  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.
