# Controllable Diversity in Normalization-Based Implicit Ensembles via Softmax-Temperature Modulation

arXiv: https://arxiv.org/abs/2607.23860 · HTML: https://mlsanity.com/paper/2607.23860/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 54 of its 62 references that are in the map.
- First posted on arXiv 26 Jul 2026.

## Builds on (most cited first, 30 of 54)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Evidential Deep Learning to Quantify Classification Uncertainty](https://mlsanity.com/paper/1806.01768/) (2018)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Regularizing Neural Networks by Penalizing Confident Output Distributions](https://mlsanity.com/paper/1701.06548/) (2017)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [BatchEnsemble: An Alternative Approach to Efficient Ensemble and Lifelong Learning](https://mlsanity.com/paper/2002.06715/) (2020)
- [Improving Adversarial Robustness via Promoting Ensemble Diversity](https://mlsanity.com/paper/1901.08846/) (2019)
- [Understanding Measures of Uncertainty for Adversarial Example Detection](https://mlsanity.com/paper/1803.08533/) (2018)
- [Training independent subnetworks for robust prediction](https://mlsanity.com/paper/2010.06610/) (2020)
- [Efficient and Scalable Bayesian Neural Nets with Rank-1 Factors](https://mlsanity.com/paper/2005.07186/) (2020)
- [AdaShare: Learning What To Share For Efficient Deep Multi-Task Learning](https://mlsanity.com/paper/1911.12423/) (2019)
- [Epistemic Neural Networks](https://mlsanity.com/paper/2107.08924/) (2021)

## Abstract (arXiv)

Deep ensembles provide the most reliable uncertainty estimates in deep learning, but their cost grows linearly with the number of members. Implicit ensembles lower this cost by sharing a single backbone across members. Member diversity is a primary determinant of ensemble quality, yet no implicit ensemble can shape it during training; existing methods fix it at initialisation or build it into the architecture. We introduce σN-Ens, a normalisation-based implicit ensemble that treats each member as a task in a multi-task architecture and modulates the shared backbone through sigmoid-bounded scalers. We also introduce a softmax-temperature regulariser, which shapes the equilibrium level of sharing between members and traces the accuracy-calibration frontier. Because only normalisation layers are replicated, the mechanism can wrap convolutional and transformer backbones alike, also allowing pretrained models to be adapted through a short fine-tune. We frame the epistemic uncertainty such an ensemble expresses as modulation uncertainty, and explain why its calibration holds under input corruption, and why its out-of-distribution detection is weaker. Our method is evaluated across ResNets and transformers on CIFAR-10/100, ImageNet and SST-2. σN-Ens matches or outperforms deep ensembles at a fraction of their parameter cost, scales with ensemble size where partitioning methods collapse, and maintains calibration under distribution shift.