Multilayer perceptron

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A multilayer perceptron (MLP) is a name for a modern feedforward artificial neural network, consisting of fully connected neurons with a nonlinear kind of activation function, organized in at least three layers, notable for being able to distinguish data that is not linearly separable.[1] It is a misnomer because the original perceptron used a Heaviside step function, instead of a nonlinear kind of activation function (used by modern networks).

Modern feedforward networks are trained using the backpropagation method[2][3][4][5][6] and are colloquially referred to as the "vanilla" neural networks.[7]


  • In 1967, a deep-learning network, which used stochastic gradient descent for the first time, able to classify non-linearily separable pattern classes, was published by Shun'ichi Amari.[14] Amari's student Saito conducted the computer experiments, using a five-layered feedforward network with two learning layers.
  • In 1985, an experimental analysis of the technique was conducted by David E. Rumelhart et al..[6] Many improvements to the approach have been made in subsequent decades,[10].
  • In 2021, a very simple NN architecture combining two deep MLPs with skip connections and layer normalizations was designed and called MLP-Mixer; its realizations featuring 19 to 431 millions of parameters were shown to be comparable to vision transformers of similar size on ImageNet and similar image classification tasks.[22]

Mathematical foundations[edit]

Activation function[edit]

If a multilayer perceptron has a linear activation function in all neurons, that is, a linear function that maps the weighted inputs to the output of each neuron, then linear algebra shows that any number of layers can be reduced to a two-layer input-output model. In MLPs some neurons use a nonlinear activation function that was developed to model the frequency of action potentials, or firing, of biological neurons.

The two historically common activation functions are both sigmoids, and are described by


The first is a hyperbolic tangent that ranges from −1 to 1, while the other is the logistic function, which is similar in shape but ranges from 0 to 1. Here is the output of the th node (neuron) and is the weighted sum of the input connections. Alternative activation functions have been proposed, including the rectifier and softplus functions. More specialized activation functions include radial basis functions (used in radial basis networks, another class of supervised neural network models).

In recent developments of deep learning the rectified linear unit (ReLU) is more frequently used as one of the possible ways to overcome the numerical problems related to the sigmoids.


The MLP consists of three or more layers (an input and an output layer with one or more hidden layers) of nonlinearly-activating nodes. Since MLPs are fully connected, each node in one layer connects with a certain weight to every node in the following layer.


Learning occurs in the perceptron by changing connection weights after each piece of data is processed, based on the amount of error in the output compared to the expected result. This is an example of supervised learning, and is carried out through backpropagation, a generalization of the least mean squares algorithm in the linear perceptron.

We can represent the degree of error in an output node in the th data point (training example) by , where is the desired target value for th data point at node , and is the value produced by the perceptron at node when the th data point is given as an input.

The node weights can then be adjusted based on corrections that minimize the error in the entire output for the th data point, given by


Using gradient descent, the change in each weight is

where is the output of the previous neuron , and is the learning rate, which is selected to ensure that the weights quickly converge to a response, without oscillations. In the previous expression, denotes the partial derivate of the error according to the weighted sum of the input connections of neuron .

The derivative to be calculated depends on the induced local field , which itself varies. It is easy to prove that for an output node this derivative can be simplified to

where is the derivative of the activation function described above, which itself does not vary. The analysis is more difficult for the change in weights to a hidden node, but it can be shown that the relevant derivative is


This depends on the change in weights of the th nodes, which represent the output layer. So to change the hidden layer weights, the output layer weights change according to the derivative of the activation function, and so this algorithm represents a backpropagation of the activation function.[23]


  1. ^ Cybenko, G. 1989. Approximation by superpositions of a sigmoidal function Mathematics of Control, Signals, and Systems, 2(4), 303–314.
  2. ^ a b Linnainmaa, Seppo (1970). The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors (Masters) (in Finnish). University of Helsinki. pp. 6–7.
  3. ^ a b Kelley, Henry J. (1960). "Gradient theory of optimal flight paths". ARS Journal. 30 (10): 947–954. doi:10.2514/8.5282.
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  11. ^ Huang, Guang-Bin; Zhu, Qin-Yu; Siew, Chee-Kheong (2006). "Extreme learning machine: theory and applications". Neurocomputing. 70 (1): 489–501. CiteSeerX doi:10.1016/j.neucom.2005.12.126. S2CID 116858.
  12. ^ Ivakhnenko, A. G. (1973). Cybernetic Predicting Devices. CCM Information Corporation.
  13. ^ Ivakhnenko, A. G.; Grigorʹevich Lapa, Valentin (1967). Cybernetics and forecasting techniques. American Elsevier Pub. Co.
  14. ^ Amari, Shun'ichi (1967). "A theory of adaptive pattern classifier". IEEE Transactions. EC (16): 279–307.
  15. ^ Rodríguez, Omar Hernández; López Fernández, Jorge M. (2010). "A Semiotic Reflection on the Didactics of the Chain Rule". The Mathematics Enthusiast. 7 (2): 321–332. doi:10.54870/1551-3440.1191. S2CID 29739148. Retrieved 2019-08-04.
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  19. ^ Bengio, Yoshua; Ducharme, Réjean; Vincent, Pascal; Janvin, Christian (March 2003). "A neural probabilistic language model". The Journal of Machine Learning Research. 3: 1137–1155.
  20. ^ Vaswani, Ashish; Shazeer, Noam; Parmar, Niki; Uszkoreit, Jakob; Jones, Llion; Gomez, Aidan N; Kaiser, Łukasz; Polosukhin, Illia (2017). "Attention is All you Need". Advances in Neural Information Processing Systems. 30. Curran Associates, Inc.
  21. ^ Geva, Mor; Schuster, Roei; Berant, Jonathan; Levy, Omer (2021). "Transformer Feed-Forward Layers Are Key-Value Memories". Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing. pp. 5484–5495. doi:10.18653/v1/2021.emnlp-main.446. S2CID 229923720.
  22. ^ "Papers with Code – MLP-Mixer: An all-MLP Architecture for Vision".
  23. ^ Haykin, Simon (1998). Neural Networks: A Comprehensive Foundation (2 ed.). Prentice Hall. ISBN 0-13-273350-1.

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