Activation function

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Logistic activation function

In artificial neural networks, the activation function of a node defines the output of that node given an input or set of inputs. A standard integrated circuit can be seen as a digital network of activation functions that can be "ON" (1) or "OFF" (0), depending on input. This is similar to the linear perceptron in neural networks. However, only nonlinear activation functions allow such networks to compute nontrivial problems using only a small number of nodes, and such activation functions are called nonlinearities.[1]

Classification of activation functions[edit]

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

An activation function is saturating if . It is nonsaturating if it is not saturating. Non-saturating activation functions, such as ReLU, may be better than saturating activation functions, as they don't suffer from vanishing gradient.[2]

Ridge activation functions[edit]

Ridge functions are multivariate functions acting on a linear combination of the input variables. Often used examples include:

  • Linear activation: ,
  • ReLU activation: ,
  • Heaviside activation: ,
  • Logistic activation: .

In biologically inspired neural networks, the activation function is usually an abstraction representing the rate of action potential firing in the cell.[3] In its simplest form, this function is binary—that is, either the neuron is firing or not. The function looks like , where is the Heaviside step function.

A line of positive slope may be used to reflect the increase in firing rate that occurs as input current increases. Such a function would be of the form .

Rectified linear unit and Gaussian error linear unit activation functions

Neurons also cannot fire faster than a certain rate, motivating sigmoid activation functions whose range is a finite interval.

Radial activation functions[edit]

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 the following functions:

  • Gaussian:
  • Multiquadratics:
  • Inverse multiquadratics:
  • Polyharmonic splines

where is the vector representing the function center and and are parameters affecting the spread of the radius.

Folding activation functions[edit]

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.

Comparison of activation functions[edit]

There are numerous activation functions. Hinton et al.'s seminal 2012 paper on automatic speech recognition uses a logistic sigmoid activation function.[4] The seminal 2012 AlexNet computer vision architecture uses the ReLU activation function, as did the seminal 2015 computer vision architecture ResNet. The seminal 2018 language processing model BERT uses a smooth version of the ReLU, the GELU.[5]

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

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.
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.

Table of activation functions[edit]

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, Derivative of , Range Order of continuity
Identity Activation identity.svg
Binary step Activation binary step.svg
Logistic, sigmoid, or soft step Activation logistic.svg
Hyperbolic tangent (tanh) Activation tanh.svg
Rectified linear unit (ReLU)[8] Activation rectified linear.svg
Gaussian Error Linear Unit (GELU)[5] Visualization of the Gaussian Error Linear Unit (GELU)
Softplus[9] Activation softplus.svg
Exponential linear unit (ELU)[10] Activation elu.svg
with parameter
Scaled exponential linear unit (SELU)[11] Activation selu.png
with parameters and
Leaky rectified linear unit (Leaky ReLU)[12] Activation prelu.svg
Parametric rectified linear unit (PReLU)[13] Activation prelu.svg
with parameter
Sigmoid linear unit (SiLU,[5] Sigmoid shrinkage,[14] SiL,[15] or Swish-‍1[16]) Swish Activation Function
Gaussian Activation gaussian.svg

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    for i = 1, …, J [1][2]
^ Here, is the Kronecker delta.
^ For instance, could be iterating through the number of kernels of the previous neural network layer while iterates through the number of kernels of the current layer.

See also[edit]


  1. ^ Hinkelmann, Knut. "Neural Networks, p. 7" (PDF). University of Applied Sciences Northwestern Switzerland. Archived from the original (PDF) on 2018-10-06. Retrieved 2018-10-06.
  2. ^ Krizhevsky, Alex; Sutskever, Ilya; Hinton, Geoffrey E. (2017-05-24). "ImageNet classification with deep convolutional neural networks". Communications of the ACM. 60 (6): 84–90. doi:10.1145/3065386. ISSN 0001-0782. S2CID 195908774.
  3. ^ Hodgkin, A. L.; Huxley, A. F. (1952-08-28). "A quantitative description of membrane current and its application to conduction and excitation in nerve". The Journal of Physiology. 117 (4): 500–544. doi:10.1113/jphysiol.1952.sp004764. PMC 1392413. PMID 12991237.
  4. ^ Hinton, Geoffrey; Deng, Li; Deng, Li; Yu, Dong; Dahl, George; Mohamed, Abdel-rahman; Jaitly, Navdeep; Senior, Andrew; Vanhoucke, Vincent; Nguyen, Patrick; Sainath, Tara; Kingsbury, Brian (2012). "Deep Neural Networks for Acoustic Modeling in Speech Recognition". IEEE Signal Processing Magazine. 29 (6): 82–97. doi:10.1109/MSP.2012.2205597. S2CID 206485943.
  5. ^ a b c Hendrycks, Dan; Gimpel, Kevin (2016). "Gaussian Error Linear Units (GELUs)". arXiv:1606.08415 [cs.LG].
  6. ^ Cybenko, G. (December 1989). "Approximation by superpositions of a sigmoidal function" (PDF). Mathematics of Control, Signals, and Systems. 2 (4): 303–314. doi:10.1007/BF02551274. ISSN 0932-4194. S2CID 3958369.
  7. ^ Snyman, Jan (3 March 2005). Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms. Springer Science & Business Media. ISBN 978-0-387-24348-1.
  8. ^ Nair, Vinod; Hinton, Geoffrey E. (2010), "Rectified Linear Units Improve Restricted Boltzmann Machines", 27th International Conference on International Conference on Machine Learning, ICML'10, USA: Omnipress, pp. 807–814, ISBN 9781605589077
  9. ^ Glorot, Xavier; Bordes, Antoine; Bengio, Yoshua (2011). "Deep sparse rectifier neural networks" (PDF). International Conference on Artificial Intelligence and Statistics.
  10. ^ Clevert, Djork-Arné; Unterthiner, Thomas; Hochreiter, Sepp (2015-11-23). "Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)". arXiv:1511.07289 [cs.LG].
  11. ^ Klambauer, Günter; Unterthiner, Thomas; Mayr, Andreas; Hochreiter, Sepp (2017-06-08). "Self-Normalizing Neural Networks". Advances in Neural Information Processing Systems. 30 (2017). arXiv:1706.02515.
  12. ^ Maas, Andrew L.; Hannun, Awni Y.; Ng, Andrew Y. (June 2013). "Rectifier nonlinearities improve neural network acoustic models". Proc. ICML. 30 (1). S2CID 16489696.
  13. ^ He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (2015-02-06). "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification". arXiv:1502.01852 [cs.CV].
  14. ^ Atto, Abdourrahmane M.; Pastor, Dominique; Mercier, Grégoire (2008), "Smooth sigmoid wavelet shrinkage for non-parametric estimation" (PDF), 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, doi:10.1109/ICASSP.2008.4518347, S2CID 9959057
  15. ^ Elfwing, Stefan; Uchibe, Eiji; Doya, Kenji (2018). "Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning". Neural Networks. 107: 3–11. arXiv:1702.03118. doi:10.1016/j.neunet.2017.12.012. PMID 29395652. S2CID 6940861.
  16. ^ Ramachandran, Prajit; Zoph, Barret; Le, Quoc V (2017). "Searching for Activation Functions". arXiv:1710.05941 [cs.NE].
  17. ^ Goodfellow, Ian J.; Warde-Farley, David; Mirza, Mehdi; Courville, Aaron; Bengio, Yoshua (2013). "Maxout Networks". JMLR Workshop and Conference Proceedings. 28 (3): 1319–1327. arXiv:1302.4389.