Indeep learning, amultilayer perceptron (MLP) is a kind of modernfeedforwardneural network consisting of fully connected neurons with nonlinearactivation functions, organized in layers, notable for being able to distinguish data that is notlinearly separable.^[1]

Modern neural networks are trained usingbackpropagation^{[2][3][4][5][6]} and are colloquially referred to as "vanilla" networks.^[7] MLPs grew out of an effort to improve onsingle-layer perceptrons, which could only be applied to linearly separable data. A perceptron traditionally used aHeaviside step function as its nonlinear activation function. However, the backpropagation algorithm requires that modern MLPs usecontinuous activation functions such assigmoid orReLU.^[8]

Multilayer perceptrons form the basis of deep learning,^[9] and areapplicable across a vast set of diverse domains.^[10]

• In 1943,Warren McCulloch andWalter Pitts proposed the binaryartificial neuron as a logical model of biological neural networks.^[11]
• In 1958,Frank Rosenblatt proposed the multilayeredperceptron model, consisting of an input layer, a hidden layer with randomized weights that did not learn, and an output layer with learnable connections.^[12]
• In 1962, Rosenblatt published many variants and experiments on perceptrons in his bookPrinciples of Neurodynamics, including up to 2 trainable layers by "back-propagating errors".^[13] However, it was not the backpropagation algorithm, and he did not have a general method for training multiple layers.

• In 1965,Alexey Grigorevich Ivakhnenko and Valentin Lapa publishedGroup Method of Data Handling. It was one of the firstdeep learning methods, used to train an eight-layer neural net in 1971.^[14][15][16]

• In 1967,Shun'ichi Amari reported^[17] the first multilayered neural network trained bystochastic gradient descent, was able to classify non-linearily separable pattern classes. Amari's student Saito conducted the computer experiments, using a five-layered feedforward network with two learning layers.^[16]

• Backpropagation was independently developed multiple times in early 1970s. The earliest published instance wasSeppo Linnainmaa's master thesis (1970).^[18][19][16]Paul Werbos developed it independently in 1971,^[20] but had difficulty publishing it until 1982.^[21]

• In 1986,David E. Rumelhart et al. popularized backpropagation.^[22][23]

• In 2003, interest in backpropagation networks returned due to the successes ofdeep learning being applied tolanguage modelling byYoshua Bengio with co-authors.^[24]

• 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 tovision transformers of similar size onImageNet and similarimage classification tasks.^[25]

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

The two historically common activation functions are bothsigmoids, and are described by

${\displaystyle y(v_i)=\tanh (v_i)\text{and}y(v_i)=(1+e^{-v_i}{)}^{-1}}$.

The first is ahyperbolic tangent that ranges from −1 to 1, while the other is thelogistic function, which is similar in shape but ranges from 0 to 1. Here${\displaystyle y_i}$ is the output of the${\displaystyle i}$th node (neuron) and${\displaystyle v_i}$ is the weighted sum of the input connections. Alternative activation functions have been proposed, including therectifier and softplus functions. More specialized activation functions includeradial basis functions (used inradial basis networks, another class of supervised neural network models).

In recent developments ofdeep learning therectified linear unit (ReLU) is more frequently used as one of the possible ways to overcome the numericalproblems related to the sigmoids.

The MLP consists of three or more layers (an input and an output layer with one or morehidden layers) of nonlinearly-activating nodes. Since MLPs are fully connected, each node in one layer connects with a certain weight${\displaystyle w_{ij}}$ 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 ofsupervised learning, and is carried out throughbackpropagation, a generalization of theleast mean squares algorithm in the linear perceptron.

We can represent the degree of error in an output node${\displaystyle j}$ in the${\displaystyle n}$th data point (training example) by${\displaystyle e_j(n)=d_j(n)-y_j(n)}$, where${\displaystyle d_j(n)}$ is the desired target value for${\displaystyle n}$th data point at node${\displaystyle j}$, and${\displaystyle y_j(n)}$ is the value produced by the perceptron at node${\displaystyle j}$ when the${\displaystyle n}$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${\displaystyle n}$th data point, given by

${\displaystyle \mathcal{E}(n)=\frac{1}{2}\sum _{\text{output node }j}{e}_j^2(n)}$.

Usinggradient descent, the change in each weight${\displaystyle w_{ij}}$ is

${\displaystyle \mathrm{\Delta }{w}_{ji}(n)=-\eta \frac{\mathrm{\partial }\mathcal{E}(n)}{\mathrm{\partial }{v}_j(n)}{y}_i(n)}$

where${\displaystyle y_i(n)}$ is the output of the previous neuron${\displaystyle i}$, and${\displaystyle \eta }$ is thelearning rate, which is selected to ensure that the weights quickly converge to a response, without oscillations. In the previous expression,${\displaystyle \frac{\mathrm{\partial }\mathcal{E}(n)}{\mathrm{\partial }{v}_j(n)}}$ denotes the partial derivate of the error${\displaystyle \mathcal{E}(n)}$ according to the weighted sum${\displaystyle v_j(n)}$ of the input connections of neuron${\displaystyle i}$.

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

${\displaystyle -\frac{\mathrm{\partial }\mathcal{E}(n)}{\mathrm{\partial }{v}_j(n)}=e_j(n){\varphi }^{\mathrm{\prime }}(v_j(n))}$

where${\displaystyle {\varphi }^{\mathrm{\prime }}}$ 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

${\displaystyle -\frac{\mathrm{\partial }\mathcal{E}(n)}{\mathrm{\partial }{v}_j(n)}={\varphi }^{\mathrm{\prime }}(v_j(n))\sum _k-\frac{\mathrm{\partial }\mathcal{E}(n)}{\mathrm{\partial }{v}_k(n)}{w}_{kj}(n)}$.

This depends on the change in weights of the${\displaystyle k}$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.^[26]

• Weka: Open source data mining software with multilayer perceptron implementation.
• Neuroph Studio documentation, implements this algorithm and a few others.

• Classification algorithms
• Neural network architectures

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