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

• URL: http://arxiv.org/abs/2307.04779v1
• Date: Mon, 10 Jul 2023 07:50:09 GMT
• Title: Law of Large Numbers for Bayesian two-layer Neural Network trained with Variational Inference
• Authors: Arnaud Descours (LMBP), Tom Huix (X), Arnaud Guillin (LMBP), Manon Michel (LMBP), \'Eric Moulines (X), Boris Nectoux (LMBP)
• Abstract summary: We provide a rigorous analysis of training by variational inference (VI) of Bayesian neural networks. We prove a law of large numbers for three different training schemes. An important result is that all methods converge to the same mean-field limit.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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