mlp

Multilayer perceptron neural network. An MLP consists of several layers of nodes, interconnected through weighted acyclic arcs from each preceding layer to the following, without lateral or feedback connections. Each node calculates a transformed weighted linear combination of its inputs (output activations from the preceding layer), with one of the weights acting as a trainable bias connected to a constant input. The transformation, called activation function, is a bounded non-decreasing (non-linear) function, such as the sigmoid functions (ranges from 0 to 1). Another popular activation function is hyperbolic tangent which is actually equivalent to the sigmoid function in shape but ranges from -1 to 1. More specialized activation functions include radial basis functions which are used in RBF networks.

The representational capabilities of a MLP are determined by the range of mappings it may implement through weight variation. Single layer perceptrons are capable of solving only linearly separable problems. With the sigmoid function as activation function, the single-layer network is identical to the logistic regression model.

The universal approximation theorem for neural networks states that every continuous function that maps intervals of real numbers to some output interval of real numbers can be approximated arbitrarily closely by a multi-layer perceptron with just one hidden layer. This result holds only for restricted classes of activation functions, which are extremely complex and NOT smooth for subtle mathematical reasons. On the other hand, smoothness is important for gradient descent learning. Besides, the proof is not constructive regarding the number of neurons required or the settings of the weights. Therefore, complex systems will have more layers of neurons with some having increased layers of input neurons and output neurons in practice.

The most popular algorithm to train MLPs is back-propagation, which is a gradient descent method. Based on chain rule, the algorithm propagates the error back through the network and adjusts the weights of each connection in order to reduce the value of the error function by some small amount. For this reason, back-propagation can only be applied on networks with differentiable activation functions.

During error back propagation, we usually times the gradient with a small number η, called learning rate, which is carefully selected to ensure that the network converges to a local minimum of the error function fast enough, without producing oscillations. One way to avoid oscillation at large η, is to make the change in weight dependent on the past weight change by adding a momentum term.

Although the back-propagation algorithm may performs gradient descent on the total error of all instances in a batch way, the learning rule is often applied to each instance separately in an online way or stochastic way. There exists empirical indication that the stochastic way results in faster convergence.

In practice, the problem of over-fitting has emerged. This arises in convoluted or over-specified systems when the capacity of the network significantly exceeds the needed free parameters. There are two general approaches for avoiding this problem: The first is to use cross-validation and similar techniques to check for the presence of over-fitting and optimally select hyper-parameters such as to minimize the generalization error. The second is to use some form of regularization, which emerges naturally in a Bayesian framework, where the regularization can be performed by selecting a larger prior probability over simpler models; but also in statistical learning theory, where the goal is to minimize over the "empirical risk" and the "structural risk".

For neural networks, the input patterns usually should be scaled/standardized. Commonly, each input variable is scaled into interval [0, 1] or to have mean 0 and standard deviation 1.

For penalty functions and output units, the following natural pairings are recommended:

• linear output units and a least squares penalty function.

• a two-class cross-entropy penalty function and a logistic activation function.

• a multi-class cross-entropy penalty function and a softmax activation function.

By assigning a softmax activation function on the output layer of the neural network for categorical target variables, the outputs can be interpreted as posterior probabilities, which are very useful.

Parameters