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Perceptron

Perceptron. f act (a 1 , . . ., a m ) = sig (a 1 , . . . , a m ) f learn (w 0j1,old , d, x 011 … x 0k , c) = w 0j1,old + c * (d – sig( x oi * w 0i )) * x oi c=0.2. Perceptron. Consider a perceptron with 3 input nodes x 01 , x 02 and x 03

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Perceptron

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  1. Perceptron • fact(a1, . . ., am) = sig (a1, . . . , am) • flearn(w0j1,old, d, x011 … x0k, c) = w0j1,old + c * (d – sig(xoi * w0i)) * xoi • c=0.2

  2. Perceptron • Consider a perceptron with 3 input nodes x01, x02 andx03 • w011 = 0.3, w021 = 0.4 and w031 = -0.3 • Input • (1,1,1)  1 • (3.2, 1.4, 9.1)  -1 • (8, 7.7, 3.1)  -1 • (1.5, 2.4, 5.5)  1

  3. x01 w011 x02 x11 w021 w031 x03

  4. First Data Item • w011 = 0.3, w021 = 0.4 and w031 = -0.3 • Input (1,1,1)  1 • fact = sig(1*0.3 + 1*0.4 + 1*-0.3) = sig(0.4) = 1 • Correct! No change

  5. Second Data Item • w011 = 0.3, w021 = 0.5 and w031 = -0.3 • Input (3.2, 1.4, 9.1)  -1 • fact = sig(3.2 * 0.3 + 1.4*0.4 + 9.1*-0.3) = sig(0.96 + 0.56 + -2.73) = sig(-1.85) = -1 • Correct! No change

  6. Third Data Item • w011 = 0.3, w021 = 0.4 and w031 = -0.3 • Input (8, 7.7, 3.1)  -1 • fact = sig(8 * 0.3 + 7.7 * 0.4 + 3.1 * -0.3) = sig(2.4 + 3.08 + -0.93) = sig(4.55) = 1 • Wrong! Change!

  7. Change the weights • w011,new = 0.3 + 0.2 * -2 * 8 = -2.9 • w021,new = 0.4 + 0.2 * -2 * 7.7 = -2.58 • w031,new = -0.3 + 0.2 * -2 * 3.1 = -1.54

  8. Fourth Data Item • w011 = -2.9, w021 = -2.58 and w031 = -1.54 • Input (1.5, 2.4, 5.5)  1 • fact = sig(1.5 * -2.9 + 2.4 * -2.58 + 5.5 * -1.54) = sig(-4.35 + -6.19 + -8.47) = sig(-19.01) = -1 • Wrong! Change!

  9. Change the weights • w011,new = -2.9 + 0.2 * 2 * 1.5 = -2.3 • w021,new = -2.58 + 0.2 * 2 * 2.4 = -1.62 • w031,new = -1.54 + 0.2 * 2 * 5.5 = 0.66

  10. Begin Again : First Data Item • w011 = -2.3, w021 = -1.62 and w031 = 0.66 • Input (1,1,1)  1 • fact = sig(1*-2.3 + 1*-1.62 + 1*0.66) = sig(-3.26) = -1 • Wrong! Change!

  11. Change the weights • w011,new = -2.3 + 0.2 * 2 * 1 = -1.9 • w021,new = -1.62 + 0.2 * 2 * 1 = -1.22 • w031,new = 0.66 + 0.2 * 2 * 1 = 1.06

  12. Begin Again : Second Data Item • w011 = -1.9, w021 = -1.22 and w031 = 1.06 • Input (3.2, 1.4, 9.1)  -1 • fact = sig(-1.9*3.2 + -1.22*1.4 + 9.1*1.06) = sig(-6.08 + -1.71 + 9.646) = sig(1.856) = 1 • Wrong! Change!

  13. Change the weights • w011,new = -1.9 + 0.2 * -2 * 3.2 = -3.18 • w021,new = -1.22 + 0.2 * -2 * 1.4 = -1.78 • w031,new = 1.06 + 0.2 * -2 * 9.1 = -2.58

  14. Begin Again : Third Data Item • w011 = -3.18, w021 = -1.78 and w031 = -2.58 • Input (8, 7.7, 3.1)  -1 • fact = sig(8*-3.18 + 7.7*-1.78 + 3.1*-2.58) = sig(-25.44 + -13.71 + -8.00) = sig(-47.15) = -1 • Correct! Don’t Change!

  15. Begin Again : Fourth Data Item • w011 = -3.18, w021 = -1.78 and w031 = -2.58 • Input (1.5, 2.4, 5.5)  1 • fact = sig(1.5*-3.18 + 2.4*-1.78 + 5.5*-2.58) = sig(-4.77 + -4.27 + -14.19) = sig(-23.23) = -1 • Wrong! Change!

  16. Change the weights • w011,new = -3.18 + 0.2 * 2 * 1.5 = -2.58 • w021,new = -1.78 + 0.2 * 2 * 2.4 = -0.82 • w031,new = -2.58 + 0.2 * 2 * 5.5 = -0.38

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