# Neural network

An**artificial neural network**, more commonly known as a

**neural network**or

**neural net**for short, is a mathematical model for information processing based on a connectionist approach to computation. The original inspiration for the technique was from examination of bioelectrical networks in the brain formed by neurons and their synapses. In a neural network model, simple nodes (or "neurons", or "units") are connected together to form a network of nodes - hence the term "neural network".

See also: biological neural networks.

## Structure

Most researchers today would agree that artificial neural networks are quite different from the brain in terms of structure. Like the brain, however, a neural net is a massively parallel collection of small and simple processing units where the interconnections form a large part of the network's intelligence; however, in terms of scale, a brain is massively larger than a neural network, and the units used in a neural network are typically far simpler than neurons. Nevertheless, certain functions that seem exclusive to the brain such as dreaming and learning, have been replicated on a simpler scale, with neural networks.

### Models

A typical *feedforward* neural network will consist of a set of nodes. Some of these are designated *input nodes*, some *output nodes*, and those in between *hidden nodes*. There are also connections between the neurons, with a number referred to as a *weight* associated with each connection. When the network is in operation, a value will be applied to each input node - the values being fed in by a human operator, or from environmental sensors, or perhaps from some other program. Each node then passes its given value to the connections leading out from it, and on each connection the value is multiplied by the weight associated with that connection. Each node in the next layer then receives a value which is the sum of the values produced by the connections leading into it, and in each node a simple computation is performed on the value - a sigmoid function is typical. This process is then repeated, with the results being passed through subsequent layers of nodes until the output nodes are reached. Early models (circa 1970) had a fixed number of layers. More recently, genetic algorithms are used to evolve the neural structure. See: Neuroevolution.

### Calculations

It should be noted that the sigmoid curve is used as a transfer function because it has the effect of "squashing" the inputs into the range [0,1]. Other functions with similar features can be used, most commonly tanh which has an output range of [-1,1]. The sigmoid function has the additional benefit of having an extremely simple derivative function for backpropagating errors through a feed-forward neural network.

Typically the weights in a neural network are initially set to small random values; this represents the network knowing nothing. As the training process proceeds, these weights will converge to values allowing them to perform a useful computation. Thus it can be said that the neural network commences knowing nothing and moves on to gain some real knowledge.

### Usefulness

Neural networks are particularly useful for dealing with bounded real-valued data, where a real-valued output is desired; in this way neural networks will perform classification by degrees, and are capable of expressing values equivalent to "not sure". If the neural network is trained using the cross-entropy error function (see Bishop's book) and if the neural network output has a sigmoidal non-linear, then the outputs will be estimates of the true posterior probability of a class.

## Real life applications

In real life applications, neural networks perform particularly well on the following common tasks:

Other kinds of neural networks, in particular Continuous-Time Recurrent Neural Networks (CTRNN), are used in conjunction with genetic algorithms(GAs) to produce robot controllers. The genome is then constitued of the networks parameters and the fitness of a network is the adequacy of the behaviour exhibited by the controlled robot (or often by a simulation of this behaviour).## Types of neural networks

### Single layer perceptron

The earliest kind of neural network is a *single-layer perceptron* network, which consists of a single layer of output nodes; the inputs are fed directly to the outputs via a series of weights. In this way it can be considered the simplest kind of feedforward network. The sum of the products of the weights and the inputs is calculated in each node, and if the value is above some threshold (typically 0) the neuron fires and takes the value 1; otherwise it takes the value -1. Neurons with this kind of activation function are also called *McCulloch-Pitts neurons* or *threshold neurons*. In the literature the term *perceptron* often refers to networks consisting of just one of these units. They were described by Warren McCulloch and Walter Pitts in the 1940s.

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.

Single-unit perceptrons are only capable of learning linearly separable patterns; in 1969 in a famous monograph entitled *Perceptrons* by Marvin Minsky and Seymour Papert showed that it was impossible for a single layer perceptron network to learn an XOR function. They conjectured (incorrectly) that a similar result would hold for a multi-layer perceptron network. Although a single threshold unit is quite limited in its computational power, it has been shown that networks of parallel threshold units can approximate any continuous function from a compact interval of the real numbers into the interval [-1, 1]. This very recent result can be found in [Auer, Burgsteiner, Maass: The p-delta learning rule for parallel perceptrons, 2001 (state Jan 2003: submitted for publication)].

A single-layer neural network can compute a continuous output instead of a step function.
A common choice is the so-called logistic function, 1/(1+exp(-*x*)).
With this choice, the single-layer network is identical to the logistic regression model,
widely used in statistical modeling.

### Multi-layer perceptron

This class of networks consists of multiple layers of computational units, usually interconnected in a feedforward way. This means that each neuron in one layer has directed connections to the neurons of the subsequent layer. In many applications the units of these networks apply a sigmoid function as an activation function.

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, e.g. for the sigmoidal functions.

Multi-layer networks use a variety of learning techniques, the most popular being *backpropagation*. Here the output values are compared with the correct answer to compute the value of some predefined error-function. By various techniques the error is then fed back through the network. Using this information, the algorithm adjusts the weights of each connection in order to reduce the value of the error-function by some small amount. After repeating this process for a sufficiently large number of training cycles the network will usually converge to some state where the error of the calculations is small. In this case one says that the network has *learned* a certain target function. To adjust weights properly one applies a general method for nonlinear optimization task that is called gradient descent. For this the derivation of the error-function with respect to the network weights is calculated and the weights are then changed such that the error decreases (thus going downhill on the surface of the error function). For this reason backpropagation can only be applied on networks with differentiable activation function.

In general the problem of reaching a network that performs well, even on examples that were not used as training examples, is a quite subtle issue that requires additional techniques. This is especially important for cases where only very limited numbers of training examples are available. The danger is that the network overfits the training data and fails to capture the true statistical process generating the data. Statistical learning theory is concerned with training classifiers on a limited amount of data. In the context of neural networks a simple heuristic, called early stopping, often ensures that the network will generalize well to examples not in the training set.

Other typical problems of the back-propagation algorithm are the speed of convergence and the possibility to end up in a local minimum of the error function. Today there are practical solutions that make backpropagation in multi-layer perceptrons the solution of choice for many machine learning tasks.

### Recurrent network

### Hopfield network

The Hopfield net is a recurrent neural network in which all connections are symmetric. Invented by John Hopfield (1982), this network has the property that its dynamics are guaranteed to converge. If the connections are trained using Hebbian learning then the Hopfield network can perform robust content-addressable memory, robust to connection alteration.

### Boltzmann machine

The Boltzmann machine can be thought of as a noisy Hopfield network. Invented by Geoff Hinton and Terry Sejnowski (1985), the Boltzmann machine was important because it was one of the first neural networks in which learning of latent variables (hidden units) was demonstrated. Boltzmann machine learning was slow to simulate, but the Contrastive Divergence algorithm of Geoff Hinton (introduced about 2000) allows models including Boltzmann machines and Product of Experts to be trained much faster.

### Support vector machine

A support vector machine (SVM) is a recently developed form of machine learning algorithm. The training of SVMs is based on quadratic programming, a form of optimization that (usually) has only one global minimum. Therefore, and because SVMs have means to reduce the danger of overfitting, some practitioners prefer SVM training to neural network training.

### Committee of machines

### Self-organizing map

The Self-organizing map (SOM), sometimes referred to as "Kohonen map" due to its invention by Professor Teuvo Kohonen, is an unsupervised learning technique that reduces the dimensionality of data through the use of a self-organizing neural network. A probabilistic version of SOM is the Generative Topographic Map (GTM) of Bishop, Svensen and Williams.

### Instantaneously trained networks

Instantaneously trained neural networks (ITNN) are also called "Kak networks" after their inventor Subhash Kak. They were inspired by the phenomenon of short-term learning that seems to occur instantaneously. In these networks the weights of the hidden and the output layers are mapped directly from the training vector data. Ordinarily, they work on binary data but versions for continuous data that require small additional processing are also available.

### Spiking neural networks

### Data representation

*(Some notes on statistics and other representation problems in here)*

## Relation to optimization techniques

Analysis of many neural network techniques reveals a close relationship to mathematical optimization techniques.

For instance, multi-layer perceptron back-propagation can be substituted with more general global optimization techniques. The objective in training a ANN is, given some set of pairs of data and output, { (d_{0}, o_{0}) , (d_{1},o_{1}), ... } to minimize some error function ||E||^{2}, where E(x_{i}) = F(w,x_{i}) - o_{i}. Here F is the neural network function which given a vector of weights w and an input vector produces an output vector for the network. Thus as well as using backpropagation to train the network, it's also possible to use global optimization techniques to produce a weight vector w.

For very large data sets, using more advanced optimization techniques is often slower than using gradient descent, if the weights of the network are updated by gradient descent after each training example. This is because one sweep of gradient descent through the training set can make a large amount of progress, while the same amount of computational effort can only compute a true gradient at one setting of the parameter vector.

*See also:* Time delay neural network (TDNN), systolic array, systolic automata, NEAT

## External links

- An online textbook on neural networks
- Open Directory link
- FAQs of the newsgroup comp.ai.neural-nets
- An online textbook on Information Theory, Bayesian inference, and neural networks
- University of Texas Neural Network Research Group's archive of papers, demos, and software
- An Overview of Neural Networks

## Bibliography

- Bishop, C.M. (1995)
*Neural Networks for Pattern Recognition*, Oxford: Oxford University Press. ISBN 0-19-853849-9 (hardback) or ISBN 0-19-853864-2 (paperback) - Duda, R.O., Hart, P.E., Stork, D.G. (2001)
*Pattern classification (2nd edition)*, Wiley, ISBN 0471056693 - Gurney, K. (1997)
*An Introduction to Neural Networks*London: Routledge. ISBN 1-85728-673-1 (hardback) or ISBN 1-85728-503-4 (paperback) - Hertz, J., Palmer, R.G., Krogh. A.S. (1990)
*Introduction to the theory of neural computation*, Perseus Books. ISBN 0201515601