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Brief Introduction to Deep Learning + Solving XOR using ANNs

This presentation gives a very simple introduction to deep learning in addition to a step-by-step example showing how to solve the XOR non-linear problem using multi-layer artificial neural networks that has both input, hidden, and output layers.
Deep learning is based on artificial neural networks and it aims to analyze large amounts of data that are not easily analyzed using conventional models. It creates a large neural network with several hidden layers and several neurons within each layer and usually may take days for its learning.
Many beginners in artificial neural networks have a problem in understanding how hidden layers are useful and what is the best number of hidden layers and best number of neurons or nodes within each layer.

أحمد فوزي جاد Ahmed Fawzy Gad
قسم تكنولوجيا المعلومات Information Technology (IT) Department
كلية الحاسبات والمعلومات Faculty of Computers and Information (FCI)
جامعة المنوفية, مصر Menoufia University, Egypt
Teaching Assistant/Demonstrator
ahmed.fawzy@ci.menofia.edu.eg

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Brief Introduction to Deep Learning + Solving XOR using ANNs

  1. 1. Brief Introduction to Deep Learning + Solving XOR using ANN MENOUFIA UNIVERSITY FACULTY OF COMPUTERS AND INFORMATION ‫المنوفية‬ ‫جامعة‬ ‫الحاسبات‬ ‫كلية‬‫والمعلومات‬ ‫المنوفية‬ ‫جامعة‬ Ahmed Fawzy Gad ahmed.fawzy@ci.menofia.edu.eg
  2. 2. Classification Example BA 01 1 10 00 0 11
  3. 3. Neural Networks Input Hidden Output BA 01 1 10 00 0 11
  4. 4. Neural Networks BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
  5. 5. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
  6. 6. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
  7. 7. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
  8. 8. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
  9. 9. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
  10. 10. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
  11. 11. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
  12. 12. = +0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5
  13. 13. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Network Architecture!!
  14. 14. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 = + 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5
  15. 15. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5
  16. 16. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5
  17. 17. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 Input OutputHidden
  18. 18. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 Input OutputHidden
  19. 19. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 Input OutputHidden
  20. 20. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 BA 01 1 10 00 0 11 Input Output
  21. 21. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 BA 01 1 10 00 0 11 Input Output
  22. 22. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 BA 01 1 10 00 0 11 Input Output
  23. 23. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 BA 01 1 10 00 0 11 Input Output A B
  24. 24. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 BA 01 1 10 00 0 11 Input Output A B
  25. 25. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 BA 01 1 10 00 0 11 Input Output A B
  26. 26. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 BA 01 1 10 00 0 11 Input Output A B
  27. 27. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 BA 01 1 10 00 0 11 Input Output A B 1/0
  28. 28. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 Input Output A B 1/0 𝑾 𝟏 𝑾 𝟐
  29. 29. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 = + 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5
  30. 30. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5
  31. 31. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5
  32. 32. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 Input Output A B 1/0 𝑾 𝟑 𝑾 𝟒
  33. 33. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 A B 1/0 𝑾 𝟑 𝑾 𝟒 A B 1/0 𝑾 𝟏 𝑾 𝟐 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5
  34. 34. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 A B 𝑾 𝟑 𝑾 𝟒
  35. 35. A B 𝑾 𝟏 𝑾 𝟐 A B 𝑾 𝟑 𝑾 𝟒 𝑾 𝟓 𝑾 𝟔
  36. 36. 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 A B 𝑾 𝟑 𝑾 𝟒 𝑾 𝟓 𝑾 𝟔 A 𝑾 𝟏 𝑾 𝟐 B 𝑾 𝟑 𝑾 𝟒
  37. 37. 𝑾 𝟓 𝑾 𝟔 A 𝑾 𝟏 𝑾 𝟐 B 𝑾 𝟑 𝑾 𝟒
  38. 38. . . . . . .
  39. 39. . . .
  40. 40. . . . . . .
  41. 41. . . .
  42. 42. . . . . . .
  43. 43. . . . . . .
  44. 44. . . .
  45. 45. . . .
  46. 46. Input Output BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Hidden Weights=𝑾𝒊 𝑾 𝟓 𝑾 𝟔 1/0 𝒀𝒋 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  47. 47. Activation Function Output 𝒀𝒋 BA 01 1 10 00 0 11 Input Hidden 1/0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  48. 48. Activation Function Output 𝒀𝒋 BA 01 1 10 00 0 11 Input Hidden 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1/0 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  49. 49. Activation Function Output 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 1/0 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  50. 50. Activation Function Components Output 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 1/0 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  51. 51. Activation Function Inputs Output s 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 1/0 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  52. 52. Activation Function Inputs Output s 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 1/0 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒 s=SOP(𝑿𝒊, 𝑾𝒊)
  53. 53. Activation Function Inputs Output s 𝑿𝒊=Inputs 𝑾𝒊=Weights 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 1/0 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒 s=SOP(𝑿𝒊, 𝑾𝒊)
  54. 54. Activation Function Inputs Output s 𝑿𝒊=Inputs 𝑾𝒊=Weights 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 1/0 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒 s=SOP(𝑿𝒊, 𝑾𝒊)
  55. 55. 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒 Activation Function Inputs Output s 𝑿𝒊=Inputs 𝑾𝒊=Weights 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊 s=SOP(𝑿𝒊, 𝑾𝒊) 1/0
  56. 56. Activation Function Inputs Output s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒 S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊 1/0 Each Hidden/Output Layer Neuron has its SOP.
  57. 57. Activation Function Inputs Output s 𝑿 𝟏 𝑿 𝟐 𝑺 𝟏=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑) 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒 S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊 1/0
  58. 58. Activation Function Inputs Output s 𝑿 𝟏 𝑿 𝟐 𝑺 𝟐=(𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒) 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒 S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊 1/0
  59. 59. Activation Function Inputs Output s 𝑿 𝟏 𝑿 𝟐 𝑺 𝟑=(𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔) 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒 S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊 1/0
  60. 60. Activation Function Outputs Output F(s)s 𝑿 𝟏 𝑿 𝟐 Class Label 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 1/0 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  61. 61. Activation Function Outputs Output F(s)s 𝑿 𝟏 𝑿 𝟐 Class Label 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Input Hidden 1/0 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  62. 62. Activation Functions Piecewise Linear Sigmoid Binary
  63. 63. Activation Functions Which activation function to use? Outputs Class Labels Activation Function TWO Class Labels TWO Outputs One that gives two outputs. Which activation function to use? 𝑪𝒋𝒀𝒋 BA 01 1 10 00 0 11 BA 01 1 10 00 0 11
  64. 64. Activation Functions Piecewise Linear Sigmoid BinaryBinary
  65. 65. Activation Function Output F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 BA 01 1 10 00 0 11 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1/0 Input Hidden 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  66. 66. Bias Input Output BA 01 1 10 00 0 11 F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 1/0 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  67. 67. Bias Hidden Layer Neurons Input Output BA 01 1 10 00 0 11 F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒 +1 𝒃 𝟏 +1 𝒃 𝟐 1/0
  68. 68. Bias Output Layer Neurons Input Output BA 01 1 10 00 0 11 F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 +1 𝒃 𝟑 1/0 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  69. 69. All Bias Values Input Output BA 01 1 10 00 0 11 F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 +1 𝒃 𝟏 +1 𝒃 𝟐 1/0 +1 𝒃 𝟑 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  70. 70. Bias Add Bias to SOP Input Output BA 01 1 10 00 0 11 F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 +1 𝒃 𝟏 +1 𝒃 𝟐 𝑺 𝟏=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑) 𝑺 𝟐=(𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒) 𝑺 𝟑=(𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔) 1/0 +1 𝒃 𝟑 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  71. 71. Bias Add Bias to SOP Input Output BA 01 1 10 00 0 11 F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 +1 𝒃 𝟏 +1 𝒃 𝟐 𝑺 𝟏=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑) 1/0 +1 𝒃 𝟑 𝑺 𝟏=(+𝟏𝒃 𝟏+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑) 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  72. 72. Bias Add Bias to SOP Input Output BA 01 1 10 00 0 11 F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 +1 𝒃 𝟏 +1 𝒃 𝟐 𝑺 𝟐=(𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒) 1/0 +1 𝒃 𝟑 𝑺 𝟐=(+𝟏𝒃 𝟐+𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒) 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  73. 73. Bias Add Bias to SOP Input Output BA 01 1 10 00 0 11 F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 +1 𝒃 𝟏 +1 𝒃 𝟐 𝑺 𝟑=(𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔) 1/0 +1 𝒃 𝟑 𝑺 𝟑=(+𝟏𝒃 𝟑+𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔) 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  74. 74. Learning Rate 𝟎 ≤ η ≤ 𝟏 F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 +1 𝒃 𝟏 +1 𝒃 𝟐 1/0 +1 𝒃 𝟑 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒
  75. 75. Other Parameters Step n 𝒏 = 𝟎, 𝟏, 𝟐, … F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 +1 𝒃 𝟏 +1 𝒃 𝟐 1/0 +1 𝒃 𝟑 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+…) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, …) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, …)
  76. 76. Other Parameters Desired Output 𝒅𝒋 𝒏 = 𝟎, 𝟏, 𝟐, … 𝒅 𝒏 = 𝟏, 𝒙 𝒏 𝒃𝒆𝒍𝒐𝒏𝒈𝒔 𝒕𝒐 𝑪𝟏 (𝟏) 𝟎, 𝒙 𝒏 𝒃𝒆𝒍𝒐𝒏𝒈𝒔 𝒕𝒐 𝑪𝟐 (𝟎) BA 01 1 10 00 0 11 F(s)s 𝑿 𝟏 𝑿 𝟐 bin 𝒀𝒋 +1 𝒃 𝟏 +1 𝒃 𝟐 1/0 +1 𝒃 𝟑 𝑾 𝟓 𝑾 𝟔 A B 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 𝑾 𝟒 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+…) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, …) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, …)
  77. 77. Neural Networks Training Steps Weights Initialization Inputs Application Sum of Inputs-Weights Products Activation Function Response Calculation Weights Adaptation Back to Step 2 1 2 3 4 5 6
  78. 78. Regarding 5th Step: Weights Adaptation • If the predicted output Y is not the same as the desired output d, then weights are to be adapted according to the following equation: 𝑾 𝒏 + 𝟏 = 𝑾 𝒏 + η 𝒅 𝒏 − 𝒀 𝒏 𝑿(𝒏) Where 𝑾 𝒏 = [𝒃 𝒏 , 𝑾 𝟏(𝒏), 𝑾 𝟐(𝒏), 𝑾 𝟑(𝒏), … , 𝑾 𝒎(𝒏)]
  79. 79. Neural Networks Training Example Step n=0 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=0: η = .001 𝑋 𝑛 = 𝑋 0 = +1, +1, +1,1, 0 𝑊 𝑛 = 𝑊 0 = 𝑏1, 𝑏2, 𝑏3, 𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝑤5, 𝑤6 = −1.5, −.5, −.5, 1, 1, 1, 1, −2, 1 𝑑 𝑛 = 𝑑 0 = 1 BA 01 1 => 1 10 00 0 => 0 11
  80. 80. Neural Networks Training Example Step n=0 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin BA 01 1 => 1 10 00 0 => 0 11
  81. 81. Neural Networks Training Example Step n=0 – SOP – 𝑺 𝟏 𝑺 𝟏=(+𝟏𝒃 𝟏+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑) =+1*-1.5+1*1+0*1 =-.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  82. 82. Neural Networks Training Example Step n=0 – Output – 𝑺 𝟏 𝒀 𝑺 𝟏 = = 𝑩𝑰𝑵 𝑺 𝟏 = 𝑩𝑰𝑵 −. 𝟓 = 𝟎 𝒃𝒊𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 𝟎, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  83. 83. Neural Networks Training Example Step n=0 – SOP – 𝑺 𝟐 𝑺 𝟐=(+𝟏𝒃 𝟐+𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒) =+1*-.5+1*1+0*1 =.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  84. 84. Neural Networks Training Example Step n=0 – Output – 𝑺 𝟐 𝒀 𝑺2 = = 𝑩𝑰𝑵 𝑺2 = 𝑩𝑰𝑵 . 𝟓 = 1 𝒃𝒊𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  85. 85. Neural Networks Training Example Step n=0 – SOP – 𝑺 𝟑 𝑺 𝟑=(+𝟏𝒃 𝟑+𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔) =+1*-.5+0*-2+1*1 =.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  86. 86. Neural Networks Training Example Step n=0 – Output – 𝑺 𝟑 𝒀 𝑺3 = = 𝑩𝑰𝑵 𝑺3 = 𝑩𝑰𝑵 . 𝟓 = 1 𝒃𝒊𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 𝟎, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  87. 87. Neural Networks Training Example Step n=0 - Output 𝒀 𝒏 = 𝒀 𝟎 = 𝒀 𝑺3 = 1 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  88. 88. Neural Networks Training Example Step n=0 Predicted Vs. Desired 𝒀 𝒏 = 𝒀 𝟎 = 1 𝐝 𝒏 = 𝒅 𝟎 = 1 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  89. 89. Neural Networks Training Example Step n=1 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=1: η = .001 𝑋 𝑛 = 𝑋 1 = +1, +1, +1,0, 1 𝑊 𝑛 = 𝑊 1 = 𝑊 0 = 𝑏1, 𝑏2, 𝑏3, 𝑤1, 𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝑤5, 𝑤6 = −1.5, −.5, −.5, 1, 1, 1, 1, −2, 1 𝑑 𝑛 = 𝑑 1 = +1 BA 01 1 => 1 10 00 0 => 0 11
  90. 90. Neural Networks Training Example Step n=1 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin BA 01 1 => 1 10 00 0 => 0 11
  91. 91. Neural Networks Training Example Step n=1 – SOP – 𝑺 𝟏 𝑺 𝟏=(+𝟏𝒃 𝟏+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑) =+1*-1.5+0*1+1*1 =-.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  92. 92. Neural Networks Training Example Step n=1 – Output – 𝑺 𝟏 𝒀 𝑺 𝟏 = = 𝑩𝑰𝑵 𝑺 𝟏 = 𝑩𝑰𝑵 −. 𝟓 = 𝟎 𝒃𝒊𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 𝟎, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  93. 93. Neural Networks Training Example Step n=1 – SOP – 𝑺 𝟐 𝑺 𝟐=(+𝟏𝒃 𝟐+𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒) =+1*-.5+0*1+1*1 =.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  94. 94. Neural Networks Training Example Step n=1 – Output – 𝑺 𝟐 𝒀 𝑺2 = = 𝑩𝑰𝑵 𝑺2 = 𝑩𝑰𝑵 . 𝟓 = 1 𝒃𝒊𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  95. 95. Neural Networks Training Example Step n=1 – SOP – 𝑺 𝟑 𝑺 𝟑=(+𝟏𝒃 𝟑+𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔) =+1*-.5+0*-2+1*1 =.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  96. 96. Neural Networks Training Example Step n=1 – Output – 𝑺 𝟑 𝒀 𝑺3 = = 𝑩𝑰𝑵 𝑺3 = 𝑩𝑰𝑵 . 𝟓 = 1 𝒃𝒊𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 𝟎, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  97. 97. Neural Networks Training Example Step n=1 - Output 𝒀 𝒏 = 𝒀 𝟏 = 𝒀 𝑺3 = 1 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  98. 98. Neural Networks Training Example Step n=1 Predicted Vs. Desired 𝒀 𝒏 = 𝒀 𝟏 = 1 𝐝 𝒏 = 𝒅 𝟏 = 1 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −2 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  99. 99. Neural Networks Training Example Step n=2 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=2: η = .001 𝑋 𝑛 = 𝑋 2 = +1, +1, +1,0, 0 𝑊 𝑛 = 𝑊 2 = 𝑊 1 = 𝑏1, 𝑏2, 𝑏3, 𝑤1, 𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝑤5, 𝑤6 = −1.5, −.5, −.5, 1, 1, 1, 1, −2, 1 𝑑 𝑛 = 𝑑 2 = 0 BA 01 1 => 1 10 00 0 => 0 11
  100. 100. Neural Networks Training Example Step n=2 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin BA 01 1 => 1 10 00 0 => 0 11
  101. 101. Neural Networks Training Example Step n=2 – SOP – 𝑺 𝟏 𝑺 𝟏=(+𝟏𝒃 𝟏+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑) =+1*-1.5+0*1+0*1 =-1.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  102. 102. Neural Networks Training Example Step n=2 – Output – 𝑺 𝟏 𝒀 𝑺 𝟏 = = 𝑩𝑰𝑵 𝑺 𝟏 = 𝑩𝑰𝑵 −𝟏. 𝟓 = 𝟎 𝒃𝒊n 𝒔 = +𝟏, 𝒔 ≥ 𝟎 𝟎, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  103. 103. Neural Networks Training Example Step n=2 – SOP – 𝑺 𝟐 𝑺 𝟐=(+𝟏𝒃 𝟐+𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒) =+1*-.5+0*1+0*1 =-.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  104. 104. Neural Networks Training Example Step n=2 – Output – 𝑺 𝟐 𝒀 𝑺2 = = 𝑺𝑮𝑵 𝑺2 = 𝑺𝑮𝑵 −. 𝟓 =0 𝒃𝒊𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  105. 105. Neural Networks Training Example Step n=2 – SOP – 𝑺 𝟑 𝑺 𝟑=(+𝟏𝒃 𝟑+𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔) =+1*-.5+0*-2+0*1 =-.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  106. 106. Neural Networks Training Example Step n=2 – Output – 𝑺 𝟑 𝒀 𝑺3 = = 𝑩𝑰𝑵 𝑺3 = 𝑩𝑰𝑵 −. 𝟓 = 𝟎 𝒃𝒊𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 𝟎, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  107. 107. Neural Networks Training Example Step n=2 - Output 𝒀 𝒏 = 𝒀 𝟐 = 𝒀 𝑺3 = 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  108. 108. Neural Networks Training Example Step n=2 Predicted Vs. Desired 𝒀 𝒏 = 𝒀 𝟐 = 𝟎 𝐝 𝒏 = 𝒅 𝟐 = 𝟎 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  109. 109. Neural Networks Training Example Step n=3 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=3: η = .001 𝑋 𝑛 = 𝑋 3 = +1, +1, +1,1, 1 𝑊 𝑛 = 𝑊 3 = 𝑊 2 = 𝑏1, 𝑏2, 𝑏3, 𝑤1, 𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝑤5, 𝑤6 = −1.5, −.5, −.5, 1, 1, 1, 1, −2, 1 𝑑 𝑛 = 𝑑 3 = 0 BA 01 1 => 1 10 00 0 => 0 11
  110. 110. Neural Networks Training Example Step n=3 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin BA 01 1 => 1 10 00 0 => 0 11
  111. 111. Neural Networks Training Example Step n=3 – SOP – 𝑺 𝟏 𝑺 𝟏=(+𝟏𝒃 𝟏+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑) =+1*-1.5+1*1+1*1 =.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  112. 112. Neural Networks Training Example Step n=3 – Output – 𝑺 𝟏 𝒀 𝑺 𝟏 = = 𝑩𝑰𝑵 𝑺 𝟏 = 𝑩𝑰𝑵 . 𝟓 = 1 𝒃𝒊𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 𝟎, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  113. 113. Neural Networks Training Example Step n=3 – SOP – 𝑺 𝟐 𝑺 𝟐=(+𝟏𝒃 𝟐+𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒) =+1*-.5+1*1+1*1 =1.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  114. 114. Neural Networks Training Example Step n=3 – Output – 𝑺 𝟐 𝒀 𝑺2 = = 𝑩𝑰𝑵 𝑺2 = 𝑩𝑰𝑵 𝟏. 𝟓 = 1 𝒃𝒊𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  115. 115. Neural Networks Training Example Step n=3 – SOP – 𝑺 𝟑 𝑺 𝟑=(+𝟏𝒃 𝟑+𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔) =+1*-.5+1*-2+1*1 =-1.5 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  116. 116. Neural Networks Training Example Step n=3 – Output – 𝑺 𝟑 𝒀 𝑺3 = = 𝑩𝑰𝑵 𝑺3 = 𝑩𝑰𝑵 −𝟏. 𝟓 = 𝟎 𝒃𝒊𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 𝟎, 𝒔 < 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  117. 117. Neural Networks Training Example Step n=3 - Output 𝒀 𝒏 = 𝒀 𝟑 = 𝒀 𝑺3 = 𝟎 BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  118. 118. Neural Networks Training Example Step n=3 Predicted Vs. Desired 𝒀 𝒏 = 𝒀 𝟑 = 𝟎 𝐝 𝒏 = 𝒅 𝟑 = 𝟎 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation BA 01 1 => 1 10 00 0 => 0 11 s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin
  119. 119. Final Weights s 𝑿 𝟏 𝑿 𝟐 𝒀𝒋 +1 −𝟏. 𝟓 +1 −. 𝟓 1/0 +1 −. 𝟓 −𝟐 +𝟏 A B +𝟏 +𝟏 +𝟏 +𝟏 bin Current weights predicted the desired outputs.

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