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

arXiv: https://arxiv.org/abs/2307.04779 · HTML: https://mlsanity.com/paper/2307.04779/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 10 Jul 2023.

## Builds on (most cited first, 18 of 18)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Fast and Scalable Bayesian Deep Learning by Weight-Perturbation in Adam](https://mlsanity.com/paper/1806.04854/) (2018)
- [A Systematic Comparison of Bayesian Deep Learning Robustness in Diabetic Retinopathy Tasks](https://mlsanity.com/paper/1912.10481/) (2019)
- [Uncertainty Quantification with Statistical Guarantees in End-to-End Autonomous Driving Control](https://mlsanity.com/paper/1909.09884/) (2019)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Scaling Hamiltonian Monte Carlo Inference for Bayesian Neural Networks with Symmetric Splitting](https://mlsanity.com/paper/2010.06772/) (2020)
- [Wide Mean-Field Bayesian Neural Networks Ignore the Data](https://mlsanity.com/paper/2202.11670/) (2022)
- [Quantitative Propagation of Chaos for SGD in Wide Neural Networks](https://mlsanity.com/paper/2007.06352/) (2020)
- [Infinite-width limit of deep linear neural networks](https://mlsanity.com/paper/2211.16980/) (2022)
- [On properties of the Generalized Wasserstein distance](https://mlsanity.com/paper/1304.7014/) (2013)
- [Control to flocking of the kinetic Cucker-Smale model](https://mlsanity.com/paper/1411.4687/) (2014)

## Built on by (most cited first, 1 of 1)

- [Central Limit Theorem for Bayesian Neural Network trained with Variational Inference](https://mlsanity.com/paper/2406.09048/) (2024)

## Abstract (arXiv)

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.