1. NN – cont. Alexandra I. Cristea USI intensive course Adaptive Systems April-May 2003

2. We have seen how the neuron computes, lets see –What it can compute? –How it can learn?

3. What does the neuron compute?

4. Perceptron, discrete neuron First, simple case: –no hidden layers –Only one neuron –Get rid of threshold – b becomes w 0 –Y – Boolean function : > 0 fires 0 doesnt fire

5. Threshold function f f (w0 = - t = -1)

6. Y = X1 or X2 1 X1 X2 0 0 1 1 1 0 Y X1 X2

7. Y = X1 and X2 0,5 X1 X2 0 0 0 0 1 0 Y X1 X2

8. Y = or(x1,…,xn) w1=w2=…=wn=1

9. Y = and(x1,…,xn) w1=w2=…=wn=1/n

10. What are we actually doing? X1 X2 0 -1 1 1 1 0 Y X1 X2 0 0 0 0 1 0 Y X1 X2 0 0 1 1 1 0 Y w0+w1*X1+w2*X2 0=-1; 7; 9 0=-1; 0,7; 0,9 0=1; 7; 9 X1 X2

11. x1 x2 w0+w1*x1+w2*x2 w0= - 1 w1= - 0,67 w2= 1 Linearly Separable Set

12. w0+w1*x1+w2*x2 Linearly Separable Set x1 x2 w0= - 1 w1= 0,25 w2= - 0,1

13. w0+w1*x1+w2*x2 Linearly Separable Set x1 x2 w0= - 1 w1= 0,25 w2= 0,04

14. w0+w1*x1+w2*x2 Linearly Separable Set x1 x2 w0= - 1 w1= 0,167 w2= 0,1

15. Non-linearly separable Set

16. w0+w1*x1+w2*x2 Non Linearly Separable Set x1 x2 w0= w1= w2=

17. w0+w1*x1+w2*x2 Non Linearly Separable Set x1 x2 w0= w1= w2=

18. w0+w1*x1+w2*x2 Non Linearly Separable Set x1 x2 w0= w1= w2=

19. w0+w1*x1+w2*x2 Non Linearly Separable Set x1 x2 w0= w1= w2=

20. Perceptron Classification Theorem A finite set X can be classified correctly by a one-layer perceptron if and only if it is linearly separable.

21. w0+w1*x1+w2*x2 Typical non-linearly separable set: Y=XOR(x1,x2) x1 x2 0,01,0 0,1 1,1 Y=1 Y=0

22. How does the neuron learn?

23. Learning: weight computation W1* X1 W * X2= X2 X1 W1*X1 W2*X2

24. Perceptron Learning Rule incremental version FOR i:= 0 TO n DO wi:=random initial value ENDFOR; REPEAT select a pair (x,t) in X; (* each pair must have a positive probability of being selected *) IF w T * x' > 0 THEN y:=1 ELSE y:=0 ENDIF; IF y t THEN FOR i:= 0 TO n DO wi:= wi + (t-y) xi' ENDFOR ENDIF; UNTIL X is correctly classified ROSENBLATT (1962)

25. Idea Perceptron Learning Rule w x w new w new =w + x t=1 y=0 (w T x 0) w niew x w x x w new =w - x wi:= wi + (t-y) xi' w changes in the direction of the input +- t=0 y=1 (w T x>0)

27. For multi-layered perceptrons w. continuous neurons, a simple and successful learning algorithm exists.

28. BKP:Error Input Output Hidden layer d d d d e1=d1 y1 e2=d2 y2 e3=d3 y3 e4=d4 y4 Hiddenlayer error error

29. Synapse W weight neuron1 neuron2 y1 value y2 w*y1 value Value (y1,y2)= Internal activation Forward propagation Weight serves as amplifier!

30. Inverse Synapse W weight neuron1 neuron2 ?? e1= ?? value e2 value Value(e1,e2)= Error Backward propagation Weight serves as amplifier!

31. Inverse Synapse W weight neuron1 neuron2 w e2 e1=w e2 value e2 value Value(e1,e2)= Error Backward propagation Weight serves as amplifier!

32. BKP:Error Input Output Hidden layer d d d d e1=d1 y1 e2=d2 y2 e3=d3 y3 e4=d4 y4 Hiddenlayer error error O2 O1 I1 O2, I2

33. Backpropagation to hidden layer Input I1 Output O1 Hidden layer ee j i e i j,i Backpropagation e e e O2, I2

34. Update rule for 2 weight types I2 hidden layer, O1 system output I1 system input, O2 hidden layer (simplification f=1 for repeater, e.g.) Δ =α(d[i]-y[i]) f(S[i])f(S[i]) = =αe[i] f(S[i]) (simplification f=1 for repeater, e.g.) S[i] = j w[j, ](t)h[j] Δ =α i e[i] [j,i] f(S[j])f(S[j]) =α ee[j]f(S[j]) S[j] = k w[k,j](t)x[k]

35. Backpropagation algorithm FOR s := 1 TO r DO Ws := initial matrix (often random); REPEAT select a pair (x,t) in X; y 0 :=x; # forward phase: compute the actual output ys of the network with input x FOR s := 1 TO r DO y s := F(Ws y s-1 ) END; # yr is the output vector of the network # backpropagation phase: propagate the errors back through the network # and adapt the weights of all layers d r := F r (t - y r ) ; FOR s := r TO 2 DO d s-1 := F s-1 ' Ws T d s ; Ws := Ws + d s y s-1 T ; END; W1 := W1 + d 1 y 0 T UNTIL stop criterion

36. Conclusion We have seen binary function representation with single layer perceptron We have seen a learning algorithm for SLP We have seen a learning algorithm for MLP (BP) So, neurons can represent knowledge AND learn!