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arXiv · 2307.04779

Law of Large Numbers for Bayesian two-layer Neural Network trained with Variational Inference

Arnaud Descours·Tom Huix·Arnaud Guillin·Manon Michel·Éric Moulines·Boris Nectoux
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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.

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Arnaud Descours, Tom Huix, Arnaud Guillin, Manon Michel, Éric Moulines, Boris Nectoux. 2023-07-10. Law of Large Numbers for Bayesian two-layer Neural Network trained with Variational Inference. https://arxiv.org/abs/2307.04779

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