Central Limit Theorem for Bayesian Neural Network trained with Variational Inference
This provides theoretical insights into the statistical properties of variational inference methods for Bayesian neural networks, which is incremental but important for researchers in machine learning theory.
The paper tackles the problem of understanding the fluctuation behavior of Bayesian neural networks trained with variational inference by rigorously deriving Central Limit Theorems for two-layer networks in the infinite-width limit, showing that idealized and Bayes-by-Backprop schemes have similar fluctuations, while Minimal VI differs but remains more efficient due to computational gains.
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.