Inartificial neural networks, theactivation function of a node 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 isnonlinear.^[1]

Modern activation functions include the logistic (sigmoid) function used in the 2012speech recognition model developed byHinton et al;^[2] theReLU used in the 2012AlexNet computer vision model^[3][4] and in the 2015ResNet model; and the smooth version of the ReLU, theGELU, which was used in the 2018BERT model.^[5]

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.^[6] This is known as theUniversal 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, smallerlearning rates are typically necessary.^{[citation needed]}

Continuously differentiable

This property is desirable for enabling gradient-based optimization methods (ReLU is not continuously differentiable and has some issues with gradient-based optimization, but it is still possible). 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.^[7]

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 thesoftplus makes it suitable for predicting variances invariational autoencoders.

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

An activation function${\displaystyle f}$ issaturating if${\displaystyle \underset{|v|\to \mathrm{\infty }}{lim}|\mathrm{\nabla }f(v)|=0}$. It isnonsaturating if${\displaystyle \underset{|v|\to \mathrm{\infty }}{lim}|\mathrm{\nabla }f(v)|\ne 0}$. Non-saturating activation functions, such asReLU, may be better than saturating activation functions, because they are less likely to suffer from thevanishing gradient problem.^[8]

Ridge functions are multivariate functions acting on a linear combination of the input variables. Often used examples include:^{[clarification needed]}

• Linear activation:${\displaystyle \varphi (\mathbf{v})=a+{\mathbf{v}}^{\prime }\mathbf{b}}$,
• ReLU activation:${\displaystyle \varphi (\mathbf{v})=max(0,a+{\mathbf{v}}^{\prime }\mathbf{b})}$,
• Heaviside activation:${\displaystyle \varphi (\mathbf{v})=1_{a+{\mathbf{v}}^{\prime }\mathbf{b}>0}}$,
• Logistic activation:${\displaystyle \varphi (\mathbf{v})=(1+\exp (-a-{\mathbf{v}}^{\prime }\mathbf{b}){)}^{-1}}$.

Inbiologically inspired neural networks, the activation function is usually an abstraction representing the rate ofaction potential firing in the cell.^[9] In its simplest form, this function isbinary—that is, either theneuron is firing or not. Neurons also cannot fire faster than a certain rate, motivatingsigmoid activation functions whose range is a finite interval.

The function looks like${\displaystyle \varphi (\mathbf{v})=U(a+{\mathbf{v}}^{\prime }\mathbf{b})}$, where${\displaystyle U}$ is theHeaviside step function.

If a line has a positiveslope, 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${\displaystyle \varphi (\mathbf{v})=a+{\mathbf{v}}^{\prime }\mathbf{b}}$.

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

• Gaussian:${\displaystyle \varphi (\mathbf{v})=\exp \left(-\frac{\Vert \mathbf{v}-\mathbf{c}{\Vert }^2}{2{\sigma }^2}\right)}$
• Multiquadratics:${\displaystyle \varphi (\mathbf{v})=\sqrt{\Vert \mathbf{v}-\mathbf{c}{\Vert }^2+a^2}}$
• Inverse multiquadratics:${\displaystyle \varphi (\mathbf{v})={\left(\Vert \mathbf{v}-\mathbf{c}{\Vert }^2+a^2\right)}^{-\frac{1}{2}}}$
• Polyharmonic splines

where${\displaystyle \mathbf{c}}$ is the vector representing the functioncenter and${\displaystyle a}$ and${\displaystyle \sigma }$ are parameters affecting the spread of the radius.

Periodic functions can serve as activation functions. Usually thesinusoid is used, as any periodic function is decomposable into sinusoids by theFourier transform.^[10]

Quadratic activation maps${\displaystyle x\mapsto x^2}$.^[11][12]

Folding activation functions are extensively used in thepooling layers inconvolutional neural networks, and in output layers ofmulticlass classification networks. These activations performaggregation over the inputs, such as taking themean,minimum ormaximum. In multiclass classification thesoftmax activation is often used.

The following table compares the properties of several activation functions that are functions of onefoldx from the previous layer or layers:

Name	Plot	Function,${\displaystyle g(x)}$	Derivative of${\displaystyle g}$,${\displaystyle g^{\prime }(x)}$	Range	Order of continuity
Identity		${\displaystyle x}$	${\displaystyle 1}$	${\displaystyle (-\mathrm{\infty },\mathrm{\infty })}$	${\displaystyle C^{\mathrm{\infty }}}$
Binary step			${\displaystyle 0}$	${\displaystyle \{0,1\}}$	${\displaystyle C^{-1}}$
Logistic, sigmoid, or soft step		${\displaystyle \sigma (x)\doteq \frac{1}{1+e^{-x}}}$	${\displaystyle g(x)(1-g(x))}$	${\displaystyle (0,1)}$	${\displaystyle C^{\mathrm{\infty }}}$
Hyperbolic tangent (tanh)		${\displaystyle \tanh (x)\doteq \frac{e^x-e^{-x}}{e^x+e^{-x}}}$	${\displaystyle 1-g(x{)}^2}$	${\displaystyle (-1,1)}$	${\displaystyle C^{\mathrm{\infty }}}$
Soboleva modified hyperbolic tangent (smht)		${\displaystyle \mathrm{smht}(x)\doteq \frac{e^{ax}-e^{-bx}}{e^{cx}+e^{-dx}}}$		${\displaystyle (-1,1)}$	${\displaystyle C^{\mathrm{\infty }}}$
Softsign		${\displaystyle \frac{x}{1+|x|}}$	${\displaystyle \frac{1}{(1+|x|{)}^2}}$	${\displaystyle (-1,1)}$	${\displaystyle C^1}$
Rectified linear unit (ReLU)[13]				${\displaystyle [0,\mathrm{\infty })}$	${\displaystyle C^0}$
Gaussian Error Linear Unit (GELU)[5]		where${\displaystyle \mathrm{e}\mathrm{r}\mathrm{f}}$ is thegaussian error function.	${\displaystyle \mathrm{\Phi }(x)+\frac{1}{2}x\varphi (x)}$ where${\displaystyle \varphi (x)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{x}^2}}$ is the probability density function of standard gaussian distribution.	${\displaystyle (-0.17\dots ,\mathrm{\infty })}$	${\displaystyle C^{\mathrm{\infty }}}$
Softplus[14]		${\displaystyle \ln \left(1+e^x\right)}$	${\displaystyle \frac{1}{1+e^{-x}}}$	${\displaystyle (0,\mathrm{\infty })}$	${\displaystyle C^{\mathrm{\infty }}}$
Exponential linear unit (ELU)[15]		with parameter${\displaystyle \alpha }$		${\displaystyle (-\alpha ,\mathrm{\infty })}$
Scaled exponential linear unit (SELU)[16]		with parameters${\displaystyle \lambda =1.0507}$ and${\displaystyle \alpha =1.67326}$		${\displaystyle (-\lambda \alpha ,\mathrm{\infty })}$	${\displaystyle C^0}$
Leaky rectified linear unit (Leaky ReLU)[17]				${\displaystyle (-\mathrm{\infty },\mathrm{\infty })}$	${\displaystyle C^0}$
Parametric rectified linear unit (PReLU)[18]		with parameter${\displaystyle \alpha }$		${\displaystyle (-\mathrm{\infty },\mathrm{\infty })}$	${\displaystyle C^0}$
Rectified Parametric Sigmoid Units (flexible, 5 parameters)		${\displaystyle \alpha (2x{1}_{\{x⩾\lambda \}}-g_{\lambda ,\sigma ,\mu ,\beta }(x))+(1-\alpha )g_{\lambda ,\sigma ,\mu ,\beta }(x)}$ where${\displaystyle g_{\lambda ,\sigma ,\mu ,\beta }(x)=\frac{(x-\lambda )1_{\{x⩾\lambda \}}}{1+e^{-\mathrm{sgn}(x-\mu ){\left(\frac{|x-\mu |}{\sigma }\right)}^{\beta }}}}$[19]	${\displaystyle -}$	${\displaystyle (-\mathrm{\infty },+\mathrm{\infty })}$	${\displaystyle C^0}$
Sigmoid linear unit (SiLU,[5] Sigmoid shrinkage,[20] SiL,[21] orSwish-‍1[22])		${\displaystyle \frac{x}{1+e^{-x}}}$	${\displaystyle \frac{1+e^{-x}+x{e}^{-x}}{{\left(1+e^{-x}\right)}^2}}$	${\displaystyle [-0.278\dots ,\mathrm{\infty })}$	${\displaystyle C^{\mathrm{\infty }}}$
Exponential Linear Sigmoid SquasHing (ELiSH)[23]				${\displaystyle [-0.881\dots ,\mathrm{\infty })}$	${\displaystyle C^1}$
Gaussian		${\displaystyle e^{-x^2}}$	${\displaystyle -2x{e}^{-x^2}}$	${\displaystyle (0,1]}$	${\displaystyle C^{\mathrm{\infty }}}$
Sinusoid		${\displaystyle \sin x}$	${\displaystyle \cos x}$	${\displaystyle [-1,1]}$	${\displaystyle C^{\mathrm{\infty }}}$

The following table lists activation functions that are not functions of a singlefoldx from the previous layer or layers:

Name	Equation,${\displaystyle g_i\left(\overrightarrow{x}\right)}$	Derivatives,${\displaystyle \frac{\mathrm{\partial }{g}_i\left(\overrightarrow{x}\right)}{\mathrm{\partial }{x}_j}}$	Range	Order of continuity
Softmax	${\displaystyle \frac{e^{x_i}}{\sum _{j=1}^J{e}^{x_j}}}$    fori = 1, …,J	${\displaystyle g_i\left(\overrightarrow{x}\right)\left({\delta }_{ij}-g_j\left(\overrightarrow{x}\right)\right)}$[1][2]	${\displaystyle (0,1)}$	${\displaystyle C^{\mathrm{\infty }}}$
Maxout[24]	${\displaystyle \underset{i}{max}{x}_i}$		${\displaystyle (-\mathrm{\infty },\mathrm{\infty })}$	${\displaystyle C^0}$

^ Here,${\displaystyle {\delta }_{ij}}$ is theKronecker delta.

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

Inquantum neural networks programmed on gate-modelquantum 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 eachperceptron at each layer. The quantum properties loaded within the circuit such assuperposition can be preserved by creating theTaylor 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.^[25]

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

• Kunc, Vladimír; Kléma, Jiří (14 February 2024),Three Decades of Activations: A Comprehensive Survey of 400 Activation Functions for Neural Networks,arXiv:2402.09092
• Nwankpa, Chigozie; Ijomah, Winifred; Gachagan, Anthony; Marshall, Stephen (8 November 2018). "Activation Functions: Comparison of trends in Practice and Research for Deep Learning".arXiv:1811.03378 [cs.LG].
• Dubey, Shiv Ram; Singh, Satish Kumar; Chaudhuri, Bidyut Baran (2022)."Activation functions in deep learning: A comprehensive survey and benchmark".Neurocomputing.503. Elsevier BV:92–108.arXiv:2109.14545.doi:10.1016/j.neucom.2022.06.111.ISSN 0925-2312.

• Artificial neural networks

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