a VI procedure, as long as the variance of the MCMC estimator can be dealt with (e.g., by running many parallel chains).