Activation function
Artificial neural network node function
title: "Activation function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["artificial-neural-networks"] description: "Artificial neural network node function" topic_path: "general/artificial-neural-networks" source: "https://en.wikipedia.org/wiki/Activation_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Artificial neural network node function ::
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::figure[src="https://upload.wikimedia.org/wikipedia/commons/8/88/Logistic-curve.svg" caption="Logistic activation function"] ::
In artificial neural networks, the activation 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 is nonlinear.
Modern activation functions include the logistic (sigmoid) function used in the 2012 speech recognition model developed by Hinton et al; the ReLU used in the 2012 AlexNet computer vision model and in the 2015 ResNet model; and the smooth version of the ReLU, the GELU, which was used in the 2018 BERT model.
Comparison of activation functions
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. 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. ; 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.
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.
Mathematical details
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 \lim_{|v|\to \infty} |\nabla f(v)| = 0. It is nonsaturating if \lim_{|v|\to \infty} |\nabla f(v)| \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.
Ridge activation functions
Main article: Ridge function
Ridge functions are multivariate functions acting on a linear combination of the input variables. Often used examples include:
- Linear activation: \phi (\mathbf v)=a +\mathbf v'\mathbf b,
- ReLU activation: \phi (\mathbf v)=\max(0,a +\mathbf v'\mathbf b),
- Heaviside activation: \phi (\mathbf v)=1_{a +\mathbf v'\mathbf b0},
- Logistic activation: \phi(\mathbf v) = (1+\exp(-a-\mathbf v'\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. 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. This biological limitation motivates the use of sigmoid activation functions, whose output range is confined to a finite interval.
The function looks like \phi(\mathbf v)=U(a + \mathbf v'\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 \phi(\mathbf v)=a+\mathbf v'\mathbf b.
::figure[src="https://upload.wikimedia.org/wikipedia/commons/4/42/ReLU_and_GELU.svg" caption="Rectified linear unit and Gaussian error linear unit activation functions"] ::
Radial activation functions
Main article: 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: ,\phi(\mathbf v)=\exp\left(-\frac{|\mathbf v - \mathbf c|^2}{2\sigma^2}\right)
- Multiquadratics: ,\phi(\mathbf v) = \sqrt{|\mathbf v - \mathbf c|^2 + a^2}
- Inverse multiquadratics: ,\phi(\mathbf v) = \left(|\mathbf v-\mathbf c|^2 + a^2\right)^{-\frac{1}{2}}
- Polyharmonic splines where \mathbf c is the vector representing the function center and a and \sigma are parameters affecting the spread of the radius.
Other examples
Periodic functions can serve as activation functions. Usually the sinusoid is used, as any periodic function is decomposable into sinusoids by the Fourier transform.
Quadratic activation maps x \mapsto x^2.
Folding activation functions
Main article: 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.
Table of activation functions
The following table compares the properties of several activation functions that are functions of one fold x from the previous layer or layers: ::data[format=table]
| Name | Plot | Function, g(x) | Derivative of g, g'(x) | Range | Order of continuity |
|---|---|---|---|---|---|
| Identity | [[File:Activation identity.svg]] | x | 1 | (-\infty,\infty) | C^\infty |
| Binary step | [[File:Activation binary step.svg]] | \begin{cases} | 0 | {0,1} | C^{-1} |
| Logistic, sigmoid, or softstep | [[File:Activation logistic.svg]] | \sigma(x) \doteq \frac{1}{1 + e^{-x}} | g(x)(1 - g(x)) | (0,1) | C^\infty |
| Hyperbolic tangent (tanh) | [[File:Activation tanh.svg]] | \tanh(x) \doteq \frac{e^x - e^{-x}}{e^x + e^{-x}} | 1 - g(x)^2 | (-1,1) | C^\infty |
| Soboleva modified hyperbolic tangent (smht) | [[File:SMHTAF size.png]] | \operatorname{smht}(x) \doteq \frac {e^{ax} - e^{-bx}} {e^{cx} + e^{-dx}} | (-1,1) | C^\infty | |
| Softsign | \frac{x}{1+ | x | } | \frac{1}{(1+ | |
| Rectified linear unit (ReLU) | [[File:Activation rectified linear.svg]] | \begin{align} | \begin{cases} | [0,\infty) | C^0 |
| Gaussian Error Linear Unit (GELU) | [[File:Activation gelu.png | Visualization of the Gaussian Error Linear Unit (GELU) | 120px]] | \begin{align} | \Phi(x) + \frac 12 x\phi(x) where \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac 12 x^2} is the probability density function of standard gaussian distribution. |
| Softplus | [[File:Activation softplus.svg]] | \ln\left(1 + e^x\right) | \frac{1}{1 + e^{-x}} | (0,\infty) | C^\infty |
| Exponential linear unit (ELU) | [[File:Activation elu.svg]] | \begin{cases} | \begin{cases} | (-\alpha,\infty) | \begin{cases} |
| Scaled exponential linear unit (SELU) | [[File:Activation selu.png]] | \lambda \begin{cases} | \lambda \begin{cases} | (-\lambda\alpha,\infty) | C^0 |
| Leaky rectified linear unit (Leaky ReLU) | [[File:Activation prelu.svg]] | \begin{cases} | \begin{cases} | (-\infty,\infty) | C^0 |
| Parametric rectified linear unit (PReLU) | [[File:Activation prelu.svg]] | \begin{cases} | \begin{cases} | (-\infty,\infty) | C^0 |
| Rectified Parametric Sigmoid Units (flexible, 5 parameters) | [[File:RePSU.svg | 120px]] | \alpha (2x {1}{ { x \geqslant \lambda } } - g{\lambda, \sigma, \mu, \beta}(x)) + (1-\alpha) g_{\lambda, \sigma, \mu, \beta}(x) | last1=Atto | first2=Sylvie |
| Sigmoid linear unit (SiLU, Sigmoid shrinkage, | last1=Atto | first2=Dominique | last2=Pastor | first3=Grégoire | last3=Mercier |
| Exponential Linear Sigmoid SquasHing (ELiSH) | [[File:Elish_activation_function.png | 100px]] | \begin{cases} | \begin{cases} \frac{2e^{2x}+e^{3x}-e^x}{e^{2x}+2e^x+1} & \text{if }x | [-0.881\ldots, \infty) |
| Gaussian | [[File:Activation gaussian.svg]] | e^{-x^2} | -2xe^{-x^2} | (0,1] | C^\infty |
| Sinusoid | \sin x | \cos x | [-1, 1] | C^\infty | |
| :: |
The following table lists activation functions that are not functions of a single fold x from the previous layer or layers:
::data[format=table]
| Name | Equation, g_i\left(\vec{x}\right) | Derivatives, \frac{\partial g_i\left(\vec{x}\right)}{\partial x_j} | Range | Order of continuity |
|---|---|---|---|---|
| Softmax | i}} = 1, …, J | g_i\left(\vec{x}\right)\left(\delta_{ij} - g_j\left(\vec{x}\right)\right) | (0,1) | C^\infty |
| Maxout | \max_i x_i | \begin{cases} | (-\infty,\infty) | C^0 |
| :: |
: Here, \delta_{ij} is the Kronecker delta. : 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.
Quantum activation functions
Main article: 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.
References
References
- Hinkelmann, Knut. "Neural Networks, p. 7".
- (2012). "Deep Neural Networks for Acoustic Modeling in Speech Recognition". IEEE Signal Processing Magazine.
- (24 May 2017). "ImageNet classification with deep convolutional neural networks". Communications of the ACM.
- (30 June 2019). "Dorsal Hand Vein Recognition by Convolutional Neural Networks: Feature Learning and Transfer Learning Approaches". International Journal of Intelligent Engineering and Systems.
- (2016). "Gaussian Error Linear Units (GELUs)".
- Cybenko, G.. (December 1989). "Approximation by superpositions of a sigmoidal function". Mathematics of Control, Signals, and Systems.
- Snyman, Jan. (3 March 2005). ["Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms"]({{google books). Springer Science & Business Media.
- (24 May 2017). "ImageNet classification with deep convolutional neural networks". Communications of the ACM.
- (28 August 1952). "A quantitative description of membrane current and its application to conduction and excitation in nerve". The Journal of Physiology.
- Gerstein, G. L.. (1964). "Representation of firing activity by nerve cell ensembles". IEEE Transactions on Biomedical Engineering.
- (2020). "Implicit Neural Representations with Periodic Activation Functions". Curran Associates, Inc..
- Flake, Gary William. (1998). "Square Unit Augmented Radially Extended Multilayer Perceptrons". Springer.
- (3 July 2018). "On the Power of Over-parametrization in Neural Networks with Quadratic Activation". PMLR.
- (2010). "27th International Conference on International Conference on Machine Learning". Omnipress.
- (2011). "Deep sparse rectifier neural networks".
- (23 November 2015). "Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)".
- (8 June 2017). "Self-Normalizing Neural Networks". Advances in Neural Information Processing Systems.
- (June 2013). "Rectifier nonlinearities improve neural network acoustic models". Proc. ICML.
- (6 February 2015). "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification".
- (2018). "Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning". Neural Networks.
- (2017). "Searching for Activation Functions".
- (2 August 2018). "The Quest for the Golden Activation Function".
- (2013). "Maxout Networks". JMLR Workshop and Conference Proceedings.
- (2022). "Quantum activation functions for quantum neural networks". Quantum Information Processing.
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