# Variational Inference of overparameterized Bayesian Neural Networks: a theoretical and empirical study

arXiv: https://arxiv.org/abs/2207.03859 · HTML: https://mlsanity.com/paper/2207.03859/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 57 references that are in the map.
- First posted on arXiv 8 Jul 2022.

## Builds on (most cited first, 30 of 38)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [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)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [Pitfalls of In-Domain Uncertainty Estimation and Ensembling in Deep Learning](https://mlsanity.com/paper/2002.06470/) (2020)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Practical Deep Learning with Bayesian Principles](https://mlsanity.com/paper/1906.02506/) (2019)
- [Multiplicative Normalizing Flows for Variational Bayesian Neural Networks](https://mlsanity.com/paper/1703.01961/) (2017)
- [Fast and Scalable Bayesian Deep Learning by Weight-Perturbation in Adam](https://mlsanity.com/paper/1806.04854/) (2018)
- [Functional Variational Bayesian Neural Networks](https://mlsanity.com/paper/1903.05779/) (2019)
- [Bayesian Neural Network Priors Revisited](https://mlsanity.com/paper/2102.06571/) (2021)
- [Priors in Bayesian Deep Learning: A Review](https://mlsanity.com/paper/2105.06868/) (2021)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [Noisy Natural Gradient as Variational Inference](https://mlsanity.com/paper/1712.02390/) (2017)
- [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)
- [All You Need is a Good Functional Prior for Bayesian Deep Learning](https://mlsanity.com/paper/2011.12829/) (2020)
- [The promises and pitfalls of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1811.10072/) (2018)
- [Scaling Hamiltonian Monte Carlo Inference for Bayesian Neural Networks with Symmetric Splitting](https://mlsanity.com/paper/2010.06772/) (2020)
- [Global inducing point variational posteriors for Bayesian neural networks and deep Gaussian processes](https://mlsanity.com/paper/2005.08140/) (2020)

## Built on by (most cited first, 5 of 5)

- [Scalable Stochastic Gradient Riemannian Langevin Dynamics in Non-Diagonal Metrics](https://mlsanity.com/paper/2303.05101/) (2023)
- [On permutation symmetries in Bayesian neural network posteriors: a variational perspective](https://mlsanity.com/paper/2310.10171/) (2023)
- [Nonparametric Automatic Differentiation Variational Inference with Spline Approximation](https://mlsanity.com/paper/2403.06302/) (2024)
- [Central Limit Theorem for Bayesian Neural Network trained with Variational Inference](https://mlsanity.com/paper/2406.09048/) (2024)
- [The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks](https://mlsanity.com/paper/2610.09132/) (2026)

## Abstract (arXiv)

This paper studies the Variational Inference (VI) used for training Bayesian Neural Networks (BNN) in the overparameterized regime, i.e., when the number of neurons tends to infinity. More specifically, we consider overparameterized two-layer BNN and point out a critical issue in the mean-field VI training. This problem arises from the decomposition of the lower bound on the evidence (ELBO) into two terms: one corresponding to the likelihood function of the model and the second to the Kullback-Leibler (KL) divergence between the prior distribution and the variational posterior. In particular, we show both theoretically and empirically that there is a trade-off between these two terms in the overparameterized regime only when the KL is appropriately re-scaled with respect to the ratio between the the number of observations and neurons. We also illustrate our theoretical results with numerical experiments that highlight the critical choice of this ratio.