Artificial neural network

An artificial neural network (ANN), also called a simulated neural network (SNN) or commonly just neural network (NN) is an interconnected group of artificial neurons that uses a mathematical or computational model for information processing based on a connectionist approach to computation. In most cases an ANN is an adaptive system that changes its structure based on external or internal information that flows through the network.

In more practical terms neural networks are non-linear statistical data modeling tools. They can be used to model complex relationships between inputs and outputs or to find patterns in data.



Background
There is no precise agreed definition among researchers as to what a neural network is, but most would agree that it involves a network of simple processing elements (neurons) which can exhibit complex global behaviour, determined by the connections between the processing elements and element parameters. The original inspiration for the technique was from examination of the central nervous system and the neurons (and their axons, dendrites and synapses) which constitute one of its most significant information processing elements (see Neuroscience). In a neural network model, simple nodes (called variously "neurons", "neurodes", "PEs" ("processing elements") or "units") are connected together to form a network of nodes &mdash; hence the term "neural network". While a neural network does not per se have to be adaptive, its practical use comes with algorithms designed to alter the strength (weights) of the connections in the network to produce a desired signal flow.

These networks are also similar to the biological neural networks in the sense that functions are performed collectively and in parallel by the units, rather than there being a clear delineation of sub-tasks to which various units are assigned (see also connectionism). Currently, the term ANN tends to refer mostly to neural network models employed in statistics and artificial intelligence. Neural network models designed with emulation of the central nervous system (CNS) in mind are a subject of theoretical neuroscience.

In modern software implementations of artificial neural networks the approach inspired by biology has more or less been abandoned for a more practical approach based on statistics and signal processing. In some of these systems neural networks, or parts of neural networks (such as artificial neurons) are used as components in larger systems that combine both adaptive and non-adaptive elements. While the more general approach of such adaptive systems is more suitable for real-world problem solving, it has far less to do with the traditional artificial intelligence connectionist models. What they do however have in common is the principle of non-linear, distributed, parallel and local processing and adaptation.

Models
Neural network models in artificial intelligence are usually referred to as artificial neural networks (ANNs); these essentially simple mathematical models defining a function $$ f : X \rightarrow Y $$. Each type of ANN model corresponds to a class of such functions.

The network in artificial neural network
The word network in the term 'artificial neural network' arises because the function $$f(x)$$ is defined as a composition of other functions $$g_i(x)$$, which can further be defined as a composition of other functions. This can be conveniently represented as a network structure, with arrows depicting the dependencies between variables. A widely used type of composition is the nonlinear weighted sum, where $$f(x) = K\big(\sum_i (w_i g_i(x)\big) $$, where $$K$$ is some predefined function, such as the hyperbolic tangent. It will be convenient for the following to refer to a collection of functions $$g_i$$ as simply a vector $$g = (g_1, g_2, \ldots, g_n)$$.



This figure depicts such a decomposition of $$f$$, with dependencies between variables indicated by arrows. These can be interpreted in two ways.

The first view is the functional view: the input $$x$$ is transformed into a 3-dimensional vector $$h$$, which is then transformed into a 2-dimensional vector $$g$$, which is finally transformed into $$f$$. This view is most commonly encountered in the context of optimization.

The second view is the probabilistic view: the random variable $$F = f(G) $$ depends upon the random variable $$G = g(H)$$, which depends upon $$H=h(X)$$, which depends upon the random variable $$X$$. This view is most commonly encountered in the context of graphical models.

The two views are largely equivalent. In either case, for this particular network architecture, the components of individual layers, i.e. the components of $$g$$ are independent of each other given their input $$h$$. This naturally enables a degree of parallelism in the implementation.



Networks such as the previous one are commonly called feedforward, because their graph is a directed acyclic graph. Networks with cycles are commonly called recurrent. Such networks are commonly depicted in the manner shown at the top of the figure above, where $$f$$ is shown as being dependent upon itself. However, there is an implied temporal dependence which is not shown. What this actually means in practice is that the value of $$f$$ at some point in time $$t$$ depends upon the values of $$f$$ at one or more other points in time. The graphical model at the bottom of the figure illustrates the case the value of $$f$$ at time $$t$$ only depends upon its last value. Models such as these, which have no dependencies in the future, are called causal models.

graphical models

Learning
However interesting such functions may be in themselves, what has attracted the most interest in neural networks is the possibility of learning, which in practice means the following:

Given a specific task to solve, and a class of functions $$F$$, learning means using a set of observations, in order to find $$f^* \in F$$ which solves the task in an optimal sense.

This entails defining a cost function $$C : F \rightarrow \mathbb{R}$$ such that, for the optimal solution $$f^*$$, $$C(f^*) \leq C(f)$$ $$\forall f \in F$$

The cost function $$C$$ is an important concept in learning, as it is a measure of how far away we are from an optimal solution to the problem that we want to solve. Learning algorithms search through the solution space in order to find a function that has the smallest possible cost.

For applications where the solution is dependent on some data, the cost must necessarily be a function of the observations, otherwise we would not be modelling anything related to the data. It is frequently defined as a statistic to which only approximations can be made. As a simple example consider the problem of finding the model $$f$$ which minimises $$C=E[|f(x) - y|^2]$$, for data pairs $$(x,y)$$ drawn from some distribution $$\mathcal{D}$$. In practical situations we would only have $$N$$ samples from $$\mathcal{D}$$ and thus, for the above example, we would only minimise $$\hat{C}=\frac{1}{N}\sum_{i=1}^N |f(x_i)-y_i|^2$$. Thus, the cost is minimised over a sample of the data rather than the true data distribution.

When $$N \rightarrow \infty$$ some form of online learning must be used, where the cost is partially minimised as each new example is seen. While online learning is often used when $$\mathcal{D}$$ is fixed, it most useful in the case where the distribution changes slowly over time. In neural network methods, some form of online learning is frequently also used for finite datasets.

See also: Optimization (mathematics), Statistical Estimation, Machine Learning

Parameterized functions and learning rules
Usually, but not always, in artificial neural networks, the function $$f$$ is defined by a vector of parameters $$w$$; this allows the optimisation to take place in the parameter space. The combination of a particular cost function with some parameterised function and a specific optimisation method, leads to a set of update equations for the parameters. This is commonly referred to as the learning rule.

Some commonly used neural network functions are:
 * $$f(x) = \tanh(w'x + w_0)$$, with $$w, x \in \mathbb{R}^n$$ and $$w_0 \in \mathbb{R}$$
 * $$f(x) = \exp(w_0\|x-w\|^2)$$, with $$w, x \in \mathbb{R}^n$$$$w_0 \in \mathbb{R}$$
 * $$f(x) = \frac{\exp(w_i)}{\sum_j \exp{w_j}}x_i$$, with $$ x_i, w_i \in \mathbb{R}$$

Choosing a cost function
While it is possible to arbitrarily define some ad hoc cost function, frequently a particular cost will be used either because it has desirable properties (such as convexity) or because it arises naturally from a particular formulation of the problem (i.e. in a probabilistic formulation the posterior probability of the model can be used as an inverse cost). Ultimately, the cost function will depend on the task we wish to perform. The three main categories of learning tasks are overviewed below.

Learning paradigms
There are three major learning paradigms, each corresponding to a particular abstract learning task. These are supervised learning, unsupervised learning and reinforcement learning. Usually any given type of network architecture can be employed in any of those tasks.

Supervised learning
In supervised learning, we are given a set of example pairs $$ (x, y), x \in X, y \in Y$$ and the aim is to find a function f in the allowed class of functions that matches the examples. In other words, we wish to infer the mapping implied by the data; the cost function is related to the mismatch between our mapping and the data and it implicitly contains prior knowledge about the problem domain.

A commonly used cost is the mean-squared error which tries to minimise the average error between the network's output, f(x), and the target value y over all the example pairs. When one tries to minimise this cost using gradient descent for the class of neural networks called Multi-Layer Perceptrons, one obtains the well-known backpropagation algorithm for training neural networks.

Tasks that fall within the paradigm of supervised learning are pattern recognition (also known as classification) and regression (also known as function approximation). The supervised learning paradigm is also applicable to sequential data, i.e. for speech and for gesture recognition.

Unsupervised learning
In unsupervised learning we are given some data $$x$$, and the cost function to be minimised can be any function of the data $$x$$ and the network's output, $$f$$.

The cost function is dependent on the task (what we are trying to model) and our a priori assumptions (the implicit properties of our model, its parameters and the observed variables).

As a trivial example, consider the model $$f(x) = a$$, where $$a$$ is a constant and the cost $$C=(E[x] - f(x))^2$$. Minimising this cost will give us a value of $$a$$ that is equal to the mean of the data. The cost function can be much more complicated. Its form depends on the application: For example in compression it could be related to the mutual information between x and y. In statistical modelling, it could be related to the posterior probability of the model given the data. (Note that in both of those examples those quantities would be maximised rather than minimised)

Tasks that fall within the paradigm of unsupervised learning are in general estimation problems; the applications include clustering, the estimation of statistical distributions, compression and filtering.

Reinforcement learning
In reinforcement learning, data $$x$$ is usually not given, but generated by an agent's interactions with the environment. At each point in time $$t$$, the agent performs an action $$y_t$$ and the environment generates an observation $$x_t$$ and an instantaneous cost $$c_t$$, according to some (usually unknown) dynamics. The aim is to discover a policy for selecting actions that minimises some measure of a long-term cost, i.e. the expected cumulative cost. The environment's dynamics and the long-term cost for each policy are usually unknown, but can be estimated.

More formally, the environment is defined as a Markov decision process (MDP) with states $$s \in S$$ and the following probability distributions: the instantaneous cost distribution $$P(c_t|s_t)$$, the observation distribution $$P(x_t|s_t)$$ and the transition $$P(s_{t+1}|s_t, y_t)$$, while a policy is defined as conditional distribution over actions given the observations. Taken together, the two define a Markov chain (MC). The aim is to discover the policy that minimises the cost, i.e. the MC for which the cost is minimal.

ANNs are frequently used in reinforcement learning as part of the overall algorithm.

Tasks that fall within the paradigm of reinforcement learning are control problems, games and other sequential decision making tasks.

See also: dynamic programming, stochastic control

Learning algorithms
There are numerous algorithms available for training neural network models; most of them can be viewed as a straightforward application of optimization theory and statistical estimation.

Most of the algorithms used in training artificial neural networks are employing some form of gradient descent. This is done by simply taking the derivative of the cost function with respect to the network parameters and then changing those parameters in a gradient-related direction.

Evolutionary methods, simulated annealing, and expectation maximisation and non-parametric methods are among other commonly used methods for training neural networks. See also machine learning.

Employing artificial neural networks
Perhaps the greatest advantage of ANNs is their ability to be used as an arbitrary function approximation mechanism which 'learns' from observed data. However, using them is not so straightforward and a relatively good understanding of the underlying theory is essential.


 * Choice of model: This will depend on the data representation and the application. Overly complex models tend to lead to problems with learning.
 * Learning algorithm: There are numerous tradeoffs between learning algorithms. Almost any algorithm will work well with the correct hyperparameters for training on a particular fixed dataset. However selecting and tuning an algorithm for training on unseen data requires a significant amount of experimentation.
 * Robustness: If the model, cost function and learning algorithm are selected appropriately the resulting ANN can be extremely robust.

With the correct implementation ANNs can be used naturally in online learning and large dataset applications. Their simple implementation and the existence of mostly local dependencies exhibited in the structure allows for fast, parallel implementations in hardware.

Applications
The utility of artificial neural network models lies in the fact that they can be used to infer a function from observations. This is particularly useful in applications where the complexity of the data or task makes the design of such a function by hand impractical.

Real life applications
The tasks to which artificial neural networks are applied tend to fall within the following broad categories:


 * Function approximation, or regression analysis, including time series prediction and modelling.
 * Classification, including pattern and sequence recognition, novelty detection and sequential decision making.
 * Data processing, including filtering, clustering, blind source separation and compression.

Application areas include system identification and control (vehicle control, process control), game-playing and decision making (backgammon, chess, racing), pattern recognition (radar systems, face identification, object recognition and more), sequence recognition (gesture, speech, handwritten text recognition), medical diagnosis, financial applications, data mining (or knowledge discovery in databases, "KDD"), and visualisation.

Neural network software
Main article: Neural network software

Neural network software is used to simulate, research, develop and apply artificial neural networks, biological neural networks and in some cases a wider array of adaptive systems.

Feedforward neural network
The feedforward neural networks are the first and arguably simplest type of artificial neural networks devised. In this network, the information moves in only one direction, forward, from the input nodes, through the hidden nodes (if any) and to the output nodes. There are no cycles or loops in the network.

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 feed-forward 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 activated value (typically 1); otherwise it takes the deactivated value (typically -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.

A perceptron can be created using any values for the activated and deactivated states as long as the threshold value lies between the two. Most perceptrons have outputs of 1 or -1 with a threshold of 0 and there is some evidence that such networks can be trained more quickly than networks created from nodes with different activation and deactivation values.

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:


 * $$\frac{1}{1+e^{-x}}$$

With this choice, the single-layer network is identical to the logistic regression model, widely used in statistical modelling. The logistic function is also known as the sigmoid function. It has a continuous derivative, which allows it to be used in backpropagation.

Multi-layer perceptron


This class of networks consists of multiple layers of computational units, usually interconnected in a feed-forward way. 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 back-propagation. 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 non-linear 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 back-propagation can only be applied on networks with differentiable activation functions.

In general the problem of teaching a network that performs well, even on samples that were not used as training samples, is a quite subtle issue that requires additional techniques. This is especially important for cases where only very limited numbers of training samples are available. The danger is that the network overfits the training data and fails to capture the true statistical process generating the data. Computational 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 back-propagation in multi-layer perceptrons the solution of choice for many machine learning tasks.

ADALINE
ADAptive LInear NEuron or later called Adapter Linear Element. It was developed by Professor Bernard Widrow and his graduate student Ted Hoff at Stanford University in 1960. It's based on McCulloch-Pitts model. It consists of a weight, a bias and a summation function.

Operation: $$y_i=wx_i+b$$

Its adaptation is defined through a cost function (error metric) of the residual $$e=d_i-(b+wx_i)$$ where $$d_i$$ is the desired input. With the MSE error metric $$E=\frac{1}{2N}\sum_i^N e_i^2$$ the adapted weight and bias become: $$b=\frac{\sum_i x_i^2\sum_i d_i - \sum_i x_i \sum_i x_i d_i}{N(\sum_i(x_i - \bar x)^2)}$$ and $$w=\frac{\sum_i(x_i - \bar x)(d_i - \bar d)}{\sum_i(x_i - \bar x)^2}$$

While the Adaline is through this capable of simple linear regression, it has limited practical use.

There is an extension of the Adaline, called the Multiple Adaline (MADALINE)that consists of two or more adalines serially connected.

Radial basis function (RBF)
Radial Basis Functions are powerful techniques for interpolation in multidimensional space. A RBF is a function which has built into a distance criterion with respect to a centre. Radial basis functions have been applied in the area of neural networks where they may be used as a replacement for the sigmoidal hidden layer transfer function in multilayer perceptrons. RBF networks have 2 layers of processing: In the first, input is mapped onto each RBF in the 'hidden' layer. The RBF chosen is usually a Gaussian. In regression problems the output layer is then a linear combination of hidden layer values representing mean predicted output. The interpretation of this output layer value is the same as a regression model in statistics. In classification problems the output layer is typically a sigmoid function of a linear combination of hidden layer values, representing a posterior probability. Performance in both cases is often improved by shrinkage techniques, known as ridge regression in classical statistics and known to correspond to a prior belief in small parameter values (and therefore smooth output functions) in a Bayesian framework.

RBF networks have the advantage of not suffering from local minima in the same way as multilayer perceptrons. This is because the only parameters that are adjusted in the learning process are the linear mapping from hidden layer to output layer. Linearity ensures that the error surface is quadratic and therefore has a single easily found minimum. In regression problems this can be found in one matrix operation. In classification problems the fixed non-linearity introduced by the sigmoid output function is most efficiently dealt with using iterated reweighted least squares.

RBF networks have the disadvantage of requiring good coverage of the input space by radial basis functions. RBF centres are determined with reference to the distribution of the input data, but without reference to the prediction task. As a result, representational resources may be wasted on areas of the input space that are irrelevant to the learning task. A common solution is to associate each data point with its own centre, although this can make the linear system to be solved in the final layer rather large, and requires shrinkage techniques to avoid overfitting.

Associating each input datum with an RBF leads naturally to kernel methods such as Support Vector Machines and Gaussian Processes (the RBF is the kernel function). All three approaches use a non-linear kernel function to project the input data into a space where the learning problem can be solved using a linear model. Like Gaussian Processes, and unlike SVMs, RBF networks are typically trained in a Maximum Likelihood framework by maximizing the probability (minimizing the error) of the data under the model. SVMs take a different approach to avoiding overfitting by avoiding maximizing instead a margin. RBF networks are outperformed in most classification applications by SVMs. In regression applications they can be competitive when the dimensionality of the input space is relatively small.

Kohonen self-organizing network
The self-organizing map (SOM) invented by Teuvo Kohonen uses a form of unsupervised learning. A set of artificial neurons learn to map points in an input space to coordinates in an output space. The input space can have different dimensions and topology from the output space, and the SOM will attempt to preserve these.

Recurrent network
Contrary to feedforward networks, recurrent neural network (RNs) are models with bi-directional data flow. While a feedforward network propagates data linearly from input to output, RNs also propagate data from later processing stages to earlier stages.

Simple recurrent network
A simple recurrent network (SRN) is a variation on the multi-layer perceptron, sometimes called an "Elman network" due to its invention by Jeff Elman. A three-layer network is used, with the addition of a set of "context units" in the input layer. There are connections from the middle (hidden) layer to these context units fixed with a weight of one. At each time step, the input is propagated in a standard feed-forward fashion, and then a learning rule (usually back-propagation) is applied. The fixed back connections result in the context units always maintaining a copy of the previous values of the hidden units (since they propagate over the connections before the learning rule is applied). Thus the network can maintain a sort of state, allowing it to perform such tasks as sequence-prediction that are beyond the power of a standard multi-layer perceptron.

In a fully recurrent network, every neuron receives inputs from every other neuron in the network. These networks are not arranged in layers. Usually only a subset of the neurons receive external inputs in addition to the inputs from all the other neurons, and another disjunct subset of neurons report their output externally as well as sending it to all the neurons. These distinctive inputs and outputs perform the function of the input and output layers of a feed-forward or simple recurrent network, and also join all the other neurons in the recurrent processing.

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

Stochastic neural networks
A stochastic neural network differs from a regular neural network in the fact that it introduces random variations into the network. In a probabilistic view of neural networks, such random variations can be viewed as a form of statistical sampling, such as Monte Carlo sampling.

Boltzmann machine
The Boltzmann machine can be thought of as a noisy Hopfield network. Invented by Geoff Hinton and Terry Sejnowski in 1985, the Boltzmann machine is important because it is one of the first neural networks to demonstrate learning of latent variables (hidden units). Boltzmann machine learning was slow to simulate, but the contrastive divergence algorithm of Geoff Hinton (circa 2000) allows models including Boltzmann machines and product of experts to be trained much faster.

Modular neural networks
Biological studies showed that the human brain functions not as one single massive network, but as a collection of small networks. This realisation gave birth to the concept of modular neural networks, in which several small networks cooperate or compete to solve problems.

Committee of machines
A committee of machines (CoM) is a collection of different neural networks that together "vote" on a given example. This generally gives a much better result compared to other neural network models. In fact in many cases, starting with the same architecture and training but different initial random weights gives vastly different networks. A CoM tends to stabilize the result.

The CoM is similar to the general machine learning bagging method, except that the necessary variety of machines in the committee is obtained by training from different random starting weights rather than training on different randomly selected subsets of the training data.

Associative Neural Network (ASNN)
Is an extension of the committee of machines that goes beyond a simple/weighted average of different models. ASNN represents a combination of an ensemble of feed-forward neural networks and the k-nearest neighbour technique (kNN). It uses the correlation between ensemble responses as a measure of distance amid the analysed cases for the kNN. This corrects the bias of the neural network ensemble. An associative neural network has a memory that can coincide with the training set. If new data becomes available, the network instantly improves its predictive ability and provides data approximation (self-learn the data) without a need to retrain the ensemble. Another important feature of ASNN is the possibility to interpret neural network results by analysis of correlations between data cases in the space of models. The method can be used on-line or downloaded at www.vcclab.org.

Other types of networks
These special networks do not fit in any of the previous categories.

Instantaneously trained networks
Instantaneously trained neural networks (ITNNs) 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
Spiking (or pulsed) neural networks (SNNs) are models which explicitly take into account the timing of inputs. The network input and output are usually represented as series of spikes (delta function or more complex shapes). SNNs have an advantage of being able to continuously process information. They are often implemented as recurrent networks.

Networks of spiking neurons -- and the temporal correlations of neural assemblies in such networks -- have been used to model figure/ground separation and region linking in the visual system (see e.g. Reitboeck et.al.in Haken and Stadler: Synergetics of the Brain. Berlin, 1989).

Gerstner and Kistler have a freely-available online textbook on Spiking Neuron Models.

Spiking neural networks with axonal conduction delays exhibit polychronization, and hence could have a potentially unlimited memory capacity.

In June 2005 IBM announced construction of a Blue Gene supercomputer dedicated to the simulation of a large recurrent spiking neural network.

Dynamic neural networks
Dynamic neural networks not only deal with nonlinear multivariate behaviour, but also include (learning of) time-dependent behaviour such as various transient phenomena and delay effects. Meijer has a Ph.D. thesis online where regular feedforward perception networks are generalized with differential equations, using variable time step algorithms for learning in the time domain and including algorithms for learning in the frequency domain (in that case linearized around a set of static bias points).

Cascading neural networks
These neural networks begin their training without any hidden neurons. As the output error reaches a predefined error threshold, the networks add a new hidden neuron. The new hidden neuron is connected to all input nodes, as well as, all previous hidden neurons. Training terminates when a suitable error threshold is reached or when the maximum number of hidden neurons is added.

Capacity
Artificial neural network models have a property called 'capacity', which roughly corresponds to their ability to model any given function. It is related to the amount of information that can be stored in the network and to the notion of complexity.

Convergence
Nothing can be said in general about convergence, since it depends on a number of factors. Firstly, there may exist many local minima. This depends on the cost function and the model. Secondly, the optimisation method used might not be guaranteed to converge when far away from a local minimum. Thirdly, for a very large amount of data or parameters, some methods become impractical. In general, however, it has been found that theoretical guarantees regarding convergence are not always a very reliable guide to practical application.

Generalisation and statistics
In applications where the goal is to create a system that generalises well in unseen examples, the problem of overtraining has emerged. This arises in overcomplex or overspecified systems when the capacity of the network significantly exceeds the needed free parameters. There are two schools of thought for avoiding this problem: The first is to use cross-validation and similar techniques to check for the presence of overtraining and optimally select hyperparameters such as to minimise the generalisation error. The second is to use some form of regularisation. This is a concept that emerges naturally in a probabilistic (Bayesian) framework, where the regularisation can be performed by putting a larger prior probability over simpler models; but also in statistical learning theory, where the goal is to minimise over two quantities: the 'empirical risk' and the 'structural risk', which roughly correspond to the error over the training set and the predicted error in unseen data due to overfitting. Supervised neural networks that use an MSE cost function can use formal statistical methods to determine the confidence of the trained model. The MSE on a validation set can be used as an estimate for variance. This value can then be used to calculate the confidence interval of the output of the network, assuming a normal distribution. A confidence analysis made this way is statistically valid as long as the output probability distribution stays the same and the network is not modified.

By assigning a softmax activation function on the output layer of the neural network (or a softmax component in a component based neural network) for categorical target variables, the outputs can be interpreted as posterior probabilities. This is very useful in classification as it gives a certainty measure on classifications.

The softmax activation function: $$y_i=\frac{e^x_i}{\sum_{j=1}^c e^x_j}$$

Dynamical properties
Various techniques originally developed for studying disordered magnetic systems (spin glasses) have been successfully applied to simple neural network architectures, such as the perceptron. Influential work by E. Gardner and B. Derrida has revealed many interesting properties about perceptrons with real-valued synaptic weights, while later work by W. Krauth and M. Mezard has extended these principles to binary-valued synapses.

Patents

 * Arima, et al., ,"Neural network integrated circuit device having self-organizing function". March 8, 1994.