# Efficient Activation Function Optimization through Surrogate Modeling

arXiv: https://arxiv.org/abs/2301.05785 · HTML: https://mlsanity.com/paper/2301.05785/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 69 references that are in the map.
- First posted on arXiv 13 Jan 2023.

## Builds on (most cited first, 30 of 43)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [CutMix: Regularization Strategy to Train Strong Classifiers with Localizable Features](https://mlsanity.com/paper/1905.04899/) (2019)
- [RandAugment: Practical automated data augmentation with a reduced search space](https://mlsanity.com/paper/1909.13719/) (2019)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Neural Architecture Search with Reinforcement Learning](https://mlsanity.com/paper/1611.01578/) (2016)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [AutoAugment: Learning Augmentation Policies from Data](https://mlsanity.com/paper/1805.09501/) (2018)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [CoAtNet: Marrying Convolution and Attention for All Data Sizes](https://mlsanity.com/paper/2106.04803/) (2021)
- [Neural Architecture Search without Training](https://mlsanity.com/paper/2006.04647/) (2020)
- [Separable Self-attention for Mobile Vision Transformers](https://mlsanity.com/paper/2206.02680/) (2022)
- [A Kronecker-factored approximate Fisher matrix for convolution layers](https://mlsanity.com/paper/1602.01407/) (2016)
- [Mish: A Self Regularized Non-Monotonic Activation Function](https://mlsanity.com/paper/1908.08681/) (2019)
- [Activation Functions: Comparison of trends in Practice and Research for Deep Learning](https://mlsanity.com/paper/1811.03378/) (2018)
- [Fisher-Rao Metric, Geometry, and Complexity of Neural Networks](https://mlsanity.com/paper/1711.01530/) (2017)
- [Universal Statistics of Fisher Information in Deep Neural Networks: Mean Field Approach](https://mlsanity.com/paper/1806.01316/) (2018)
- [How Powerful are Performance Predictors in Neural Architecture Search?](https://mlsanity.com/paper/2104.01177/) (2021)
- [A survey on modern trainable activation functions](https://mlsanity.com/paper/2005.00817/) (2020)
- [A Survey on Neural Architecture Search](https://mlsanity.com/paper/1905.01392/) (2019)
- [Pad'e Activation Units: End-to-end Learning of Flexible Activation Functions in Deep Networks](https://mlsanity.com/paper/1907.06732/) (2019)
- [Learning Activation Functions to Improve Deep Neural Networks](https://mlsanity.com/paper/1412.6830/) (2014)
- [Catastrophic Fisher Explosion: Early Phase Fisher Matrix Impacts Generalization](https://mlsanity.com/paper/2012.14193/) (2020)
- [Evolving Normalization-Activation Layers](https://mlsanity.com/paper/2004.02967/) (2020)

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

- [Activation Function Design Sustains Plasticity in Continual Learning](https://mlsanity.com/paper/2509.22562/) (2025)
- [Efficient Search for Customized Activation Functions with Gradient Descent](https://mlsanity.com/paper/2408.06820/) (2024)
- [Beyond Gaussian Initializations: Signal Preserving Weight Initialization for Odd-Sigmoid Activations](https://mlsanity.com/paper/2509.23085/) (2025)
- [Neural Architecture Search for global multi-step Forecasting of Energy Production Time Series](https://mlsanity.com/paper/2511.00035/) (2025)
- [A Composite Activation Function for Learning Stable Binary Representations](https://mlsanity.com/paper/2605.11558/) (2026)

## Abstract (arXiv)

Carefully designed activation functions can improve the performance of neural networks in many machine learning tasks. However, it is difficult for humans to construct optimal activation functions, and current activation function search algorithms are prohibitively expensive. This paper aims to improve the state of the art through three steps: First, the benchmark datasets Act-Bench-CNN, Act-Bench-ResNet, and Act-Bench-ViT were created by training convolutional, residual, and vision transformer architectures from scratch with 2,913 systematically generated activation functions. Second, a characterization of the benchmark space was developed, leading to a new surrogate-based method for optimization. More specifically, the spectrum of the Fisher information matrix associated with the model's predictive distribution at initialization and the activation function's output distribution were found to be highly predictive of performance. Third, the surrogate was used to discover improved activation functions in several real-world tasks, with a surprising finding: a sigmoidal design that outperformed all other activation functions was discovered, challenging the status quo of always using rectifier nonlinearities in deep learning. Each of these steps is a contribution in its own right; together they serve as a practical and theoretical foundation for further research on activation function optimization.