Activation function: Difference between revisions

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{{Short description|Artificial neural network node function}}

{{About||the formalism used to approximate the influence of an extracellular electrical field on neurons|activating function|a linear system’s transfer functionansferfunction|transfer function}}

{{Machine learning}}

{{Use dmy dates|date=August 2025}}

[[File:Logistic-curve.svg|thumb|Logistic activation function]]

The '''activation function''' of a node in an [[artificial neural network]] is a function that calculates the output of the node based on its individual inputs and their weights. Nontrivial problems can be solved using only a few nodes if the activation function is ''nonlinear''.<ref>{{Cite web|url=http://didattica.cs.unicam.it/lib/exe/fetch.php?media=didattica:magistrale:kebi:ay_1718:ke-11_neural_networks.pdf|title=Neural Networks, p. 7|last=Hinkelmann|first=Knut|website=University of Applied Sciences Northwestern Switzerland|access-date=6 October 2018-10-06|archive-date=6 October 2018-10-06|archive-url=https://web.archive.org/web/20181006235506/http://didattica.cs.unicam.it/lib/exe/fetch.php?media=didattica:magistrale:kebi:ay_1718:ke-11_neural_networks.pdf|url-status=dead}}</ref>

Modern activation functions include the logistic ([[Sigmoid function|sigmoid]]) function used in the 2012 [[speech recognition]] model developed by [[Geoffrey Hinton|Hinton]] et al;<ref>{{Cite journal |last1=Hinton |first1=Geoffrey |last2=Deng |first2=Li |last3=Deng |first3=Li |last4=Yu |first4=Dong |last5=Dahl |first5=George |last6=Mohamed |first6=Abdel-rahman |last7=Jaitly |first7=Navdeep |last8=Senior |first8=Andrew |last9=Vanhoucke |first9=Vincent |last10=Nguyen |first10=Patrick |last11=Sainath |first11=Tara|author11-link= Tara Sainath |last12=Kingsbury |first12=Brian |year=2012 |title=Deep Neural Networks for Acoustic Modeling in Speech Recognition |journal=IEEE Signal Processing Magazine |volume=29 |issue=6 |pages=82–97 |doi=10.1109/MSP.2012.2205597|s2cid=206485943 }}</ref> the [[ReLU]] used in the 2012 [[AlexNet]] computer vision model<ref>{{Cite journal |last1=Krizhevsky |first1=Alex |last2=Sutskever |first2=Ilya |last3=Hinton |first3=Geoffrey E. |date=2017-05-24 May 2017 |title=ImageNet classification with deep convolutional neural networks |url=https://dl.acm.org/doi/10.1145/3065386 |journal=Communications of the ACM |language=en |volume=60 |issue=6 |pages=84–90 |doi=10.1145/3065386 |issn=0001-0782}}</ref><ref>{{Cite journal |last1=King Abdulaziz University |last2=Al-johania |first2=Norah |last3=Elrefaei |first3=Lamiaa |last4=Benha University |date=2019-06-30 June 2019 |title=Dorsal Hand Vein Recognition by Convolutional Neural Networks: Feature Learning and Transfer Learning Approaches |url=http://www.inass.org/2019/2019063019.pdf |journal=International Journal of Intelligent Engineering and Systems |volume=12 |issue=3 |pages=178–191 |doi=10.22266/ijies2019.0630.19}}</ref> and in the 2015 [[Residual neural network|ResNet]] model; and the smooth version of the ReLU, the [[ReLU#Gaussian-error linear unit (GELU)|GELU]], which was used in the 2018 [[BERT (language model)|BERT]] model.<ref name="ReferenceA">{{Cite arXiv |eprint=1606.08415 |title=Gaussian Error Linear Units (GELUs) |last1=Hendrycks |first1=Dan |last2=Gimpel |first2=Kevin |year=2016 |class=cs.LG}}</ref>

==Comparison of activation functions==

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; Nonlinear: When the activation function is non-linear, then a two-layer neural network can be proven to be a universal function approximator.<ref>{{Cite journal|author1-link=George Cybenko|last=Cybenko|first=G.|date=December 1989|title=Approximation by superpositions of a sigmoidal function|journal=Mathematics of Control, Signals, and Systems|language=en|volume=2|issue=4|pages=303–314|doi=10.1007/BF02551274|bibcode=1989MCSS....2..303C |s2cid=3958369|issn=0932-4194|url=https://hal.archives-ouvertes.fr/hal-03753170/file/Cybenko1989.pdf }}</ref> This is known as the [[Universal approximation theorem|Universal Approximation Theorem]]. The identity activation function does not satisfy this property. When multiple layers use the identity activation function, the entire network is equivalent to a single-layer model.

; Range: When the range of the activation function is finite, gradient-based training methods tend to be more stable, because pattern presentations significantly affect only limited weights. When the range is infinite, training is generally more efficient because pattern presentations significantly affect most of the weights. In the latter case, smaller [[learning rate]]s are typically necessary.{{citation needed|date=January 2016}}

; Continuously differentiable: This property is desirable for enabling gradient-based optimization methods ([[Rectifier (neural networks)|ReLU]] is not continuously differentiable and has some issues with gradient-based optimization, but it is still possible) for enabling gradient-based optimization methods. The binary step activation function is not differentiable at 0, and it differentiates to 0 for all other values, so gradient-based methods can make no progress with it.<ref>{{cite book|url={{google books |plainurl=y |id=0tFmf_UKl7oC}}|title=Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms|last=Snyman|first=Jan|date=3 March 2005|publisher=Springer Science & Business Media|isbn=978-0-387-24348-1}}</ref>

These properties do not decisively influence performance, nor are they the only mathematical properties that may be useful. For instance, the strictly positive range of the [[softplus]] makes it suitable for predicting variances in [[Autoencoder#Variational autoencoder (VAE)|variational autoencoders]].

== Mathematical details ==

The most common activation functions can be divided into three categories: [[ridge function]]s, [[radial function]]s and [[fold function]]s.

An activation function <math>f</math> is '''saturating''' if <math>\lim_{|v|\to \infty} |\nabla f(v)| = 0</math>. It is '''nonsaturating''' if it is <math>\lim_{|v|\to \infty} |\nabla f(v)| \neq 0</math>. Non-saturating activation functions, such as [[ReLU]], may be better than saturating activation functions, because they are less likely to suffer from the [[vanishing gradient problem]].<ref>{{Cite journal |last1=Krizhevsky |first1=Alex |last2=Sutskever |first2=Ilya |last3=Hinton |first3=Geoffrey E. |date=2017-05-24 May 2017 |title=ImageNet classification with deep convolutional neural networks |journal=Communications of the ACM |volume=60 |issue=6 |pages=84–90 |doi=10.1145/3065386 |s2cid=195908774 |issn=0001-0782|doi-access=free }}</ref>

=== Ridge activation functions ===

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* [[logistic function|Logistic]] activation: <math>\phi(\mathbf v) = (1+\exp(-a-\mathbf v'\mathbf b))^{-1}</math>.

In [[Biological neural network|biologically inspired neural networks]], the activation function is usually an abstraction representing the rate of [[action potential]] firing in the cell.<ref>{{Cite journal|last1=Hodgkin|first1=A. L.|last2=Huxley|first2=A. F.|date=1952-08-28 August 1952|title=A quantitative description of membrane current and its application to conduction and excitation in nerve |journal=The Journal of Physiology|volume=117|issue=4|pages=500–544 |pmc=1392413|pmid=12991237|doi=10.1113/jphysiol.1952.sp004764}}</ref> In its simplest form, this function is [[Binary function|binary]]—that is, either the [[neuron]] is firing or not. Neurons also cannot fire faster than a certain rate, motivating [[sigmoid function|sigmoid]] activation functions whose range is a finite interval.

The function looks like <math>\phi(\mathbf v)=U(a + \mathbf v'\mathbf b)</math>, where <math>U</math> is the [[Heaviside step function]].

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Periodic functions can serve as activation functions. Usually the [[Sine wave|sinusoid]] is used, as any periodic function is decomposable into sinusoids by the [[Fourier transform]].<ref>{{Cite journal |last1=Sitzmann |first1=Vincent |last2=Martel |first2=Julien |last3=Bergman |first3=Alexander |last4=Lindell |first4=David |last5=Wetzstein |first5=Gordon |date=2020 |title=Implicit Neural Representations with Periodic Activation Functions |url=https://proceedings.neurips.cc/paper/2020/hash/53c04118df112c13a8c34b38343b9c10-Abstract.html |journal=Advances in Neural Information Processing Systems |publisher=Curran Associates, Inc. |volume=33 |pages=7462–7473|arxiv=2006.09661 }}</ref>

Quadratic activation maps <math>x \mapsto x^2</math>.<ref>{{Citation |last=Flake |first=Gary William |title=Square Unit Augmented Radially Extended Multilayer Perceptrons |date=1998 |work=Neural Networks: Tricks of the Trade |series=Lecture Notes in Computer Science |volume=1524 |pages=145–163 |editor-last=Orr |editor-first=Genevieve B. |url=https://link.springer.com/chapter/10.1007/3-540-49430-8_8 |access-date=5 October 2024-10-05 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/3-540-49430-8_8 |isbn=978-3-540-49430-0 |editor2-last=Müller |editor2-first=Klaus-Robert|url-access=subscription }}</ref><ref>{{Cite journal |last1=Du |first1=Simon |last2=Lee |first2=Jason |date=3 July 2018-07-03 |title=On the Power of Over-parametrization in Neural Networks with Quadratic Activation |url=https://proceedings.mlr.press/v80/du18a.html |journal=Proceedings of the 35th International Conference on Machine Learning |language=en |publisher=PMLR |pages=1329–1338|arxiv=1803.01206 }}</ref>

=== Folding activation functions ===

{{Main|Fold function}}

Folding activation functions are extensively used in the [[Pooling layer|pooling layers]] in [[convolutional neural network]]s, and in output layers of [[multiclass classification]] networks. These activations perform [[Aggregate function|aggregation]] over the inputs, such as taking the [[mean]], [[minimum]] or [[maximum]]. In multiclass classification the [[Softmax function|softmax]] activation is often used.

=== Table of activation functions ===

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| <math>C^\infty</math>

|-

| [[Rectifier (neural networks)#ELU|Exponential linear unit (ELU)]]<ref>{{Cite arXiv|last1=Clevert|first1=Djork-Arné|last2=Unterthiner|first2=Thomas|last3=Hochreiter|first3=Sepp|date=2015-11-23 November 2015|title=Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)|eprint=1511.07289 |class=cs.LG}}</ref>

| [[File:Activation elu.svg]]

| <math>\begin{cases}

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\end{cases}</math>

|-

| Scaled exponential linear unit (SELU)<ref>{{Cite journal |last1=Klambauer |first1=Günter |last2=Unterthiner |first2=Thomas |last3=Mayr |first3=Andreas |last4=Hochreiter |first4=Sepp |date=8 June 2017-06-08 |title=Self-Normalizing Neural Networks |journal=Advances in Neural Information Processing Systems |volume=30 |issue=2017 |arxiv=1706.02515}}</ref>

| [[File:Activation selu.png]]

| <math>\lambda \begin{cases}

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| <math>C^0</math>

|-

| Parametric rectified linear unit (PReLU)<ref>{{Cite arXiv |last1=He |first1=Kaiming |last2=Zhang |first2=Xiangyu |last3=Ren |first3=Shaoqing |last4=Sun |first4=Jian |date=6 February 2015-02-06 |title=Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification |eprint=1502.01852 |class=cs.CV}}</ref>

| [[File:Activation prelu.svg]]

| <math>\begin{cases}

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|title = Elsevier Pattern Recognition

|date = 2023

|volume=160 |pages=12–21 |doi=10.1016/j.neunet.2022.12.019 |pmid=36592526 |arxiv=2101.09948}}</ref>

| <math> - </math>

| <math>(-\infty,+\infty)</math>

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| <math>C^\infty</math>

|-

|Exponential Linear Sigmoid SquasHing (ELiSH)<ref>{{Citation |last1=Basirat |first1=Mina |title=The Quest for the Golden Activation Function |date=2 August 2018-08-02 |arxiv=1808.00783 |last2=Roth |first2=Peter M.}}</ref>

|[[File:Elish_activation_function.png|thumb|An image of the ELiSH activation function plotted over the range [-3, 3] with a minumum value of ~0.881 at x ~= -0.172.]]

|<math>\begin{cases}

\frac{e^x - 1}{1 + e^{-x}} & \text{if } x < 0 \\

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== Further reading ==

* {{Citation |last1=Kunc |first1=Vladimír |title=Three Decades of Activations: A Comprehensive Survey of 400 Activation Functions for Neural Networks |date=2024-02-14 February 2024 |arxiv=2402.09092 |last2=Kléma |first2=Jiří}}

* {{cite arXiv |last1=Nwankpa |first1=Chigozie |title=Activation Functions: Comparison of trends in Practice and Research for Deep Learning |date=8 November 2018-11-08 |eprint=1811.03378 |last2=Ijomah |first2=Winifred |last3=Gachagan |first3=Anthony |last4=Marshall |first4=Stephen|class=cs.LG }}

* {{cite journal |last1=Dubey |first1=Shiv Ram |last2=Singh |first2=Satish Kumar |last3=Chaudhuri |first3=Bidyut Baran |year=2022 |title=Activation functions in deep learning: A comprehensive survey and benchmark |journal=Neurocomputing |publisher=Elsevier BV |volume=503 |pages=92–108 |doi=10.1016/j.neucom.2022.06.111 |issn=0925-2312 |doi-access=free|arxiv=2109.14545 }}