73 Performance Factors

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With reference to the bias/variance decomposition of the MSE function [313], smaller network architectures reduce the variance component of the MSE NNs are generally plagued by high variance due to the limited training set sizes This variance is reduced by introducing bias through minimization of the network architecture Smaller networks are biased because the hypothesis space is reduced; thus limiting the available functions that can t the data The e ects of architecture selection on the bias/variance trade-o have been studied by Gedeon et al [311]

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The performance of NNs can be improved by allowing activation functions to change dynamically according to the characteristics of the training data One of the rst techniques to use adaptive activations functions was developed by Zurada [961], where the slope of the sigmoid activation function is learned together with the weights A slope parameter is kept for each hidden and output unit The lambda-learning algorithm of Zurada was extended by Engelbrecht et al [244] where the sigmoid function is given as (728) f (net, , ) = 1 + e net where is the slope of the function and the maximum range Engelbrecht et al developed learning equations to also learn the maximum ranges of the sigmoid functions, thereby performing automatic scaling By using gamma-learning, it is not necessary to scale target values to the range (0, 1) The e ect of changing the slope and range of the sigmoid function is illustrated in Figure 76

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35 standard sigmoid slope = 5 slope = 05 slope = 1, range = 3 3

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Figure 76 Adaptive Sigmoid Algorithm 71 illustrates the di erences between standard GD learning (referred to as

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7 Performance Issues (Supervised Learning)

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delta learning) and the lambda and gamma learning variations (Note that although the momentum terms are omitted below, a momentum term is usually used for the weight, lambda and gamma updates)

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Ockham s razor states that unnecessarily complex models should not be preferred to simpler ones a very intuitive principle [544, 844] A neural network (NN) model is described by the network weights Model selection in NNs consists of nding a set of weights that best performs the learning task In this sense, the data, and not just the architecture should be viewed as part of the NN model, since the data is instrumental in nding the best weights Model selection is then viewed as the process of designing an optimal NN architecture as well as the implementation of techniques to make optimal use of the available training data Following from the principle of Ockham s razor is a preference then for both simple NN architectures and optimized training data Usually, model selection techniques address only the question of which architecture best ts the task Standard error back-propagating NNs are passive learners These networks passively receive information about the problem domain, randomly sampled to form a xed size training set Random sampling is believed to reproduce the density of the true distribution However, more gain can be achieved if the learner is allowed to use current attained knowledge about the problem to guide the acquisition of training examples As passive learner, a NN has no such control over what examples are presented for learning The NN has to rely on the teacher (considering supervised learning) to present informative examples The generalization abilities and convergence time of NNs are greatly in uenced by the training set size and distribution: Literature has shown that to generalize well, the training set must contain enough information to learn the task Here lies one of the problems in model selection: the selection of concise training sets Without prior knowledge about the learning task, it is very di cult to obtain a representative training set Theoretical analysis provides a way to compute worst-case bounds on the number of training examples needed to ensure a speci ed level of generalization A widely used theorem concerns the Vapnik-Chervonenkis (VC) dimension [8, 9, 54, 152, 375, 643] This theorem states that the generalization error, EG , of a learner with VC-dimension, dV C , trained on PT random examples will, with high con dence, be no worse than a limit of order dV C /PT For NN learners, the total number of weights in a one hidden layer network is used as an estimate of the VC-dimension This means that the appropriate number of examples to ensure an EG generalization is approximately the number of weights divided by EG The VC-dimension provides overly pessimistic bounds on the number of training examples, often leading to an overestimation of the required training set size [152, 337, 643, 732, 948] Experimental results have shown that acceptable generalization performances can be obtained with training set sizes much less than that

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