  We provide a rigorous analysis of training by variational inference (VI) of
Bayesian neural networks in the two-layer and infinite-width case. We consider
a regression problem with a regularized evidence lower bound (ELBO) which is
decomposed into the expected log-likelihood of the data and the
Kullback-Leibler (KL) divergence between the a priori distribution and the
variational posterior. With an appropriate weighting of the KL, we prove a law
of large numbers for three different training schemes: (i) the idealized case
with exact estimation of a multiple Gaussian integral from the
reparametrization trick, (ii) a minibatch scheme using Monte Carlo sampling,
commonly known as Bayes by Backprop, and (iii) a new and computationally
cheaper algorithm which we introduce as Minimal VI. An important result is that
all methods converge to the same mean-field limit. Finally, we illustrate our
results numerically and discuss the need for the derivation of a central limit
theorem.
