Neural network activation functions

January 13, 2017 — August 2, 2021

machine learning
neural nets
Figure 1: The Rectified Linear Unit circa 1920. Don’t we long to be as cool as this guy?

There is a whole cottage industry in showing neural networks are reasonably universal function approximators with various nonlinearities as activations, under various conditions. Usually we take it as a given that the particular activation function is not too important.

Sometimes, we might like to play with the precise form of the nonlinearities, even making the nonlinearities themselves directly learnable, because some function shapes might have better approximation properties with respect to various assumptions on the learning problems, in a sense which I will not attempt to make rigorous now, vague hand-waving arguments being the whole point of deep learning.

I think a part of this field has been subsumed into the stability-of-dynamical-systems setting? Or we do not care because something-something BatchNorm?

The current default activation function is ReLU, i.e. \(x\mapsto \max\{0,x\}\), which has many nice properties. However, it does lead to piecewise linear spline approximators. One could regard that as a plus (Unser 2019) but OTOH that makes it hard to solve differential equations.

Sometimes, then, we want something different. Other classic activations such as \(x\mapsto\tanh x\) have fallen from favour, supplanted by ReLU. However, differentiable activations are useful, especially if higher-order gradients of the solution will be important. Many virtues of differentiable activation functions are documented Implicit Neural Representations with Periodic Activation Functions. Sitzmann et al. (2020) argues for \(x\mapsto\sin x\) on the basis of various handy properties. Ramachandran, Zoph, and Le (2017) advocate Swish, \(x\mapsto \frac{x}{1+\exp -x}.\)

Other fun things, SELU, the “self-normalizing” SELU (scaled exponential linear unit) Klambauer et al. (2017).

1 References

Agostinelli, Hoffman, Sadowski, et al. 2015. Learning Activation Functions to Improve Deep Neural Networks.” In Proceedings of International Conference on Learning Representations (ICLR) 2015.
Anil, Lucas, and Grosse. 2018. Sorting Out Lipschitz Function Approximation.”
Arjovsky, Shah, and Bengio. 2016. Unitary Evolution Recurrent Neural Networks.” In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48. ICML’16.
Balduzzi, Frean, Leary, et al. 2017. The Shattered Gradients Problem: If Resnets Are the Answer, Then What Is the Question? In PMLR.
Cho, and Saul. 2009. Kernel Methods for Deep Learning.” In Proceedings of the 22nd International Conference on Neural Information Processing Systems. NIPS’09.
Clevert, Unterthiner, and Hochreiter. 2016. Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs).” In Proceedings of ICLR.
Duch, and Jankowski. 1999. Survey of Neural Transfer Functions.”
Glorot, and Bengio. 2010. Understanding the Difficulty of Training Deep Feedforward Neural Networks. In Aistats.
Glorot, Bordes, and Bengio. 2011. Deep Sparse Rectifier Neural Networks. In Aistats.
Goodfellow, Warde-Farley, Mirza, et al. 2013. Maxout Networks.” In ICML (3).
Hayou, Doucet, and Rousseau. 2019. On the Impact of the Activation Function on Deep Neural Networks Training.” In Proceedings of the 36th International Conference on Machine Learning.
He, Zhang, Ren, et al. 2015a. Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification.” arXiv:1502.01852 [Cs].
———, et al. 2015b. Deep Residual Learning for Image Recognition.”
———, et al. 2016. Identity Mappings in Deep Residual Networks.” In arXiv:1603.05027 [Cs].
Hochreiter. 1998. The Vanishing Gradient Problem During Learning Recurrent Neural Nets and Problem Solutions.” International Journal of Uncertainty Fuzziness and Knowledge Based Systems.
Hochreiter, Bengio, Frasconi, et al. 2001. Gradient Flow in Recurrent Nets: The Difficulty of Learning Long-Term Dependencies.” In A Field Guide to Dynamical Recurrent Neural Networks.
Klambauer, Unterthiner, Mayr, et al. 2017. Self-Normalizing Neural Networks.” In Proceedings of the 31st International Conference on Neural Information Processing Systems.
Laurent. n.d. “The Multilinear Structure of ReLU Networks.”
Lederer. 2021. Activation Functions in Artificial Neural Networks: A Systematic Overview.” arXiv:2101.09957 [Cs, Stat].
Lee, Bahri, Novak, et al. 2018. Deep Neural Networks as Gaussian Processes.” In ICLR.
Maas, Hannun, and Ng. 2013. Rectifier Nonlinearities Improve Neural Network Acoustic Models.” In Proceedings of ICML.
Pascanu, Mikolov, and Bengio. 2013. On the Difficulty of Training Recurrent Neural Networks.” In arXiv:1211.5063 [Cs].
Rahaman, Baratin, Arpit, et al. 2019. On the Spectral Bias of Neural Networks.” arXiv:1806.08734 [Cs, Stat].
Ramachandran, Zoph, and Le. 2017. Searching for Activation Functions.” arXiv:1710.05941 [Cs].
Sitzmann, Martel, Bergman, et al. 2020. Implicit Neural Representations with Periodic Activation Functions.” arXiv:2006.09661 [Cs, Eess].
Srivastava, Greff, and Schmidhuber. 2015. Highway Networks.” In arXiv:1505.00387 [Cs].
Unser. 2019. A Representer Theorem for Deep Neural Networks.” Journal of Machine Learning Research.
Wisdom, Powers, Hershey, et al. 2016. Full-Capacity Unitary Recurrent Neural Networks.” In Advances in Neural Information Processing Systems.
Yang, and Salman. 2020. A Fine-Grained Spectral Perspective on Neural Networks.” arXiv:1907.10599 [Cs, Stat].