Multilayer perceptron

A multilayer perceptron (MLP) is a class of. An MLP consists of at least three layers of nodes: an input layer, a hidden layer and an output layer. Except for the input nodes, each node is a neuron that uses a nonlinear. MLP utilizes a technique called  for training. Its multiple layers and non-linear activation distinguish MLP from a linear. It can distinguish data that is not.

Multilayer perceptrons are sometimes colloquially referred to as "vanilla" neural networks, especially when they have a single hidden layer.

Activation function
If a multilayer perceptron has a linear in all neurons, that is, a linear function that maps the  to the output of each neuron, then  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, or firing, of biological neurons.

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


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

In recent developments of the  is more frequently used as one of the possible ways to overcome the numerical  related to the sigmoids.

The first is a that ranges from -1 to 1, while the other is the, which is similar in shape but ranges from 0 to 1. Here $$y_i$$ is the output of the $$i$$th node (neuron) and $$v_i$$ is the weighted sum of the input connections. Alternative activation functions have been proposed, including the functions. More specialized activation functions include (used in  another class of supervised neural network models).

Layers
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 $$w_{ij}$$ to every node in the following layer.

Learning
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, and is carried out through , a generalization of the in the linear perceptron.

We can represent the degree of error in an output node $$j$$ in the $$n$$th data point (training example) by $$e_j(n)=d_j(n)-y_j(n)$$, where $$d$$ is the target value and $$y$$ is the value produced by the perceptron. The node weights can then be adjusted based on corrections that minimize the error in the entire output, given by


 * $$\mathcal{E}(n)=\frac{1}{2}\sum_j e_j^2(n)$$.

Using, the change in each weight is


 * $$\Delta w_{ji} (n) = -\eta\frac{\partial\mathcal{E}(n)}{\partial v_j(n)} y_i(n)$$

where $$y_i$$ is the output of the previous neuron and $$\eta$$ is the , which is selected to ensure that the weights quickly converge to a response, without oscillations.

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


 * $$-\frac{\partial\mathcal{E}(n)}{\partial v_j(n)} = e_j(n)\phi^\prime (v_j(n))$$

where $$\phi^\prime$$ 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


 * $$-\frac{\partial\mathcal{E}(n)}{\partial v_j(n)} = \phi^\prime (v_j(n))\sum_k -\frac{\partial\mathcal{E}(n)}{\partial v_k(n)} w_{kj}(n)$$.

This depends on the change in weights of the $$k$$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.

Terminology
The term "multilayer perceptron" does not refer to a single perceptron that has multiple layers. Rather, it contains many perceptrons that are organized into layers. An alternative is "multilayer perceptron network". Moreover, MLP "perceptrons" are not perceptrons in the strictest possible sense. True perceptrons are formally a special case of artificial neurons that use a threshold activation function such as the. MLP perceptrons can employ arbitrary activation functions. A true perceptron performs binary classification (either this or that), an MLP neuron is free to either perform classification or regression, depending upon its activation function.

The term "multilayer perceptron" later was applied without respect to nature of the nodes/layers, which can be composed of arbitrarily defined artificial neurons, and not perceptrons specifically. This interpretation avoids the loosening of the definition of "perceptron" to mean an artificial neuron in general.

Applications
MLPs are useful in research for their ability to solve problems stochastically, which often allows approximate solutions for extremely problems like.

MLPs are universal function approximators as shown by Cybenko's theorem, so they can be used to create mathematical models by regression analysis. As is a particular case of  when the response variable is, MLPs make good classifier algorithms.

MLPs were a popular machine learning solution in the 1980s, finding applications in diverse fields such as, , and software, but thereafter faced strong competition from much simpler (and related) s. Interest in backpropagation networks returned due to the successes of.