Bridging the Gap between Variational Inference and Stochastic Gradient MCMC in Function Space

Traditional parameter-space posterior inference for Bayesian neural networks faces several challenges, such as the difficulty in specifying meaningful prior, the potential pathologies in deep models and the intractability for multi-modal posterior. To address these issues, functional variational inference (fVI) and functional Markov Chain Monte Carlo (fMCMC) are two recently emerged Bayesian inference schemes that perform posterior inference directly in function space by incorporating more informative functional priors. Similar to their parameter-space counterparts, fVI and fMCMC have their own strengths and weaknesses. For instance, fVI is computationally efficient but imposes strong distributional assumptions, while fMCMC is asymptotically exact but suffers from slow mixing in high dimensions. To inherit the complementary benefits of both schemes, this work proposes a novel hybrid inference method for functional posterior inference. Specifically, it combines fVI and fMCMC successively by an elaborate linking mechanism to form an alternating approximation process. We also provide theoretical justification for the soundness of such a hybrid inference through the lens of Wasserstein gradient flows in the function space. We evaluate our method on several benchmark tasks and observe improvements in both predictive accuracy and uncertainty quantification compared to parameter/function-space VI and MCMC.