A feedforward neural network (FNN) is one of the two broad types of artificial neural network, characterized by direction of the flow of information between its layers. Its flow is uni-directional, meaning that the information in the model flows in only one direction—forward—from the input nodes, through the hidden nodes (if any) and to the output nodes, without any cycles or loops (in contrast to recurrent neural networks, which have a bi-directional flow). Modern feedforward networks are trained using backpropagation, and are colloquially referred to as "vanilla" neural networks.

Mathematical foundations

Activation function

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 y_i is the output of the ith node (neuron) and 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 j in the nth data point (training example) by, where d_j(n) is the desired target value for nth data point at node j, and y_j(n) is the value produced at node j when the nth 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 nth data point, given by Using gradient descent, the change in each weight w_{ij} is where y_i(n) is the output of the previous neuron i, and \eta 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 v_j(n) of the input connections of neuron i. 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 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 This depends on the change in weights of the kth 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.

History

Timeline

Linear regression

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 non-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. 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 multilayer perceptron (MLP) is a misnomer for a modern feedforward artificial neural network, consisting of fully connected neurons (hence the synonym sometimes used of fully connected network (FCN)), often 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.

Other feedforward networks

Examples of other feedforward networks include convolutional neural networks and radial basis function networks, which use a different activation function.