Central limit theorem for bayesian neural network trained with variational inference
Descours, Arnaud Huix, Tom Moulines, Eric Guillin, Arnaud Michel, Manon Nectoux, Boris
Descours, Arnaud
Huix, Tom
Moulines, Eric
Guillin, Arnaud
Michel, Manon
Nectoux, Boris
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Machine Learning
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English
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Abstract
In this paper, we rigorously derive Central Limit Theorems (CLT) for Bayesian two-layer neural 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.
Citation
A. Descours, T. Huix, E. Moulines, A. Guillin, M. Michel, B. Nectoux, "Central limit theorem for bayesian neural network trained with variational inference," Stochastics and Partial Differential Equations: Analysis and Computations, pp. 1-57, 2026, https://doi.org/10.1007/s40072-025-00399-4.
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Stochastics and Partial Differential Equations: Analysis and Computations
Conference
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46 Information and Computing Sciences, 4611 Machine Learning, 49 Mathematical Sciences, 4905 Statistics
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Springer Nature