# Single-Layer Feedforward Neural Network

A Single-Layer Feedforward Neural Network is a feedforward neural network that is a single-layer neural network.

**Example(s):**- MADALINE.
- …

**Counter-Example(s):****See:**Perceptron Function, Artificial Neural Network, Backpropagation, Parallel Delta Rule .

## References

### 2017

- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Feedforward_neural_network#Single-layer_perceptron Retrieved:2017-12-17.
- The simplest 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*artificial neurons*or*linear threshold units*. In the literature the term*perceptron*often refers to networks consisting of just one of these units. A similar neuron was 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.

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*, Marvin Minsky and Seymour Papert showed that it was impossible for a single-layer perceptron network to learn an XOR function (nonetheless, it was known that multi-layer perceptrons are capable of producing any possible boolean function).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 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".

^{[1]}A multi-layer neural network can compute a continuous output instead of a step function. A common choice is the so-called logistic function: : [math]\displaystyle{ f(x) = \frac{1}{1+e^{-x}} }[/math] With this choice, the single-layer network is identical to the logistic regression model, widely used in statistical modeling. The logistic function is also known as the sigmoid function. It has a continuous derivative, which allows it to be used in backpropagation. This function is also preferred because its derivative is easily calculated: : [math]\displaystyle{ f'(x) = f(x)(1-f(x)) }[/math] .

(The fact that f satisfies the differential equation above can easily be shown by applying the Chain Rule.)

- The simplest kind of neural network is a

- ↑ Auer, Peter; Harald Burgsteiner; Wolfgang Maass (2008). "A learning rule for very simple universal approximators consisting of a single layer of perceptrons" (PDF). Neural Networks. 21 (5): 786–795. PMID:18249524.

### 2008

- (Auer et al., 2008) ⇒ Auer, P., Burgsteiner, H., & Maass, W. (2008). "A learning rule for very simple universal approximators consisting of a single layer of perceptrons". Neural networks, 21(5), 786-795. PMID:18249524
- ABSTRACT: One may argue that the simplest type of neural networks beyond a single perceptron is an array of several perceptrons in parallel. In spite of their simplicity, such circuits can compute any Boolean function if one views the majority of the binary perceptron outputs as the binary output of the parallel perceptron, and they are universal approximators for arbitrary continuous functions with values in [0,1] if one views the fraction of perceptrons that output 1 as the analog output of the parallel perceptron. Note that in contrast to the familiar model of a “multi-layer perceptron” the parallel perceptron that we consider here has just binary values as outputs of gates on the hidden layer. For a long time one has thought that there exists no competitive learning algorithm for these extremely simple neural networks, which also came to be known as committee machines. It is commonly assumed that one has to replace the hard threshold gates on the hidden layer by sigmoidal gates (or RBF-gates) and that one has to tune the weights on at least two successive layers in order to achieve satisfactory learning results for any class of neural networks that yield universal approximators. We show that this assumption is not true, by exhibiting a simple learning algorithm for parallel perceptrons - the parallel delta rule (p-delta rule). In contrast to backprop for multi-layer perceptrons, the p-delta rule only has to tune a single layer of weights, and it does not require the computation and communication of analog values with high precision. Reduced communication also distinguishes our new learning rule from other learning rules for parallel perceptrons such as MADALINE. Obviously these features make the p-delta rule attractive as a biologically more realistic alternative to backprop in biological neural circuits, but also for implementations in special purpose hardware. We show that the p-delta rule also implements gradient descent-with regard to a suitable error measure-although it does not require to compute derivatives. Furthermore it is shown through experiments on common real-world benchmark datasets that its performance is competitive with that of other learning approaches from neural networks and machine learning. It has recently been shown [Anthony, M. (2007). On the generalization error of fixed combinations of classifiers. Journal of Computer and System Sciences 73(5), 725-734; Anthony, M. (2004). On learning a function of perceptrons. In: Proceedings of the 2004 IEEE international joint conference on neural networks (pp. 967-972): Vol. 2] that one can also prove quite satisfactory bounds for the generalization error of this new learning rule.