(a) Small

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(b) Large gets stuck

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(c) Large overshoots

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(d) Small gets stuck

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Figure 75 E ect of Learning Rate Of course more complex adaptive learning rate techniques have been developed, with elaborate theoretical analysis The interested reader is referred to [170, 552, 755, 880] Momentum Stochastic learning, where weights are adjusted after each pattern presentation, has the disadvantage of uctuating changes in the sign of the error derivatives The network spends a lot of time going back and forth, unlearning what the previous steps have learned Batch learning is a solution to this problem, since weight changes are accumulated and applied only after all patterns in the training set have been presented Another solution is to keep with stochastic learning, and to add a momentum term

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73 Performance Factors

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The idea of the momentum term is to average the weight changes, thereby ensuring that the search path is in the average downhill direction The momentum term is then simply the previous weight change weighted by a scalar value If = 0, then the weight changes are not in uenced by past weight changes The larger the value of , the longer the change in the steepest descent direction has to be persevered in order to a ect the direction in which weights are adjusted A static value of 09 is usually used The optimal value of can also be determined through cross-validation Strategies have also been developed that use adaptive momentum rates, where each weight has a di erent momentum rate Fahlman developed the schedule kj (t) =

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E wkj (t) E wkj (t 1)

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E wkj (t)

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(724)

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This variation to the standard back-propagation algorithm is referred to as quickprop [253] Becker and Le Cun [57] calculated the momentum rate as a function of the second-order error derivatives: 2E = ( 2 ) 1 (725) wkj For more information on other approaches to adapt the momentum rate refer to [644, 942]

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Optimization Method

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The optimization method used to determine weight adjustments has a large in uence on the performance of NNs While GD is a very popular optimization method, GD is plagued by slow convergence and susceptibility to local minima (as introduced and discussed in Section 322) Improvements of GD have been made to address these problems, for example, the addition of the momentum term Also, secondorder derivatives of the objective function have been used to compute weight updates In doing so, more information about the structure of the error surface is used to direct weight changes The reader is referred to [51, 57, 533] Other approaches to improve NN training are to use global optimization algorithms instead of local optimization algorithms, for example simulated annealing [736], genetic algorithms [247, 412, 494], particle swarm optimization algorithms [157, 229, 247, 862, 864], and LeapFrog optimization [247, 799, 800]

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Architecture Selection

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Referring to one of Ockham s statements, if several networks t the training set equally well, then the simplest network (ie the network that has the smallest number of weights) will on average give the best generalization performance [844] This hypothesis has been investigated and con rmed by Sietsma and Dow [789] A network with

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7 Performance Issues (Supervised Learning)

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too many free parameters may actually memorize training patterns and may also accurately t the noise embedded in the training data, leading to bad generalization Over tting can thus be prevented by reducing the size of the network through elimination of individual weights or units The objective is therefore to balance the complexity of the network with goodness-of- t of the true function This process is referred to as architecture selection Several approaches have been developed to select the optimal architecture, ie regularization, network construction (growing) and pruning These approaches will be overviewed in more detail below Learning is not just perceived as nding the optimal weight values, but also nding the optimal architecture However, it is not always obvious what is the best architecture Finding the ultimate best architecture requires a search of all possible architectures For large networks an exhaustive search is prohibitive, since the search space consists of 2w architectures, where w is the total number of weights [602] Instead, heuristics are used to reduce the search space A simple method is to train a few networks of di erent architecture and to choose the one that results in the lowest generalization error as estimated from the generalized prediction error [603, 604] or the network information criterion [616, 617, 618] This approach is still expensive and requires many architectures to be investigated to reduce the possibility that the optimal model is not found The NN architecture can alternatively be optimized by trial and error An architecture is selected, and its performance is evaluated If the performance is unacceptable, a di erent architecture is selected This process continues until an architecture is found that produces an acceptable generalization error Other approaches to architecture selection are divided into three categories: Regularization: Neural network regularization involves the addition of a penalty term to the objective function to be minimized In this case the objective function changes to (726) E = ET + EC where ET is the usual measure of data mis t, and EC is a penalty term, penalizing network complexity (network size) The constant controls the in uence of the penalty term With the changed objective function, the NN now tries to nd a locally optimal trade-o between data-mis t and network complexity Neural network regularization has been studied rigorously by Girosi et al [318], and Williams [910] Several penalty terms have been developed to reduce network size automatically 2 wi , is intended to drive small during training Weight decay, where EC = 1 2 weights to zero [79, 346, 435, 491] It is a simple method to implement, but su ers from penalizing large weights at the same rate as small weights To solve this problem, Hanson and Pratt [346] propose the hyperbolic and exponential penalty functions which penalize small weights more than large weights Nowlan and Hinton [633] developed a more complicated soft weight sharing, where the distribution of weight values is modeled as a mixture of multiple Gaussian distributions A narrow Gaussian is responsible for small weights, while a broad Gaussian is responsible for large weights Using this scheme, there is less pressure on large weights to be reduced Weigend et al [895] propose weight elimination where the penalty function

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