Activation function

A special class of activation functions known as radial basis functions (RBFs) are used in RBF networks, which are extremely efficient as universal function approximators. These activation functions can take many forms, but they are usually found as one of three functions:

• Gaussian: ${\displaystyle \,\phi (v_{i})=\exp \left(-{\frac {\|v_{i}-c_{i}\|^{2}}{2\sigma ^{2}}}\right)}$
• Multiquadratics: ${\displaystyle \,\phi (v_{i})={\sqrt {\|v_{i}-c_{i}\|^{2}+a^{2}}}}$
• Inverse multiquadratics: ${\displaystyle \,\phi (v_{i})=(\|v_{i}-c_{i}\|^{2}+a^{2})^{-1/2}}$

where ${\displaystyle c_{i}}$ is the vector representing the function center and ${\displaystyle a}$ and ${\displaystyle \sigma }$ are parameters affecting the spread of the radius.

A computationally efficient Radial Basis Function has been proposed,[4] called Square-law based RBF kernel (SQ-RBF) which eliminates the exponential term as found in Gaussian RBF.

• SQ-RBF: ${\displaystyle f(x)={\begin{cases}1-{\frac {x^{2}}{2}}&:|x|\leq 1\\{\frac {(2-|x|)^{2}}{2}}&:1\leq |x|\leq 2\\0&:|x|\geq 2\end{cases}}}$

Support-vector machines (SVMs) can effectively utilize a class of activation functions that includes both sigmoids and RBFs. In this case, the input is transformed to reflect a decision boundary hyperplane based on a few training inputs called support-vectors ${\displaystyle x}$. The activation function for the hidden layer of these machines is referred to as the inner product kernel, ${\displaystyle K(v_{i},x)=\phi (v_{i})}$. The support-vectors are represented as the centers in RBFs with the kernel equal to the activation function, but they take a unique form in the perceptron as

${\displaystyle \,\phi (v_{i})=\tanh \left(\beta _{1}+\beta _{0}\sum _{j}v_{i,j}x_{j}\right)}$,

where ${\displaystyle \beta _{0}}$ and ${\displaystyle \beta _{1}}$ must satisfy certain conditions for convergence. These machines can also accept arbitrary-order polynomial activation functions where

${\displaystyle \,\phi (v_{i})=\left(1+\sum _{j}v_{i,j}x_{j}\right)^{p}}$.[5]

Activation function having types:

• Identity function
• Binary step function
• Bipolar step function
• Sigmoidal function
  • Binary sigmoidal function
  • Bipolar sigmoidal function
• Ramp function

Some desirable properties in an activation function include:

• 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.
• 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]
• Monotonic – When the activation function is monotonic, the error surface associated with a single-layer model is guaranteed to be convex.[8]
• Smooth functions with a monotonic derivative – These have been shown to generalize better in some cases.
• Approximates identity near the origin – When activation functions have this property, the neural network will learn efficiently when its weights are initialized with small random values. When the activation function does not approximate identity near the origin, special care must be used when initializing the weights.[9] In the table below, activation functions where ${\displaystyle f(0)=0}$ and ${\displaystyle f'(0)=1}$ and ${\displaystyle f'}$ is continuous at 0 are indicated as having this property.

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	Equation	Derivative (with respect to x)	Range	Order of continuity	Monotonic	Monotonic derivative	Approximates identity near the origin
Identity		${\displaystyle f(x)=x}$	${\displaystyle f'(x)=1}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{\infty }}$	Yes	Yes	Yes
Binary step		${\displaystyle f(x)={\begin{cases}0&{\text{for }}x<0\\1&{\text{for }}x\geq 0\end{cases}}}$	${\displaystyle f'(x)={\begin{cases}0&{\text{for }}x\neq 0\\?&{\text{for }}x=0\end{cases}}}$	${\displaystyle \{0,1\}}$	${\displaystyle C^{-1}}$	Yes	No	No
Logistic (a.k.a. Sigmoid or Soft step)		${\displaystyle f(x)=\sigma (x)={\frac {1}{1+e^{-x}}}}$	${\displaystyle f'(x)=f(x)(1-f(x))}$	${\displaystyle (0,1)}$	${\displaystyle C^{\infty }}$	Yes	No	No
TanH		${\displaystyle f(x)=\tanh(x)={\frac {(e^{x}-e^{-x})}{(e^{x}+e^{-x})}}}$	${\displaystyle f'(x)=1-f(x)^{2}}$	${\displaystyle (-1,1)}$	${\displaystyle C^{\infty }}$	Yes	No	Yes
SQNL[10]		${\displaystyle f(x)={\begin{cases}1&:x>2.0\\x-{\frac {x^{2}}{4}}&:0\leq x\leq 2.0\\x+{\frac {x^{2}}{4}}&:-2.0\leq x<0\\-1&:x<-2.0\end{cases}}}$	${\displaystyle f'(x)=1\mp {\frac {x}{2}}}$	${\displaystyle (-1,1)}$	${\displaystyle C^{2}}$	Yes	No	Yes
ArcTan		${\displaystyle f(x)=\tan ^{-1}(x)}$	${\displaystyle f'(x)={\frac {1}{x^{2}+1}}}$	${\displaystyle \left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)}$	${\displaystyle C^{\infty }}$	Yes	No	Yes
ArSinH		${\displaystyle f(x)=\sinh ^{-1}(x)=\ln \left(x+{\sqrt {x^{2}+1}}\right)}$	${\displaystyle f'(x)={\frac {1}{\sqrt {x^{2}+1}}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{\infty }}$	Yes	No	Yes
ElliotSig[11][12] Softsign[13][14]		${\displaystyle f(x)={\frac {x}{1+|x|}}}$	${\displaystyle f'(x)={\frac {1}{(1+|x|)^{2}}}}$	${\displaystyle (-1,1)}$	${\displaystyle C^{1}}$	Yes	No	Yes
Inverse square root unit (ISRU)[15]		${\displaystyle f(x)={\frac {x}{\sqrt {1+\alpha x^{2}}}}}$	${\displaystyle f'(x)=\left({\frac {1}{\sqrt {1+\alpha x^{2}}}}\right)^{3}}$	${\displaystyle \left(-{\frac {1}{\sqrt {\alpha }}},{\frac {1}{\sqrt {\alpha }}}\right)}$	${\displaystyle C^{\infty }}$	Yes	No	Yes
Inverse square root linear unit (ISRLU)[15]		${\displaystyle f(x)={\begin{cases}{\frac {x}{\sqrt {1+\alpha x^{2}}}}&{\text{for }}x<0\\x&{\text{for }}x\geq 0\end{cases}}}$	${\displaystyle f'(x)={\begin{cases}\left({\frac {1}{\sqrt {1+\alpha x^{2}}}}\right)^{3}&{\text{for }}x<0\\1&{\text{for }}x\geq 0\end{cases}}}$	${\displaystyle \left(-{\frac {1}{\sqrt {\alpha }}},\infty \right)}$	${\displaystyle C^{2}}$	Yes	Yes	Yes
Piecewise Linear Unit (PLU)[16]		${\displaystyle f(x)=\max(\alpha (x+c)-c,\min(\alpha (x-c)+c,x))}$	${\displaystyle f'(x)={\begin{cases}\alpha &{\text{for }}|x|>c\\1&{\text{for }}|x|<c\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$	Yes	No	Yes
Rectified linear unit (ReLU)[17]		${\displaystyle f(x)={\begin{cases}0&{\text{for }}x\leq 0\\x&{\text{for }}x>0\end{cases}}}$	${\displaystyle f'(x)={\begin{cases}0&{\text{for }}x\leq 0\\1&{\text{for }}x>0\end{cases}}}$	${\displaystyle [0,\infty )}$	${\displaystyle C^{0}}$	Yes	Yes	No
Bipolar rectified linear unit (BReLU)[18]		${\displaystyle f(x_{i})={\begin{cases}ReLU(x_{i})&{\text{if }}i{\bmod {2}}=0\\-ReLU(-x_{i})&{\text{if }}i{\bmod {2}}\neq 0\end{cases}}}$	${\displaystyle f'(x_{i})={\begin{cases}ReLU'(x_{i})&{\text{if }}i{\bmod {2}}=0\\ReLU'(-x_{i})&{\text{if }}i{\bmod {2}}\neq 0\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$	Yes	Yes	No
Leaky rectified linear unit (Leaky ReLU)[19]		${\displaystyle f(x)={\begin{cases}0.01x&{\text{for }}x<0\\x&{\text{for }}x\geq 0\end{cases}}}$	${\displaystyle f'(x)={\begin{cases}0.01&{\text{for }}x<0\\1&{\text{for }}x\geq 0\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$	Yes	Yes	No
Parameteric rectified linear unit (PReLU)[20]		${\displaystyle f(\alpha ,x)={\begin{cases}\alpha x&{\text{for }}x<0\\x&{\text{for }}x\geq 0\end{cases}}}$	${\displaystyle f'(\alpha ,x)={\begin{cases}\alpha &{\text{for }}x<0\\1&{\text{for }}x\geq 0\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$	Yes iff ${\displaystyle \alpha \geq 0}$	Yes	Yes iff ${\displaystyle \alpha =1}$
Randomized leaky rectified linear unit (RReLU)[21]		${\displaystyle f(\alpha ,x)={\begin{cases}\alpha x&{\text{for }}x<0\\x&{\text{for }}x\geq 0\end{cases}}}$	${\displaystyle f'(\alpha ,x)={\begin{cases}\alpha &{\text{for }}x<0\\1&{\text{for }}x\geq 0\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$	Yes	Yes	No
Gaussian Error Linear Unit (GELU)[22]		${\displaystyle f(x)=x\Phi (x)=x(1+{\text{erf}}(x/{\sqrt {2}}))/2}$	${\displaystyle f'(x)=\Phi (x)+x\phi (x)}$	${\displaystyle (-0.17,\infty )}$	${\displaystyle C^{\infty }}$	No	No	No
Exponential linear unit (ELU)[23]		${\displaystyle f(\alpha ,x)={\begin{cases}\alpha (e^{x}-1)&{\text{for }}x\leq 0\\x&{\text{for }}x>0\end{cases}}}$	${\displaystyle f'(\alpha ,x)={\begin{cases}f(\alpha ,x)+\alpha &{\text{for }}x\leq 0\\1&{\text{for }}x>0\end{cases}}}$	${\displaystyle (-\alpha ,\infty )}$	${\displaystyle {\begin{cases}C^{1}&{\text{when }}\alpha =1\\C^{0}&{\text{otherwise }}\end{cases}}}$	Yes iff ${\displaystyle \alpha \geq 0}$	Yes iff ${\displaystyle 0\leq \alpha \leq 1}$	Yes iff ${\displaystyle \alpha =1}$
Scaled exponential linear unit (SELU)[24]		${\displaystyle f(\alpha ,x)=\lambda {\begin{cases}\alpha (e^{x}-1)&{\text{for }}x<0\\x&{\text{for }}x\geq 0\end{cases}}}$ with ${\displaystyle \lambda =1.0507}$ and ${\displaystyle \alpha =1.67326}$	${\displaystyle f'(\alpha ,x)=\lambda {\begin{cases}\alpha (e^{x})&{\text{for }}x<0\\1&{\text{for }}x\geq 0\end{cases}}}$	${\displaystyle (-\lambda \alpha ,\infty )}$	${\displaystyle C^{0}}$	Yes	No	No
S-shaped rectified linear activation unit (SReLU)[25]		${\displaystyle f_{t_{l},a_{l},t_{r},a_{r}}(x)={\begin{cases}t_{l}+a_{l}(x-t_{l})&{\text{for }}x\leq t_{l}\\x&{\text{for }}t_{l}<x<t_{r}\\t_{r}+a_{r}(x-t_{r})&{\text{for }}x\geq t_{r}\end{cases}}}$ ${\displaystyle t_{l},a_{l},t_{r},a_{r}}$ are parameters.	${\displaystyle f'_{t_{l},a_{l},t_{r},a_{r}}(x)={\begin{cases}a_{l}&{\text{for }}x\leq t_{l}\\1&{\text{for }}t_{l}<x<t_{r}\\a_{r}&{\text{for }}x\geq t_{r}\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$	No	No	No
Adaptive piecewise linear (APL)[26]		${\displaystyle f(x)=\max(0,x)+\sum _{s=1}^{S}a_{i}^{s}\max(0,-x+b_{i}^{s})}$	${\displaystyle f'(x)=H(x)-\sum _{s=1}^{S}a_{i}^{s}H(-x+b_{i}^{s})}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$	No	No	No
SoftPlus[27]		${\displaystyle f(x)=\ln(1+e^{x})}$	${\displaystyle f'(x)={\frac {1}{1+e^{-x}}}}$	${\displaystyle (0,\infty )}$	${\displaystyle C^{\infty }}$	Yes	Yes	No
Bent identity		${\displaystyle f(x)={\frac {{\sqrt {x^{2}+1}}-1}{2}}+x}$	${\displaystyle f'(x)={\frac {x}{2{\sqrt {x^{2}+1}}}}+1}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{\infty }}$	Yes	Yes	Yes
Sigmoid Linear Unit (SiLU)[22] (AKA SiL[28] and Swish-1[29])		${\displaystyle f(x)={\frac {x}{1+e^{-x}}}}$	${\displaystyle f'(x)={\frac {1+e^{-x}+xe^{-x}}{\left(1+e^{-x}\right)^{2}}}}$	${\displaystyle [\approx -0.278,\infty )}$	${\displaystyle C^{\infty }}$	No	No	Approximates identity/2
SoftExponential[30]		${\displaystyle f(\alpha ,x)={\begin{cases}-{\frac {\ln(1-\alpha (x+\alpha ))}{\alpha }}&{\text{for }}\alpha <0\\x&{\text{for }}\alpha =0\\{\frac {e^{\alpha x}-1}{\alpha }}+\alpha &{\text{for }}\alpha >0\end{cases}}}$	${\displaystyle f'(\alpha ,x)={\begin{cases}{\frac {1}{1-\alpha (\alpha +x)}}&{\text{for }}\alpha <0\\e^{\alpha x}&{\text{for }}\alpha \geq 0\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{\infty }}$	Yes	Yes	Yes iff ${\displaystyle \alpha =0}$
Soft Clipping [31][32]		${\displaystyle f(\alpha ,x)={\frac {1}{\alpha }}\log {\frac {1+e^{\alpha x}}{1+e^{\alpha (x-1)}}}}$	${\displaystyle f'(\alpha ,x)={\frac {1}{2}}\sinh \left({\frac {p}{2}}\right){\text{sech}}\left({\frac {px}{2}}\right){\text{sech}}\left({\frac {p}{2}}(1-x)\right)}$	${\displaystyle (0,1)}$	${\displaystyle C^{\infty }}$	Yes	No	No
Squashing[32][33][34]		${\displaystyle S_{a,\lambda }^{(\beta )}(x)={\frac {1}{\lambda \beta }}\ln {\frac {1+e^{\beta \left(x-(a-\lambda /2)\right)}}{1+e^{\beta \left(x-(a+\lambda /2)\right)}}}={\frac {1}{\lambda \beta }}\ln {\frac {\sigma _{a+\lambda /2}^{(-\beta )}(x)}{\sigma _{a-\lambda /2}^{(-\beta )}(x)}}.}$	${\displaystyle {\frac {\partial S_{a,\lambda }^{(\beta )}(x)}{\partial x}}={\frac {1}{\lambda }}\left(\sigma _{a-\lambda /2}^{(\beta )}(x)-\sigma _{a+\lambda /2}^{(\beta )}(x)\right)}$	${\displaystyle (0,1)}$	${\displaystyle C^{\infty }}$	Yes	No	No
Sinusoid[35]		${\displaystyle f(x)=\sin(x)}$	${\displaystyle f'(x)=\cos(x)}$	${\displaystyle [-1,1]}$	${\displaystyle C^{\infty }}$	No	No	Yes
Sinc		${\displaystyle f(x)={\begin{cases}1&{\text{for }}x=0\\{\frac {\sin(x)}{x}}&{\text{for }}x\neq 0\end{cases}}}$	${\displaystyle f'(x)={\begin{cases}0&{\text{for }}x=0\\{\frac {\cos(x)}{x}}-{\frac {\sin(x)}{x^{2}}}&{\text{for }}x\neq 0\end{cases}}}$	${\displaystyle [\approx -.217234,1]}$	${\displaystyle C^{\infty }}$	No	No	No
Gaussian		${\displaystyle f(x)=e^{-x^{2}}}$	${\displaystyle f'(x)=-2xe^{-x^{2}}}$	${\displaystyle (0,1]}$	${\displaystyle C^{\infty }}$	No	No	No
SQ-RBF		${\displaystyle f(x)={\begin{cases}1-{\frac {x^{2}}{2}}&:|x|\leq 1\\{\frac {(2-|x|)^{2}}{2}}&:1\leq |x|\leq 2\\0&:|x|\geq 2\end{cases}}}$	${\displaystyle f'(x)={\begin{cases}-x&:|x|\leq 1\\x-2\operatorname {sgn}(x)&:1\leq |x|\leq 2\\0&:|x|\geq 2\end{cases}}}$	${\displaystyle [0,1]}$	${\displaystyle C^{0}}$	No	No	No

^ Here, H is the Heaviside step function.

^ α is a stochastic variable sampled from a uniform distribution at training time and fixed to the expectation value of the distribution at test time.

^ ^ ^ Here, ${\displaystyle \sigma }$ is the logistic function.

^ ${\displaystyle \alpha >0}$ for the range to hold true.

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

Name	Equation	Derivatives	Range	Order of continuity
Softmax	${\displaystyle f_{i}({\vec {x}})={\frac {e^{x_{i}}}{\sum _{j=1}^{J}e^{x_{j}}}}}$    for i = 1, …, J	${\displaystyle {\frac {\partial f_{i}({\vec {x}})}{\partial x_{j}}}=f_{i}({\vec {x}})(\delta _{ij}-f_{j}({\vec {x}}))}$	${\displaystyle (0,1)}$	${\displaystyle C^{\infty }}$
Maxout[36]	${\displaystyle f({\vec {x}})=\max _{i}x_{i}}$	${\displaystyle {\frac {\partial f}{\partial x_{j}}}={\begin{cases}1&{\text{for }}j={\underset {i}{\operatorname {argmax} }}\,x_{i}\\0&{\text{for }}j\neq {\underset {i}{\operatorname {argmax} }}\,x_{i}\end{cases}}}$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle C^{0}}$

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