Central Limit Theorem for Bayesian Neural Network trained with Variational Inference

TL;DR

This paper establishes Central Limit Theorems for Bayesian neural networks trained with variational inference, comparing different training schemes and analyzing their fluctuation behaviors and computational efficiencies.

Contribution

It rigorously derives CLTs for Bayesian neural networks trained by various variational inference schemes, revealing differences in their fluctuation behaviors and efficiency.

Findings

Idealized and Bayes-by-Backprop schemes have similar fluctuation behavior.

Minimal VI scheme exhibits different fluctuation behavior from the other schemes.

Minimal VI is more computationally efficient despite larger variances.

Abstract

In this paper, we rigorously derive Central Limit Theorems (CLT) for Bayesian two-layerneural networks in the infinite-width limit and trained by variational inference on a regression task. The different networks are trained via different maximization schemes of the regularized evidence lower bound: (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 computationally cheaper algorithm named Minimal VI. The latter was recently introduced by leveraging the information obtained at the level of the mean-field limit. Laws of large numbers are already rigorously proven for the three schemes that admits the same asymptotic limit. By deriving CLT, this work shows that the idealized and Bayes-by-Backprop schemes have similar fluctuation behavior, that is different from the Minimal VI one. Numerical experiments then illustrate that the Minimal VI scheme is still more efficient, in spite of bigger variances, thanks to its important gain in computational complexity.