Deep ensembles are a powerful tool in machine learning, improving both model performance and uncertainty calibration. While ensembles are typically formed by training and tuning models individually, evidence suggests that jointly tuning the ensemble can lead to better performance. This paper investigates the impact of jointly tuning weight decay, temperature scaling, and early stopping on both predictive performance and uncertainty quantification. Additionally, we propose a partially overlapping holdout strategy as a practical compromise between enabling joint evaluation and maximizing the use of data for training. Our results demonstrate that jointly tuning the ensemble generally matches or improves performance, with significant variation in effect size across different tasks and metrics. We highlight the trade-offs between individual and joint optimization in deep ensemble training, with the overlapping holdout strategy offering an attractive practical solution. We believe our findings provide valuable insights and guidance for practitioners looking to optimize deep ensemble models.

A memory efficient approach to ensembling neural networks is to share most weights among the ensembled models by means of a single reference network. We refer to this strategy as Embedded Ensembling (EE); its particular examples are BatchEnsembles and Monte-Carlo dropout ensembles. In this paper we perform a systematic theoretical and empirical analysis of embedded ensembles with different number of models. Theoretically, we use a Neural-Tangent-Kernel-based approach to derive the wide network limit of the gradient descent dynamics. In this limit, we identify two ensemble regimes - independent and collective - depending on the architecture and initialization strategy of ensemble models. We prove that in the independent regime the embedded ensemble behaves as an ensemble of independent models. We confirm our theoretical prediction with a wide range of experiments with finite networks, and further study empirically various effects such as transition between the two regimes, scaling of ensemble performance with the network width and number of models, and dependence of performance on a number of architecture and hyperparameter choices.

Ensemble methods which average over multiple neural network predictions are a simple approach to improve a model's calibration and robustness. Similarly, data augmentation techniques, which encode prior information in the form of invariant feature transformations, are effective for improving calibration and robustness. In this paper, we show a surprising pathology: combining ensembles and data augmentation can harm model calibration. This leads to a trade-off in practice, whereby improved accuracy by combining the two techniques comes at the expense of calibration. On the other hand, selecting only one of the techniques ensures good uncertainty estimates at the expense of accuracy. We investigate this pathology and identify a compounding under-confidence among methods which marginalize over sets of weights and data augmentation techniques which soften labels. Finally, we propose a simple correction, achieving the best of both worlds with significant accuracy and calibration gains over using only ensembles or data augmentation individually. Applying the correction produces new state-of-the art in uncertainty calibration across CIFAR-10, CIFAR-100, and ImageNet.

Recent approaches to efficiently ensemble neural networks have shown that strong robustness and uncertainty performance can be achieved with a negligible gain in parameters over the original network. However, these methods still require multiple forward passes for prediction, leading to a significant computational cost. In this work, we show a surprising result: the benefits of using multiple predictions can be achieved `for free' under a single model's forward pass. In particular, we show that, using a multi-input multi-output (MIMO) configuration, one can utilize a single model's capacity to train multiple subnetworks that independently learn the task at hand. By ensembling the predictions made by the subnetworks, we improve model robustness without increasing compute. We observe a significant improvement in negative log-likelihood, accuracy, and calibration error on CIFAR10, CIFAR100, ImageNet, and their out-of-distribution variants compared to previous methods.

Bayesian neural networks (BNNs) demonstrate promising success in improving the robustness and uncertainty quantification of modern deep learning. However, they generally struggle with underfitting at scale and parameter efficiency. On the other hand, deep ensembles have emerged as alternatives for uncertainty quantification that, while outperforming BNNs on certain problems, also suffer from efficiency issues. It remains unclear how to combine the strengths of these two approaches and remediate their common issues. To tackle this challenge, we propose a rank-1 parameterization of BNNs, where each weight matrix involves only a distribution on a rank-1 subspace. We also revisit the use of mixture approximate posteriors to capture multiple modes, where unlike typical mixtures, this approach admits a significantly smaller memory increase (e.g., only a 0.4% increase for a ResNet-50 mixture of size 10). We perform a systematic empirical study on the choices of prior, variational posterior, and methods to improve training. For ResNet-50 on ImageNet, Wide ResNet 28-10 on CIFAR-10/100, and an RNN on MIMIC-III, rank-1 BNNs achieve state-of-the-art performance across log-likelihood, accuracy, and calibration on the test sets and out-of-distribution variants.

Modern deep neural networks can produce badly calibrated predictions, especially when train and test distributions are mismatched. Training an ensemble of models and averaging their predictions can help alleviate these issues. We propose a simple technique to improve calibration, using a different data augmentation for each ensemble member. We additionally use the idea of `mixing' un-augmented and augmented inputs to improve calibration when test and training distributions are the same. These simple techniques improve calibration and accuracy over strong baselines on the CIFAR10 and CIFAR100 benchmarks, and out-of-domain data from their corrupted versions.

Ensembles, where multiple neural networks are trained individually and their predictions are averaged, have been shown to be widely successful for improving both the accuracy and predictive uncertainty of single neural networks. However, an ensemble's cost for both training and testing increases linearly with the number of networks, which quickly becomes untenable. In this paper, we propose BatchEnsemble, an ensemble method whose computational and memory costs are significantly lower than typical ensembles. BatchEnsemble achieves this by defining each weight matrix to be the Hadamard product of a shared weight among all ensemble members and a rank-one matrix per member. Unlike ensembles, BatchEnsemble is not only parallelizable across devices, where one device trains one member, but also parallelizable within a device, where multiple ensemble members are updated simultaneously for a given mini-batch. Across CIFAR-10, CIFAR-100, WMT14 EN-DE/EN-FR translation, and out-of-distribution tasks, BatchEnsemble yields competitive accuracy and uncertainties as typical ensembles; the speedup at test time is 3X and memory reduction is 3X at an ensemble of size 4. We also apply BatchEnsemble to lifelong learning, where on Split-CIFAR-100, BatchEnsemble yields comparable performance to progressive neural networks while having a much lower computational and memory costs. We further show that BatchEnsemble can easily scale up to lifelong learning on Split-ImageNet which involves 100 sequential learning tasks.