Activation function

Aside from their empirical performance, activation functions also have different mathematical properties:

; Nonlinear: When the activation function is non-linear, then a two-layer neural network can be proven to be a universal function approximator.{{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 }} This is known as the 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 rates are typically necessary.{{citation needed|date=January 2016}}

; Continuously differentiable: This property is desirable (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.{{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}}

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 variational autoencoders.

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

An activation function $f$ is saturating if $\backslash lim\_\{|v|\backslash to\; \backslash infty\}\; |\backslash nabla\; f(v)|\; =\; 0$. It is nonsaturating if it is $\backslash lim\_\{|v|\backslash to\; \backslash infty\}\; |\backslash nabla\; f(v)|\; \backslash neq\; 0$. 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.{{Cite journal |last1=Krizhevsky |first1=Alex |last2=Sutskever |first2=Ilya |last3=Hinton |first3=Geoffrey E. |date=2017-05-24 |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 }}

{{Main|Ridge function}}

Ridge functions are multivariate functions acting on a linear combination of the input variables. Often used examples include:{{Clarify|reason=what is a? what is v'? what is b?|date=November 2023}}

• Linear activation: $\backslash phi\; (\backslash mathbf\; v)=a\; +\backslash mathbf\; v\text{'}\backslash mathbf\; b$,
• ReLU activation: $\backslash phi\; (\backslash mathbf\; v)=\backslash max(0,a\; +\backslash mathbf\; v\text{'}\backslash mathbf\; b)$,
• Heaviside activation: $\backslash phi\; (\backslash mathbf\; v)=1\_\{a\; +\backslash mathbf\; v\text{'}\backslash mathbf\; b>0\}$,
• Logistic activation: $\backslash phi(\backslash mathbf\; v)\; =\; (1+\backslash exp(-a-\backslash mathbf\; v\text{'}\backslash mathbf\; b))^\{-1\}$.

In biologically inspired neural networks, the activation function is usually an abstraction representing the rate of action potential firing in the cell.{{Cite journal|last1=Hodgkin|first1=A. L.|last2=Huxley|first2=A. F.|date=1952-08-28|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}} In its simplest form, this function is binary—that is, either the neuron is firing or not. Neurons also cannot fire faster than a certain rate, motivating sigmoid activation functions whose range is a finite interval.

The function looks like $\backslash phi(\backslash mathbf\; v)=U(a\; +\; \backslash mathbf\; v\text{'}\backslash mathbf\; b)$, where $U$ is the Heaviside step function.

If a line has a positive slope, on the other hand, it may reflect the increase in firing rate that occurs as input current increases. Such a function would be of the form $\backslash phi(\backslash mathbf\; v)=a+\backslash mathbf\; v\text{'}\backslash mathbf\; b$.

Image:ReLU_and_GELU.svg

{{Main|Radial function}}

A special class of activation functions known as radial basis functions (RBFs) are used in RBF networks. These activation functions can take many forms, but they are usually found as one of the following functions:

• Gaussian: $\backslash ,\backslash phi(\backslash mathbf\; v)=\backslash exp\backslash left(-\backslash frac\{\backslash |\backslash mathbf\; v\; -\; \backslash mathbf\; c\backslash |^2\}\{2\backslash sigma^2\}\backslash right)$
• Multiquadratics: $\backslash ,\backslash phi(\backslash mathbf\; v)\; =\; \backslash sqrt\{\backslash |\backslash mathbf\; v\; -\; \backslash mathbf\; c\backslash |^2\; +\; a^2\}$
• Inverse multiquadratics: $\backslash ,\backslash phi(\backslash mathbf\; v)\; =\; \backslash left(\backslash |\backslash mathbf\; v-\backslash mathbf\; c\backslash |^2\; +\; a^2\backslash right)^\{-\backslash frac\{1\}\{2\}\}$
• Polyharmonic splines

where $\backslash mathbf\; c$ is the vector representing the function center and $a$ and $\backslash sigma$ are parameters affecting the spread of the radius.

Periodic functions can serve as activation functions. Usually the sinusoid is used, as any periodic function is decomposable into sinusoids by the Fourier transform.{{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 }}

Quadratic activation maps $x\; \backslash mapsto\; x^2$.{{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=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}}{{Cite journal |last1=Du |first1=Simon |last2=Lee |first2=Jason |date=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 }}

{{Main|Fold function}}

Folding activation functions are extensively used in the pooling layers in convolutional neural networks, and in output layers of multiclass classification networks. These activations perform aggregation over the inputs, such as taking the mean, minimum or maximum. In multiclass classification the softmax activation is often used.

The following table compares the properties of several activation functions that are functions of one fold {{mvar|x}} from the previous layer or layers:

|$\backslash frac\{1\}\{(1+|x|)^2\}$

|$(-\backslash infty,\; +\backslash infty)$

|$C^1$

|-

| Rectified linear unit (ReLU){{Citation|last1=Nair|first1=Vinod|last2=Hinton|first2=Geoffrey E.|date=2010|contribution=Rectified Linear Units Improve Restricted Boltzmann Machines|contribution-url=http://dl.acm.org/citation.cfm?id=3104322.3104425|title=27th International Conference on International Conference on Machine Learning|series=ICML'10|location=USA|publisher=Omnipress|pages=807–814|isbn=9781605589077}}

| File:Activation rectified linear.svg

| $\backslash begin\{align\}$

(x)^+ \doteq {} &\begin{cases}

0 & \text{if } x \le 0\\

x & \text{if } x > 0

\end{cases} \\

= {} &\max(0,x) = x \textbf{1}_{x>0}

\end{align}

| $\backslash begin\{cases\}$

0 & \text{if } x < 0\\

1 & \text{if } x > 0

\end{cases}

| $[0,\backslash infty)$

| $C^0$

|-

| Gaussian Error Linear Unit (GELU)

| File:Activation gelu.png

| $\backslash begin\{align\}$

&\frac{1}{2} x \left(1 + \text{erf}\left(\frac{x}{\sqrt{2}}\right)\right) \\

{}={} &x\Phi(x)

\end{align} where $\backslash mathrm\{erf\}$ is the gaussian error function.

| $\backslash Phi(x)\; +\; \backslash frac\; 12\; x\backslash phi(x)$ where $\backslash phi(x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; e^\{-\backslash frac\; 12\; x^2\}$ is the probability density function of standard gaussian distribution.

| $(-0.17\backslash ldots,\; \backslash infty)$

| $C^\backslash infty$

|-

| Softplus{{Cite web|url=http://proceedings.mlr.press/v15/glorot11a/glorot11a.pdf|title=Deep sparse rectifier neural networks|last1=Glorot|first1=Xavier|last2=Bordes|first2=Antoine|date=2011|website=International Conference on Artificial Intelligence and Statistics|last3=Bengio|first3=Yoshua}}

| File:Activation softplus.svg

| $\backslash ln\backslash left(1\; +\; e^x\backslash right)$

| $\backslash frac\{1\}\{1\; +\; e^\{-x\}\}$

| $(0,\backslash infty)$

| $C^\backslash infty$

|-

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

| File:Activation elu.svg

| $\backslash begin\{cases\}$

\alpha\left(e^x - 1\right) & \text{if } x \le 0\\

x & \text{if } x > 0

\end{cases}

: with parameter $\backslash alpha$

| $\backslash begin\{cases\}$

\alpha e^x & \text{if } x < 0 \\

1 & \text{if } x > 0

\end{cases}

| $(-\backslash alpha,\backslash infty)$

| $\backslash begin\{cases\}$

C^1 & \text{if } \alpha = 1 \\

C^0 & \text{otherwise}

\end{cases}

|-

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

| File:Activation selu.png

| $\backslash lambda\; \backslash begin\{cases\}$

\alpha(e^x - 1) & \text{if } x < 0\\

x & \text{if } x \ge 0

\end{cases}

:with parameters $\backslash lambda\; =\; 1.0507$ and $\backslash alpha\; =\; 1.67326$

| $\backslash lambda\; \backslash begin\{cases\}$

\alpha e^x & \text{if } x < 0\\

1 & \text{if } x \ge 0

\end{cases}

| $(-\backslash lambda\backslash alpha,\backslash infty)$

| $C^0$

|-

| Leaky rectified linear unit (Leaky ReLU){{cite journal |last1=Maas |first1=Andrew L. |last2=Hannun |first2=Awni Y. |last3=Ng |first3=Andrew Y. |s2cid=16489696 |title=Rectifier nonlinearities improve neural network acoustic models |journal=Proc. ICML |date=June 2013 |volume=30 |issue=1}}

| File:Activation prelu.svg

| $\backslash begin\{cases\}$

0.01x & \text{if } x \le 0\\

x & \text{if } x > 0

\end{cases}

| $\backslash begin\{cases\}$

0.01 & \text{if } x < 0\\

1 & \text{if } x > 0

\end{cases}

| $(-\backslash infty,\backslash infty)$

| $C^0$

|-

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

| File:Activation prelu.svg

| $\backslash begin\{cases\}$

\alpha x & \text{if } x < 0\\

x & \text{if } x \ge 0

\end{cases}

: with parameter $\backslash alpha$

| $\backslash begin\{cases\}$

\alpha & \text{if } x < 0\\

1 & \text{if } x \ge 0

\end{cases}

| $(-\backslash infty,\backslash infty)$

| $C^0$

|-

| Rectified Parametric Sigmoid Units (flexible, 5 parameters)

| File:RePSU.svg

| $\backslash alpha\; (2x\; \{1\}\_\{\; \backslash \{\; x\; \backslash geqslant\; \backslash lambda\; \backslash \}\; \}\; -\; g\_\{\backslash lambda,\; \backslash sigma,\; \backslash mu,\; \backslash beta\}(x))\; +\; (1-\backslash alpha)\; g\_\{\backslash lambda,\; \backslash sigma,\; \backslash mu,\; \backslash beta\}(x)$

where

$g\_\{\backslash lambda,\; \backslash sigma,\; \backslash mu,\; \backslash beta\}(x)\; =\; \backslash frac\{\; (x\; -\; \backslash lambda)\; \{1\}\_\{\; \backslash \{\; x\; \backslash geqslant\; \backslash lambda\; \backslash \}\; \}\; \}\{\; 1\; +\; e^\{-\; \backslash sgn(x-\backslash mu)\; \backslash left(\; \backslash frac\{\backslash vert\; x-\backslash mu\; \backslash vert\}\{\backslash sigma\}\; \backslash right)^\backslash beta\; \}\; \}$

{{Citation

|first1=Abdourrahmane M.|last1=Atto|first2=Sylvie|last2=Galichet|first3=Dominique|last3=Pastor|first4=Nicolas|last4=Méger

|contribution = On joint parameterizations of linear and nonlinear functionals in neural networks

|contribution-url = https://doi.org/10.1016/j.neunet.2022.12.019

|title = Elsevier Pattern Recognition

|date = 2023

|volume=160 |pages=12–21 |doi=10.1016/j.neunet.2022.12.019 |pmid=36592526 }}

| $-$

| $(-\backslash infty,+\backslash infty)$

| $C^0$

|-

| Sigmoid linear unit (SiLU, Sigmoid shrinkage,

{{Citation

|first1=Abdourrahmane M.|last1=Atto|first2=Dominique|last2=Pastor|first3=Grégoire|last3=Mercier

|contribution = Smooth sigmoid wavelet shrinkage for non-parametric estimation

|contribution-url = https://hal.archives-ouvertes.fr/hal-02136546/file/ICASSP_ATTO_2008.pdf

|title = 2008 IEEE International Conference on Acoustics, Speech and Signal Processing

|date = 2008

|pages=3265–3268 |doi=10.1109/ICASSP.2008.4518347|isbn=978-1-4244-1483-3 |s2cid=9959057}} SiL,{{Cite journal |arxiv = 1702.03118|title = Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning|journal = Neural Networks |volume = 107|pages = 3–11|last1 = Elfwing|first1 = Stefan|last2 = Uchibe|first2 = Eiji|last3 = Doya|first3 = Kenji|year = 2018|pmid = 29395652|doi = 10.1016/j.neunet.2017.12.012|s2cid = 6940861}} or Swish-{{zwj}}1{{Cite arXiv |eprint = 1710.05941 |title = Searching for Activation Functions|last1 = Ramachandran|first1 = Prajit|last2 = Zoph|first2 = Barret|last3 = Le|first3 = Quoc V|year = 2017|class = cs.NE}})

| File:Swish.svg

| $\backslash frac\{x\}\{1\; +\; e^\{-x\}\}$

| $\backslash frac\{1\; +\; e^\{-x\}\; +\; xe^\{-x\}\}\{\backslash left(1\; +\; e^\{-x\}\backslash right)^2\}$

| $[-0.278\backslash ldots,\; \backslash infty)$

| $C^\backslash infty$

|-

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

|File:Elish_activation_function.png

|$\backslash begin\{cases\}$

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

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

\end{cases}

|

|$[-0.881\backslash ldots,\; \backslash infty)$

|$C^1$

|-

| Gaussian

| File:Activation gaussian.svg

| $e^\{-x^2\}$

| $-2xe^\{-x^2\}$

| $(0,1]$

| $C^\backslash infty$

|-

|Sinusoid

|

|$\backslash sin\; x$

|$\backslash cos\; x$

|$[-1,\; 1]$

|$C^\backslash infty$

|}

The following table lists activation functions that are not functions of a single fold {{mvar|x}} from the previous layer or layers:

:{{note|kronecker_delta}} Here, $\backslash delta\_\{ij\}$ is the Kronecker delta.

:{{note|j}} For instance, $j$ could be iterating through the number of kernels of the previous neural network layer while $i$ iterates through the number of kernels of the current layer.

{{Main|Quantum function}}

In quantum neural networks programmed on gate-model quantum computers, based on quantum perceptrons instead of variational quantum circuits, the non-linearity of the activation function can be implemented with no need of measuring the output of each perceptron at each layer. The quantum properties loaded within the circuit such as superposition can be preserved by creating the Taylor series of the argument computed by the perceptron itself, with suitable quantum circuits computing the powers up to a wanted approximation degree. Because of the flexibility of such quantum circuits, they can be designed in order to approximate any arbitrary classical activation function.{{cite journal|doi=10.1007/s11128-022-03466-0 |issn=1570-0755 |title=Quantum activation functions for quantum neural networks|year=2022|last1=Maronese |first1=Marco|last2=Destri |first2=Claudio|last3= Prati|first3=Enrico |journal= Quantum Information Processing |volume=21|issue=4|page=128 |arxiv=2201.03700|bibcode=2022QuIP...21..128M }}

• Logistic function
• Rectifier (neural networks)
• Stability (learning theory)
• Softmax function

{{reflist|30em}}

• {{Citation |last1=Kunc |first1=Vladimír |title=Three Decades of Activations: A Comprehensive Survey of 400 Activation Functions for Neural Networks |date=2024-02-14 |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=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 }}

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