Law of Large Numbers for Bayesian two-layer Neural Network trained with Variational Inference
Abstract
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.
Keywords
Cite
@article{arxiv.2307.04779,
title = {Law of Large Numbers for Bayesian two-layer Neural Network trained with Variational Inference},
author = {Arnaud Descours and Tom Huix and Arnaud Guillin and Manon Michel and Éric Moulines and Boris Nectoux},
journal= {arXiv preprint arXiv:2307.04779},
year = {2023}
}