IV. Neural Network Learning

1. IV. Neural Network Learning

2. A.Neural Network Learning

3. Supervised Learning • Produce desired outputs for training inputs • Generalize reasonably & appropriately to other inputs • Good example: pattern recognition • Feedforward multilayer networks

4. input layer output layer hidden layers Feedforward Network . . . . . . . . . . . . . . . . . .

5. connectionweights inputs Typical Artificial Neuron output threshold

6. linearcombination activationfunction net input(local field) Typical Artificial Neuron

7. Net input: Neuron output: Equations

8. Single-Layer Perceptron . . . . . .

9. Variables x1 w1 h S Q y xj wj wn xn q

10. Single Layer Perceptron Equations

11. w2 x f w1 v 2D Weight Vector – + w

12. N-Dimensional Weight Vector + normal vector w separating hyperplane –

13. Goal of Perceptron Learning • Suppose we have training patterns x1, x2, …, xP with corresponding desired outputs y1, y2, …, yP • where xp {0, 1}n, yp  {0, 1} • We want to find w, q such thatyp = Q(wxp – q) for p = 1, …, P

14. Treating Threshold as Weight x1 w1 h S Q y xj wj wn xn q

15. x0 = = w0 Treating Threshold as Weight –1 x1 q w1 h S Q y xj wj wn xn

16. Augmented Vectors

17. Reformulation as Positive Examples

18. Adjustment of Weight Vector z5 z1 z10 z9 z11 z8 z6 z2 z3 z4 z7

19. Outline ofPerceptron Learning Algorithm • initialize weight vector randomly • until all patterns classified correctly, do: • for p = 1, …, P do: • if zp classified correctly, do nothing • else adjust weight vector to be closer to correct classification

20. Weight Adjustment

21. Improvement in Performance

22. Perceptron Learning Theorem • If there is a set of weights that will solve the problem, • then the PLA will eventually find it • (for a sufficiently small learning rate) • Note: only applies if positive & negative examples are linearly separable

23. NetLogo Simulation of Perceptron Learning Run Perceptron-Geometry.nlogo

24. Classification Power of Multilayer Perceptrons • Perceptrons can function as logic gates • Therefore MLP can form intersections, unions, differences of linearly-separable regions • Classes can be arbitrary hyperpolyhedra • Minsky & Papert criticism of perceptrons • No one succeeded in developing a MLP learning algorithm

25. Hyperpolyhedral Classes

26. . . . input layer output layer hidden layers Credit Assignment Problem How do we adjust the weights of the hidden layers? Desired output . . . . . . . . . . . . . . . . . .

27. NetLogo Demonstration ofBack-Propagation Learning Run Artificial Neural Net.nlogo

28. Control Algorithm C Adaptive System Evaluation Function (Fitness, Figure of Merit) System S F … … P1 Pk Pm Control Parameters

29. Gradient

30. F gradient ascent Gradient Ascenton Fitness Surface + –

31. Gradient Ascentby Discrete Steps F + –

32. Gradient Ascent is LocalBut Not Shortest + –

33. Gradient Ascent Process Therefore gradient ascent increases fitness (until reaches 0 gradient)

34. General Ascent in Fitness

35. General Ascenton Fitness Surface F + –

36. Fitness as Minimum Error

37. Gradient of Fitness

38. Jacobian Matrix

39. Derivative of Squared Euclidean Distance

40. Gradient of Error on qth Input

41. Recap The Jacobian depends on the specific form of the system, in this case, a feedforward neural network

42. Multilayer Notation WL–1 xq yq W1 WL–2 W2 s1 sL s2 sL–1

43. Notation • L layers of neurons labeled 1, …, L • Nl neurons in layer l • sl = vector of outputs from neurons in layer l • input layer s1 = xq (the input pattern) • output layer sL = yq (the actual output) • Wl = weights between layers l and l+1 • Problem: find how outputs yiq vary with weights Wjkl (l = 1, …, L–1)

44. Typical Neuron s1l–1 Wi1l–1 Wijl–1 hil S s sjl–1 sil WiNl–1 sNl–1

45. Error Back-Propagation

46. Delta Values

47. Output-Layer Neuron s1L–1 Wi1L–1 WijL–1 hiL S s sjL–1 tiq siL = yiq WiNL–1 Eq sNL–1

48. Output-Layer Derivatives (1)

49. Output-Layer Derivatives (2)

50. Hidden-Layer Neuron s1l s1l–1 s1l+1 W1il Wi1l–1 Wijl–1 Wkil hil S s sjl–1 skl+1 Eq sil WiNl–1 WNil sNl–1 sNl+1 sNl