# On Uniform, Bayesian, and PAC-Bayesian Deep Ensembles

arXiv: https://arxiv.org/abs/2406.05469 · HTML: https://mlsanity.com/paper/2406.05469/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 44 of its 86 references that are in the map.
- First posted on arXiv 8 Jun 2024.

## Builds on (most cited first, 30 of 44)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Snapshot Ensembles: Train 1, get M for free](https://mlsanity.com/paper/1704.00109/) (2017)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Pitfalls of In-Domain Uncertainty Estimation and Ensembling in Deep Learning](https://mlsanity.com/paper/2002.06470/) (2020)
- [Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning](https://mlsanity.com/paper/1902.03932/) (2019)
- [Evaluating Scalable Bayesian Deep Learning Methods for Robust Computer Vision](https://mlsanity.com/paper/1906.01620/) (2019)
- [Hyperparameter Ensembles for Robustness and Uncertainty Quantification](https://mlsanity.com/paper/2006.13570/) (2020)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [Why M Heads are Better than One: Training a Diverse Ensemble of Deep Networks](https://mlsanity.com/paper/1511.06314/) (2015)
- [Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1512.07666/) (2015)
- [Bayesian Neural Network Priors Revisited](https://mlsanity.com/paper/2102.06571/) (2021)
- [Tighter risk certificates for neural networks](https://mlsanity.com/paper/2007.12911/) (2020)
- [Repulsive Deep Ensembles are Bayesian](https://mlsanity.com/paper/2106.11642/) (2021)
- [Uncertainty in Neural Networks: Approximately Bayesian Ensembling](https://mlsanity.com/paper/1810.05546/) (2018)
- [Uncertainty Baselines: Benchmarks for Uncertainty & Robustness in Deep Learning](https://mlsanity.com/paper/2106.04015/) (2021)
- [A Strongly Quasiconvex PAC-Bayesian Bound](https://mlsanity.com/paper/1608.05610/) (2016)
- [A statistical theory of cold posteriors in deep neural networks](https://mlsanity.com/paper/2008.05912/) (2020)
- [Risk Bounds for the Majority Vote: From a PAC-Bayesian Analysis to a Learning Algorithm](https://mlsanity.com/paper/1503.08329/) (2015)
- [Benchmarking Bayesian Deep Learning on Diabetic Retinopathy Detection Tasks](https://mlsanity.com/paper/2211.12717/) (2022)
- [On Uncertainty, Tempering, and Data Augmentation in Bayesian Classification](https://mlsanity.com/paper/2203.16481/) (2022)

## Abstract (arXiv)

It is common practice to combine deep neural networks into ensembles. These deep ensembles can benefit from the cancellation of errors effect: Errors by ensemble members may average out, leading to better generalization performance than each individual network. Bayesian neural networks learn a posterior distribution over model parameters, and sampling and weighting networks according to this posterior yields an ensemble model referred to as a Bayes ensemble. This study reviews the argument that neither the sampling nor the weighting in Bayes ensembles are particularly well suited for increasing generalization performance, as they do not support the cancellation of errors effect. In contrast, we show that a weighted average of models, where the weights are optimized by minimizing a second-order PAC-Bayesian generalization bound, can improve generalization. It is crucial that the optimization takes correlations between models into account. This can be achieved by minimizing the tandem loss, which requires hold-out data for estimating error correlations. The tandem loss based PAC-Bayesian weighting increases robustness against correlated models and models with lower performance in an ensemble. This allows us to safely add several models from the same learning process to an ensemble, instead of using early-stopping for selecting a single weight configuration. Our experiments provide further evidence that state-of-the-art intricate Bayes ensembles do not outperform simple uniformly weighted deep ensembles in terms of classification accuracy. Additionally, we show that these Bayes ensembles cannot match the performance of deep ensembles weighted by optimizing the tandem loss, which additionally provides nonvacuous rigorous generalization guarantees.