Activation function

The activation function of a node in an

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 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]}

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.^[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 the softplus makes it suitable for predicting variances in variational autoencoders.

The most common activation functions can be divided into three categories:

An activation function ${\displaystyle f}$ is saturating if ${\displaystyle \lim _{|v|\to \infty }|\nabla f(v)|=0}$. It is nonsaturating if it is ${\displaystyle \lim _{|v|\to \infty }|\nabla f(v)|\neq 0}$. Non-saturating activation functions, such as

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

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

In

The function looks like ${\displaystyle \phi (\mathbf {v} )=U(a+\mathbf {v} '\mathbf {b} )}$, where ${\displaystyle U}$ 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 ${\displaystyle \phi (\mathbf {v} )=a+\mathbf {v} '\mathbf {b} }$.

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

where ${\displaystyle \mathbf {c} }$ is the vector representing the function center and ${\displaystyle a}$ and ${\displaystyle \sigma }$ 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.^[10]

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

Folding activation functions are extensively used in the

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

Name	Plot	Function, ${\displaystyle g(x)}$	Derivative of ${\displaystyle g}$, ${\displaystyle g'(x)}$	Range	Order of continuity
Identity		${\displaystyle x}$	${\displaystyle 1}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{\infty }}$
Binary step		${\displaystyle {\begin{cases}0&{\text{if }}x<0\\1&{\text{if }}x\geq 0\end{cases}}}$	${\displaystyle 0}$	${\displaystyle \{0,1\}}$	${\displaystyle C^{-1}}$
Logistic, sigmoid, or soft step		${\displaystyle \sigma (x)\doteq {\frac {1}{1+e^{-x}}}}$	${\displaystyle g(x)(1-g(x))}$	${\displaystyle (0,1)}$	${\displaystyle C^{\infty }}$
Hyperbolic tangent ( tanh )		${\displaystyle \tanh(x)\doteq {\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}}$	${\displaystyle 1-g(x)^{2}}$	${\displaystyle (-1,1)}$	${\displaystyle C^{\infty }}$
Soboleva modified hyperbolic tangent (smht)		${\displaystyle \operatorname {smht} (x)\doteq {\frac {e^{ax}-e^{-bx}}{e^{cx}+e^{-dx}}}}$		${\displaystyle (-1,1)}$	${\displaystyle C^{\infty }}$
Rectified linear unit (ReLU)[13]		${\displaystyle {\begin{aligned}(x)^{+}\doteq {}&{\begin{cases}0&{\text{if }}x\leq 0\\x&{\text{if }}x>0\end{cases}}\\={}&\max(0,x)=x{\textbf {1}}_{x>0}\end{aligned}}}$	${\displaystyle {\begin{cases}0&{\text{if }}x<0\\1&{\text{if }}x>0\end{cases}}}$	${\displaystyle [0,\infty )}$	${\displaystyle C^{0}}$
Gaussian Error Linear Unit (GELU)[5]		${\displaystyle {\begin{aligned}&{\frac {1}{2}}x\left(1+{\text{erf}}\left({\frac {x}{\sqrt {2}}}\right)\right)\\{}={}&x\Phi (x)\end{aligned}}}$ where ${\displaystyle \mathrm {erf} }$ is the gaussian error function.	${\displaystyle \Phi (x)+{\frac {1}{2}}x\phi (x)}$ where ${\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}}$ is the probability density function of standard gaussian distribution.	${\displaystyle (-0.17\ldots ,\infty )}$	${\displaystyle C^{\infty }}$
Softplus[14]		${\displaystyle \ln \left(1+e^{x}\right)}$	${\displaystyle {\frac {1}{1+e^{-x}}}}$	${\displaystyle (0,\infty )}$	${\displaystyle C^{\infty }}$
Exponential linear unit (ELU)[15]		${\displaystyle {\begin{cases}\alpha \left(e^{x}-1\right)&{\text{if }}x\leq 0\\x&{\text{if }}x>0\end{cases}}}$ with parameter ${\displaystyle \alpha }$	${\displaystyle {\begin{cases}\alpha e^{x}&{\text{if }}x<0\\1&{\text{if }}x>0\end{cases}}}$	${\displaystyle (-\alpha ,\infty )}$	${\displaystyle {\begin{cases}C^{1}&{\text{if }}\alpha =1\\C^{0}&{\text{otherwise}}\end{cases}}}$
Scaled exponential linear unit (SELU)[16]		${\displaystyle \lambda {\begin{cases}\alpha (e^{x}-1)&{\text{if }}x<0\\x&{\text{if }}x\geq 0\end{cases}}}$ with parameters ${\displaystyle \lambda =1.0507}$ and ${\displaystyle \alpha =1.67326}$	${\displaystyle \lambda {\begin{cases}\alpha e^{x}&{\text{if }}x<0\\1&{\text{if }}x\geq 0\end{cases}}}$	${\displaystyle (-\lambda \alpha ,\infty )}$	${\displaystyle C^{0}}$
Leaky rectified linear unit (Leaky ReLU)[17]		${\displaystyle {\begin{cases}0.01x&{\text{if }}x\leq 0\\x&{\text{if }}x>0\end{cases}}}$	${\displaystyle {\begin{cases}0.01&{\text{if }}x<0\\1&{\text{if }}x>0\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$
Parametric rectified linear unit (PReLU)[18]		${\displaystyle {\begin{cases}\alpha x&{\text{if }}x<0\\x&{\text{if }}x\geq 0\end{cases}}}$ with parameter ${\displaystyle \alpha }$	${\displaystyle {\begin{cases}\alpha &{\text{if }}x<0\\1&{\text{if }}x\geq 0\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$
Rectified Parametric Sigmoid Units (flexible, 5 parameters)		${\displaystyle \alpha (2x{1}_{\{x\geqslant \lambda \}}-g_{\lambda ,\sigma ,\mu ,\beta }(x))+(1-\alpha )g_{\lambda ,\sigma ,\mu ,\beta }(x)}$ where ${\displaystyle g_{\lambda ,\sigma ,\mu ,\beta }(x)={\frac {(x-\lambda ){1}_{\{x\geqslant \lambda \}}}{1+e^{-\operatorname {sgn}(x-\mu )\left({\frac {\vert x-\mu \vert }{\sigma }}\right)^{\beta }}}}}$ [19]	${\displaystyle -}$	${\displaystyle (-\infty ,+\infty )}$	${\displaystyle C^{0}}$
Sigmoid linear unit (SiLU,[5] Sigmoid shrinkage,[20] SiL,[21] or Swish-‍1[22])		${\displaystyle {\frac {x}{1+e^{-x}}}}$	${\displaystyle {\frac {1+e^{-x}+xe^{-x}}{\left(1+e^{-x}\right)^{2}}}}$	${\displaystyle [-0.278\ldots ,\infty )}$	${\displaystyle C^{\infty }}$
Exponential Linear Sigmoid SquasHing (ELiSH)[23]		${\displaystyle {\begin{cases}{\frac {e^{x}-1}{1+e^{-x}}}&{\text{if }}x<0\\{\frac {x}{1+e^{-x}}}&{\text{if }}x\geq 0\end{cases}}}$	${\displaystyle {\begin{cases}{\frac {2e^{2x}+e^{3x}-e^{x}}{e^{2x}+2e^{x}+1}}&{\text{if }}x<0\\{\frac {xe^{x}+e^{2x}+e^{x}}{e^{2x}+2e^{x}+1}}&{\text{if }}x\geq 0\end{cases}}}$	${\displaystyle [-0.881\ldots ,\infty )}$	${\displaystyle C^{1}}$
Gaussian		${\displaystyle e^{-x^{2}}}$	${\displaystyle -2xe^{-x^{2}}}$	${\displaystyle (0,1]}$	${\displaystyle C^{\infty }}$
Sinusoid		${\displaystyle \sin x}$	${\displaystyle \cos x}$	${\displaystyle [-1,1]}$	${\displaystyle C^{\infty }}$

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

Name	Equation, ${\displaystyle g_{i}\left({\vec {x}}\right)}$	Derivatives, ${\displaystyle {\frac {\partial g_{i}\left({\vec {x}}\right)}{\partial x_{j}}}}$	Range	Order of continuity
Softmax	${\displaystyle {\frac {e^{x_{i}}}{\sum _{j=1}^{J}e^{x_{j}}}}}$    for i = 1, …, J	${\displaystyle g_{i}\left({\vec {x}}\right)\left(\delta _{ij}-g_{j}\left({\vec {x}}\right)\right)}$[1][2]	${\displaystyle (0,1)}$	${\displaystyle C^{\infty }}$
Maxout[24]	${\displaystyle \max _{i}x_{i}}$	${\displaystyle {\begin{cases}1&{\text{if }}j={\underset {i}{\operatorname {argmax} }}\,x_{i}\\0&{\text{if }}j\neq {\underset {i}{\operatorname {argmax} }}\,x_{i}\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$

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

In

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