f E = 2(tp op ) zi,p vi netp in VS .NET

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f E = 2(tp op ) zi,p vi netp
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and is the learning rate (ie the size of the steps taken in the negative direction of the gradient) The calculation of the partial derivative of f with respect to netp (the net input for pattern p) presents a problem for all discontinuous activation functions,
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such as the step and ramp functions; zi,p is the i-th input signal corresponding to pattern p The Widrow-Ho learning rule presents a solution for the step and ramp functions, while the generalized delta learning rule assumes continuous functions that are at least once di erentiable
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Widrow-Ho Learning Rule
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For the Widrow-Ho learning rule [907], assume that f = netp Then giving E = 2(tp op )zi,p vi Weights are then updated using vi (t) = vi (t 1) + 2 (tp op )zi,p
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= 1, (221)
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The Widrow-Ho learning rule, also referred to as the least-means-square (LMS) algorithm, was one of the rst algorithms used to train layered neural networks with multiple adaptive linear neurons This network was commonly referred to as the Madaline [907, 908]
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Generalized Delta Learning Rule
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The generalized delta learning rule is a generalization of the Widrow-Ho learning rule that assumes di erentiable activation functions Assume that the sigmoid function (from equation (211)) is used Then, f = op (1 op ) netp giving E = 2(tp op )op (1 op )zi,p vi (224) (223)
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Error-Correction Learning Rule
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For the error-correction learning rule it is assumed that binary-valued activation functions are used, for example, the step function Weights are only adjusted when the neuron responds in error That is, only when (tp op ) = 1 or (tp op ) = 1, are weights adjusted using equation (222)
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1 Explain why the threshold is necessary What is the e ect of , and what will the consequences be of not having a threshold
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2 The Artificial Neuron 2 Explain what the e ects of weight changes are on the separating hyperplane 3 Explain the e ect of changing on the hyperplane that forms the decision boundary 4 Which of the following Boolean functions can be realized with a single neuron that implements a SU Justify your answer by giving weight and threshold values (a) z1 z2 z 3 (b) z1 z 2 + z 1 z2 (c) z1 + z2 where z1 z2 denotes (z1 AN D z2 ); z1 + z2 denotes (z1 OR z2 ); z 1 denotes (N OT z1 ) 5 Is it possible to use a single PU to learn problems that are not linearly separable 6 In the calculation of error, why is the error per pattern squared 7 Can errors be calculated as |tp op | instead of (tp op )2 if gradient descent is used to adjust weights 8 Is the following statement true or false: A single neuron can be used to approximate the function f (z) = z 2 Justify your answer 9 What are the advantages of using the hyperbolic tangent activation function instead of the sigmoid activation function
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Supervised Learning Neural Networks
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Single neurons have limitations in the type of functions they can learn A single neuron (implementing a SU) can be used to realize linearly separable functions only As soon as functions that are not linearly separable need to be learned, a layered network of neurons is required Training these layered networks is more complex than training a single neuron, and training can be supervised, unsupervised or through reinforcement This chapter deals with supervised training Supervised learning requires a training set that consists of input vectors and a target vector associated with each input vector The NN learner uses the target vector to determine how well it has learned, and to guide adjustments to weight values to reduce its overall error This chapter considers di erent NN types that learn under supervision These network types include standard multilayer NNs, functional link NNs, simple recurrent NNs, time-delay NNs, product unit NNs, and cascade networks These di erent architectures are rst described in Section 31 Di erent learning rules for supervised training are then discussed in Section 32 The chapter ends with a short discussion on ensemble NNs in Section 34