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tkd should vary with a change in the input xi Similarly, denotes the ax, corresponding derivative of the actual learned network The constant , deu termines the relative weight placed on fitting the training values versus the training derivatives Minimizing the cross entropy of the network with respect to the target values Consider learning a probabilistic function, such as predicting whether a loan applicant will pay back a loan based on attributes such as the applicant's age and bank balance Although the training examples exhibit only boolean target values (either a 1 or 0, depending on whether this applicant paid back the loan), the underlying target function might be best modeled by outputting the probability that the given applicant will repay the loan, rather than attempting to output the actual 1 and 0 value for each input instance Given such situations in which we wish for the network to output probability estimates, it can be shown that the best (ie, maximum likelihood) probability estimates are given by the network that minimizes the cross entropy, defined as
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Here od is the probability estimate output by the network for training example d, and td is the 1 or 0 target value for training example d 6 discusses when and why the most probable network hypothesis is the one that minimizes this cross entropy and derives the corresponding gradient descent weight-tuning rule for sigmoid units That chapter also describes other conditions under which the most probable hypothesis is the one that minimizes the sum of squared errors Altering the effective error function can also be accomplished by weight sharing, or "tying together" weights associated with different units or inputs The idea here is that different network weights are forced to take on identical values, usually to enforce some constraint known in advance to the human designer For example, Waibel et al (1989) and Lang et al (1990) describe an application of neural networks to speech recognition, in which the network inputs are the speech frequency components at different times within a 144 millisecond time window One assumption that can be made in this application is that the frequency components that identify a specific sound (eg, "eee") should be independent of the exact time that the sound occurs within the 144 millisecond window To enforce this constraint, the various units that receive input from different portions of the time window are forced to share weights The net effect is to constrain the space of potential hypotheses, thereby reducing the risk of overfitting and improving the chances for accurately generalizing to unseen situations Such weight sharing is typically implemented by first updating each of the shared weights separately within each unit that uses the weight, then replacing each instance of the shared weight by the mean of their values The result of this procedure is that shared weights effectively adapt to a different error function than do the unshared weights
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While gradient descent is one of the most general search methods for finding a hypothesis to minimize the error function, it is not always the most efficient It is not uncommon for BACKPROPAGATION to require tens of thousands of iterations through the weight update loop when training complex networks For this reason, a number of alternative weight optimization algorithms have been proposed and explored To see some of the other possibilities, it is helpful to think of a weightupdate method as involving two decisions: choosing a direction in which to alter the current weight vector and choosing a distance to move In BACKPROPAGATION the direction is chosen by taking the negative of the gradient, and the distance is determined by the learning rate constant q One optimization method, known as line search, involves a different approach to choosing the distance for the weight update In particular, once a line is chosen that specifies the direction of the update, the update distance is chosen by finding the minimum of the error function along this line Notice this can result in a very large or very small weight update, depending on the position of the point along the line that minimizes error A second method, that builds on the idea of line search, is called the conjugate gradient method Here, a sequence of line searshes is performed to search for a minimum in the error surface On the first step in this sequence, the direction chosen is the negative of the gradient On each subsequent step, a new direction is chosen so that the component of the error gradient that has just been made zero, remains zero While alternative error-minimization methods sometimes lead to improved efficiency in training the network, methods such as conjugate gradient tend to have no significant impact on the generalization error of the final network The only likely impact on the final error is that different error-minimizationprocedures may fall into different local minima Bishop (1996) contains a general discussion of several parameter optimization methods for training networks
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