机器学习第四章答案
2页1、4.1 What are the values of weights wo, wl, and w2 for the perceptron whose decision surface is illustrated in Figure 4.3? Assume the surface crosses the xl axis at -1, and the x2 axis at 2. Ans. The function of the decision surface is: 2+2x1-x2 = 0, so w0 =2, w1 = 2, w2 = -1. 4.2. Design a two-input perceptron that implements the boolean function A ? B. Design a two-layer network of perceptrons that implements A XOR B. Ans. We assume 1 for true, -1 for false. (1) A ? B: w0 = -0.8. w1 = 0.5, w2 =
2、 -0.5. x1(A) x2(B) w0+w1x1+w2x2 output -1 -1 -0.8 -1 -1 1 -1.8 -1 1 -1 0.2 1 1 1 -0.8 -1 (2) A XOR B = (A ? B) (? A B) The weights are: Hidden unit 1: w0 = -0.8, w1 = 0.5, w2 = -0.5 Hidden unit 2: w0 = -0.8, w1 = -0.5, w2 = 0.5 Output unit: w0 = 0.3, w1 = 0.5, w2 = 0.5 x1(A) x2(B) Hidden unit 1 value Hidden unit 2 value Output value-1 -1 -1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 4.3. Consider two perceptrons defined by the threshold expression 022110+xwxww. Perceptron A has weight values: w0
3、 = 1, w1 =2, w2 = 1. and perceptron B has the weight values: w0 = 0, w1 = 2, w2 = 1. True or false? Perceptron A is more-generalthan perceptron B. (more-generalthan is defined in Chapter 2.) Ans. True. For each input instance x=(x1, x2), if x is satisfied by B, which means 2x1+x20, then we have 2x1+x2 +10. Hence, x is also satisfied by the A. 4.5. Derive a gradient descent training rule for a single unit with output o, where 2211110.nnnnxwxwxwxwwo+=Ans. The gradient descent is:,.,1,)(0wnEwEwEwE?=?r)( )()()()(2122ididDdddddDdddDdddxxototwiototwiwiE-=-?-=-?=?The training rule for gradient descent is:iiiwww+=, where +-=?-=DdididddiixxotwEw)(2
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