Delta rule

The Back-Propagation Algorithm (BPA) is a supervised learning method for training ANNs, and is one of the most common forms of training techniques. It uses a gradient-descent optimization method, also referred to as the delta rule when applied to feedforward networks. A feedforward network that has employed the delta rule for training, is called a Multi-Layer Perceptron (MLP). 

This leads to a weight inerement, ealled the delta rule, for a partieular neuron... 

The delta rule given in equation (10.76) ean be modified to inelude momentum as indieated in equation (10.81). 

Henee, using the delta rule, the weight inerements for the hidden layer are... 

The general principle behind most commonly used back-propagation learning methods is the delta rule, by which an objective function involving squares of the output errors from the network is minimized. The delta rule requires that the sigmoidal function used at each neuron be continuously differentiable. This methods identifies an error associated with each neuron for each iteration involving a cause-effect pattern. Therefore, the error for each neuron in the output layer can be represented as ... 

Other rules are of course possible. One popular choice called the Delta Rule was introduced in 1960 by Widrow and Hoff ([widrowGO], [widrow62]). Their idea was to adjust the weights in proportion to the error (= A) between the desired output (= D(t)) and the actual output (= y(t)) of the perceptron ... 

We are now ready to introduce the backpropagation learning rule (also called the generalized delta rule) for multidayercd perceptrons, credited to Rumelhart and McClelland [rumel86a]. Figure 10.12 shows a schematic of the multi-layered per-ceptron s structure. Notice that the design shown, and the only kind we will consider in this chapter, is strictly feed-forward. That is to say, information always flows from the input layer to each hidden layer, in turn, and out into the output layer. There are no feedback loops anywhere in the system. 

Backpropagation is a generalized version of the delta rule, extended to multiple layers. The central assumption of BP is that when the target output and actual output at a node differ, the responsibility for the error can be divided between ... 

There are modifications to the perceptron learning rule to help effect faster convergence. The Widrow-Hoff delta rule (Widrow Hoff, 1960) multiplies the delta term by a number less than 1, called the learning rate, tv This effectively causes smaller changes to be made at each step. There are heuristic rules to decrease T] as training time increases the idea is that big changes may be taken at first and as the final solution is approached, smaller changes may be desired. 

The perceptron delta rule can be seen as one technique to achieve the function minimization of equation 5.1. There are more effective methods that will be discussed in later sections. 

One of the early problems with multilayer perceptrons was that it was not clear how to train them. The perception training rule doesn t apply directly to networks with hidden layers. Fortunately, Rumelhart and others (Rumelhart et al 1986) devised an intuitive method that quickly became adopted and revolutionized the field of artificial neural networks. The method is called back-propagation because it computes the error term as described above and propagates the error backward through the network so that weights to and from hidden units can be modified in a fashion similar to the delta rule for perceptions. 

Since multilayer perceptions use neurons that have differentiable functions, it was possible, using the chain rule of calculus, to derive a delta rule for training similar in form and function to that for perceptions. The result of this clever mathematics is a powerful and relatively efficient iterative method for multilayer perceptions. The rule for changing weights into a neuron unit becomes... 

This has the same form as the perception delta rule but the terms are a little more complicated. 

Neural network learning algorithms BP = Back-Propagation Delta = Delta Rule QP = Quick-Propagation RP = Rprop ART = Adaptive Resonance Theory, CP = Counter-Propagation. 

The basic learning mechanism for networks of multilayer neurons is the generalized delta rule, commonly referred to as back propagation. This learning rule is more complex than that employed with the simple perceptron unit because of the greater information content associated with the continuous output variable compared with the binary output of the perceptron. 

The final output from the network for our input pattern is compared with the known, correct result and a measure of the error is computed. In order to reduce this error, the weight vectors between neurons are adjusted by using the generalized delta rule and back-propagating the error from one layer to the previous layer. 

Gaussian distribution, 2 Generalized delta rule, 150 Goodness of fit, 159 Group average clustering, 106... 

I will discuss the generalized delta rule in chapter 14 in the presentation of the learning model. 

One well-known procedure for providing feedback and altering the weights is called the delta rule (3). It works in the following way. Suppose, in the initial training trial, our weights are set up so that when our network is presented with the aforementioned vector for "A", it responds that the stimulus could have been either A, B, or E. That is, its output vector is... 

According to the delta rule, after a training trial the change (c ) in the weight of an interconnection between input unit i (like our units 1-15) and output unit j (our units 16-20) depends on the activation Ipi of the input unit and the delta Tpj-Opj of the output unit ... 

Suppose our learning rate is. 9. According to the delta rule, in our example the interconnection between unit 8 and unit 17 (the output unit for "B") should change by... 

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