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55 mm ( 15) 22 to 33 mm 33 to 44 mm Data max)

.

The iPhone Software Development Kit (SDK)

12 mm ( 030/ 006) (two bonded substrates) 06 mm ( 0043/ 0030)

As shown above, the BACKPROPAGATION algorithm implements a gradient descent search through the space of possible network weights, iteratively reducing the error E between the training example target values and the network outputs Because the error surface for multilayer networks may contain many different local minima, gradient descent can become trapped in any of these As a result, BACKPROPAGATIONmultilayer networks is only guaranteed to converge toward over some local minimum in E and not necessarily to the global minimum error Despite the lack of assured convergence to the global minimum error, BACKPROPAGATION is a highly effective function approximation method in practice In many practical applications the problem of local minima has not been found to be as severe as one might fear To develop some intuition here, consider that networks with large numbers of weights correspond to error surfaces in very high dimensional spaces (one dimension per weight) When gradient descent falls into a local minimum with respect to one of these weights, it will not necessarily be in a local minimum with respect to the other weights In fact, the more weights in the network, the more dimensions that might provide "escape routes" for gradient descent to fall away from the local minimum with respect to this single weight A second perspective on local minima can be gained by considering the manner in which network weights evolve as the number of training iterations increases Notice that if network weights are initialized to values near zero, then during early gradient descent steps the network will represent a very smooth function that is approximately linear in its inputs This is because the sigmoid threshold function itself is approximately linear when the weights are close to zero (see the plot of the sigmoid function in Figure 46) Only after the weights have had time to grow will they reach a point where they can represent highly nonlinear network functions One might expect more local minima to exist in the region of the weight space that represents these more complex functions One hopes that by the time the weights reach this point they have already moved close enough to the global minimum that even local minima in this region are acceptable Despite the above comments, gradient descent over the complex error surfaces represented by ANNs is still poorly understood, and no methods are known to predict with certainty when local minima will cause difficulties Common heuristics to attempt to alleviate the problem of local minima include: Add a momentum term to the weight-update rule as described in Equation (418) Momentum can sometimes carry the gradient descent procedure through narrow local minima (though in principle it can also carry it through narrow global minima into other local minima!) Use stochastic gradient descent rather than true gradient descent As discussed in Section 4433, the stochastic approximation to gradient descent effectively descends a different error surface for each training example, re-.

to the canvas (Figure 1-11)

13 to 20 g (12-cm disc) or 6 to 9 g (8-cm disc)

8 Double-click the button on the canvas, and give the button a title 9 Drag a label from the library to the canvas (Figure 1-12)

As a second example, consider an algorithm that learns by maintaining a description of the version space, incrementally refining the version space as each new training example is encountered The CANDIDATE-ELIMINATION algorithm and the LIST-THEN-ELIMINATE algorithm from 2 are examples of such algorithms In this section we derive a worst-case bound on the number of mistakes that will be made by such a learner, for any finite hypothesis space H, assuming again that the target concept must be learned exactly To analyze the number of mistakes made while learning we must first specify precisely how the learner will make predictions given a new instance x Let us assume this prediction is made by taking a majority vote among the hypotheses in the current version space If the majority of version space hypotheses classify the new instance as positive, then this prediction is output by the learner Otherwise a negative prediction is output

120 or 80 mm ( 030)

(continued)

15 mm ( 015/ 000)

.

10 Save and exit Interface Builder 11 Select QuickStartViewControllerm in the Classes folder in Groups & Files Xcode should