Wide Mean-Field Variational Bayesian Neural Networks Ignore the Data
This work addresses a pathological behavior in variational inference for Bayesian neural networks, showing over-regularization issues that affect practitioners using these methods.
The authors proved that as the number of hidden units increases to infinity, the function-space posterior mean under mean-field variational inference in Bayesian neural networks converges to zero, ignoring the data, unlike the true posterior which becomes a Gaussian process.
Variational inference enables approximate posterior inference of the highly over-parameterized neural networks that are popular in modern machine learning. Unfortunately, such posteriors are known to exhibit various pathological behaviors. We prove that as the number of hidden units in a single-layer Bayesian neural network tends to infinity, the function-space posterior mean under mean-field variational inference actually converges to zero, completely ignoring the data. This is in contrast to the true posterior, which converges to a Gaussian process. Our work provides insight into the over-regularization of the KL divergence in variational inference.