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41 What are the values of weights wo, w l , and w2 for the perceptron whose decision surface is illustrated in Figure 43 Assume the surface crosses the x l axis at -1, and the x2 axis at 2 42 Design a two-input perceptron that implements the boolean function A A -B Design a two-layer network of perceptrons that implements A XO R B 43 Consider two perceptrons defined by the threshold expression wo w l x l + ~ 2 x > 0 2 Perceptron A has weight values
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and perceptron B has the weight values
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perceptron B (more-general~han True or false Perceptron A is more-general~han is defined in 2) 44 Implement the delta training rule for a two-input linear unit Train it to fit the target concept -2 + X I + 2x2 > 0 Plot the error E as a function of the number of training iterations Plot the decision surface after 5, 10, 50, 100, , iterations ( a ) Try this using various constant values for 17 and using a decaying learning rate of qo/i for the ith iteration Which works better (b) Try incremental and batch learning Which converges more quickly Consider both number of weight updates and total execution time 45 Derive a gradient descent training rule for a single unit with output o, where
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46 Explain informally why the delta training rule in Equation (410) is only an approximation to the true gradient descent rule of Equation (47) 47 Consider a two-layer feedforward ANN with two inputs a and b, one hidden unit c, and one output unit d This network has five weights (w,, web, wd, wdc, wdO), where w,o represents the threshold weight for unit x Initialize these weights to the values (1, l , l , l, I), then give their values after each of the first two training iterations of the BACKPROPAGATION algorithm Assume learning rate 17 = 3, momentum a! = 09, incremental weight updates, and the following training examples:
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a 1 0 b 0 1 d 1 0
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48 Revise the BACKPROPAGATION algorithm in Table 42 so that it operates on units using the squashing function tanh in place of the sigmoid function That is, assume the output of a single unit is o = t a n h ( 6 x ' ) Give the weight update rule for output layer weights and hidden layer weights Hint: tanh'(x) = 1 - tanh2(x) 49 Recall the 8 x 3 x 8 network described in Figure 47 Consider trying to train a 8 x 1x 8 network for the same task; that is, a network with just one hidden unit Notice the eight training examples in Figure 47 could be represented by eight distinct values for the single hidden unit (eg, 01,02, ,08) Could a network with just one hidden unit therefore learn the identity function defined over these training examples Hint: Consider questions such as "do there exist values for the hidden unit weights that can create the hidden unit encoding suggested above '"do there exist values for the output unit weights that could correctly decode this encoding of the input 'and "is gradient descent likely to find such weights ' 410 Consider the alternative error function described in Section 481
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Derive the gradient descent update rule for this definition of E Show that it can be implemented by multiplying each weight by some constant before performing the standard gradient descent update given in Table 42 411 Apply BACKPROPAGATION task of face recognition See World Wide Web to the URL http://wwwcscmuedu/-tomlbookhtml details, including face-image data, for BACKPROPAGATION and specific tasks code, 412 Consider deriving a gradient descent algorithm to learn target concepts corresponding to rectangles in the x , y plane Describe each hypothesis by the x and y coordinates of the lower-left and upper-right comers of the rectangle - Ilx, Ily, urn, and ury respectively An instance ( x , y ) is labeled positive by hypothesis ( l l x , l l y , u r x , u r y ) if and only if the point ( x , y ) lies inside the corresponding rectangle Define error E as in the chapter Can you devise a gradient descent algorithm to learn such rectangle hypotheses Notice that E is not a continuous function of l l x , Ily, u r x , and u r y , just as in the case of perceptron learning (Hint: Consider the two solutions used for perceptrons: (1) changing the classification rule to make output predictions continuous functions of the inputs, and (2) defining an alternative error-such as distance to the rectangle center-as in using the delta rule to train perceptrons) Does your algorithm converge to the minimum error hypothesis when the positive and negative examples are separable by a rectangle When they are not Do you
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have problems with local minima How does your algorithm compare to symbolic methods for learning conjunctions of feature constraints
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