Langevin Dynamics as Nonparametric Variational Inference
Variational inference and Markov chain Monte Carlo 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 gradient-based 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 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 may track that of VI, up to differences due to VI’s parameterization and asymptotic bias.Empirically, we find that the transient biases of these algorithms do evolve similarly.This suggests that practitioners with a limited time budget may get more accurate results by running an MCMC procedure than a VI procedure, as long as the variance of the MCMC estimator can be dealt with.
The transient behavior of gradient-based MCMC and variational inference algorithms is more similar than one might think, calling into question the claim that variational inference is faster than MCMC.