Activation function

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

${\displaystyle y(v_{i})=\tanh(v_{i})~~{\textrm {and}}~~y(v_{i})=(1+e^{-v_{i}})^{-1}}$.

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 ${\displaystyle y_{i}}$ 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 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.

Learning

Learning occurs 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.

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 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

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

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

${\displaystyle \Delta w_{ji}(n)=-\eta {\frac {\partial {\mathcal {E}}(n)}{\partial v_{j}(n)}}y_{i}(n)}$

where ${\displaystyle y_{i}(n)}$ is the output of the previous neuron ${\displaystyle i}$, and ${\displaystyle \eta }$ is the learning rate, which is selected to ensure that the weights quickly converge to a response, without oscillations. In the previous expression, ${\displaystyle {\frac {\partial {\mathcal {E}}(n)}{\partial v_{j}(n)}}}$ 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

${\displaystyle -{\frac {\partial {\mathcal {E}}(n)}{\partial v_{j}(n)}}=e_{j}(n)\phi ^{\prime }(v_{j}(n))}$

where ${\displaystyle \phi ^{\prime }}$ 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

${\displaystyle -{\frac {\partial {\mathcal {E}}(n)}{\partial v_{j}(n)}}=\phi ^{\prime }(v_{j}(n))\sum _{k}-{\frac {\partial {\mathcal {E}}(n)}{\partial v_{k}(n)}}w_{kj}(n)}$.

This depends on the change in weights of the ${\displaystyle k}$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.^[24]

Linear neural network

The simplest kind of feedforward neural network is a linear network, which consists of a single layer of output nodes; the inputs are fed directly to the outputs via a series of weights. The sum of the products of the weights and the inputs is calculated in each node. The mean squared errors between these calculated outputs and a given target values are minimized by creating an adjustment to the weights. This technique has been known for over two centuries as the method of least squares or linear regression. It was used as a means of finding a good rough linear fit to a set of points by Legendre (1805) and Gauss (1795) for the prediction of planetary movement.^{[25][26][27][12][28]}

Perceptron

If using a threshold, i.e. a linear activation function, the resulting linear threshold unit is called a perceptron. (Often the term is used to denote just one of these units.) Multiple parallel linear units are able to approximate any continuous function from a compact interval of the real numbers into the interval [−1,1] despite the limited computational power of single unit with a linear threshold function. This result can be found in Peter Auer, Harald Burgsteiner and Wolfgang Maass "A learning rule for very simple universal approximators consisting of a single layer of perceptrons".^[29]

Perceptrons can be trained by a simple learning algorithm that is usually called the delta rule. It calculates the errors between calculated output and sample output data, and uses this to create an adjustment to the weights, thus implementing a form of gradient descent.

Multilayer perceptron

A two-layer neural network capable of calculating XOR. The numbers within the neurons represent each neuron's explicit threshold. The numbers that annotate arrows represent the weight of the inputs. Note that If the threshold of 2 is met then a value of 1 is used for the weight multiplication to the next layer. Not meeting the threshold results in 0 being used. The bottom layer of inputs is not always considered a real neural network layer

A multilayer perceptron (MLP) is a misnomer 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.^[30] 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).