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Lecture 7  Artificial neural networks: Supervised learning ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Introduction, or how the brain works Machine learning involves adaptive mechanisms that enable computers to learn from experience, learn by example and learn by analogy.  Learning capabilities can improve the performance of an intelligent system over time. The most popular approaches to machine learning are  artificial neural networks  and  genetic algorithms .  This lecture is dedicated to neural networks.
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[object Object],[object Object]
Biological neural network
[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
Architecture of a typical artificial neural network
Analogy between biological and  artificial neural networks
The neuron as a simple computing element Diagram of a neuron
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Activation functions of a neuron
Can a single neuron learn a task? ,[object Object],[object Object]
Single-layer two-input perceptron
The Perceptron ,[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object]
Linear separability in the perceptrons
This is done by making small adjustments in the weights to reduce the difference between the actual and desired outputs of the perceptron.  The initial weights are randomly assigned, usually in the range [  0.5, 0.5], and then updated to obtain the output consistent with the training examples. How does the perceptron learn its classification tasks?
[object Object],[object Object],[object Object],[object Object]
The perceptron learning rule where  p  = 1, 2, 3, . . .    is the  learning rate , a positive constant less than unity. The perceptron learning rule was first proposed by Rosenblatt  in 1960.  Using this rule we can derive  the perceptron training algorithm for classification  tasks.
Step 1 : Initialisation Set initial weights  w 1 ,  w 2 ,…,  w n  and threshold    to random numbers in the range [  0.5, 0.5].  If the error,  e ( p ), is positive, we need to increase perceptron output  Y ( p ), but if it is negative, we need to decrease  Y ( p ). Perceptron’s training algorithm
Step 2 : Activation Activate the perceptron by applying inputs  x 1 ( p ),  x 2 ( p ),…,  x n ( p ) and desired output  Y d  ( p ).  Calculate the actual output at iteration  p  = 1 where  n  is the number of the perceptron inputs, and  step  is a step activation function. Perceptron’s training algorithm (continued)
Step 3 : Weight training Update the weights of the perceptron where   w i ( p ) is the weight correction at iteration  p . The weight correction is computed by the  delta rule : Step 4 : Iteration Increase iteration  p  by one, go back to  Step 2  and repeat the process until convergence. Perceptron’s training algorithm (continued)
Example of perceptron learning: the logical operation  AND
Two-dimensional plots of basic logical operations A perceptron can learn the operations  AND  and  OR , but not  Exclusive-OR .
Multilayer neural networks ,[object Object],[object Object],[object Object]
Multilayer perceptron with two hidden layers
What does the middle layer hide? ,[object Object],[object Object]
Back-propagation neural network ,[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
Three-layer back-propagation neural network
Step 1 : Initialisation Set all the weights and threshold levels of the network to random numbers uniformly distributed inside a small range: where  F i  is the total number of inputs of neuron  i  in the network.  The weight initialisation is done on a neuron-by-neuron basis. The back-propagation training algorithm
Step 2 : Activation Activate the back-propagation neural network by applying inputs  x 1 ( p ),  x 2 ( p ),…,  x n ( p ) and desired outputs  y d ,1 ( p ),  y d ,2 ( p ),…,  y d , n ( p ). ( a )  Calculate the actual outputs of the neurons in the hidden layer: where  n  is the number of inputs of neuron  j  in the hidden layer, and  sigmoid  is the  sigmoid  activation function.
( b )  Calculate the actual outputs of the neurons in the output layer: where  m  is the number of inputs of neuron  k  in the output layer. Step 2 : Activation (continued)
Step 3 : Weight training Update the weights in the back-propagation network propagating backward the errors associated with output neurons. ( a ) Calculate the error gradient for the neurons in the output layer: where Calculate the weight corrections: Update the weights at the output neurons:
( b )  Calculate the error gradient for the neurons in the hidden layer: Calculate the weight corrections: Update the weights at the hidden neurons: Step 3 : Weight training (continued)
Step 4 : Iteration Increase iteration  p  by one, go back to  Step 2  and repeat the process until the selected error criterion is satisfied. As an example, we may consider the three-layer back-propagation network.  Suppose that the network is required to perform logical operation  Exclusive-OR .  Recall that a single-layer perceptron could not do this operation.  Now we will apply the three-layer net.
Three-layer network for solving the Exclusive-OR operation
[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
[object Object],[object Object]
[object Object],[object Object],5038 . 0 0038 . 0 5 . 0 13 13 13       w w w 8985 . 0 0015 . 0 9 . 0 14 14 14       w w w 4038 . 0 0038 . 0 4 . 0 23 23 23       w w w 9985 . 0 0015 . 0 0 . 1 24 24 24       w w w 2067 . 1 0067 . 0 2 . 1 35 35 35         w w w 0888 . 1 0112 . 0 1 . 1 45 45 45       w w w 7962 . 0 0038 . 0 8 . 0 3 3 3          0985 . 0 0015 . 0 1 . 0 4 4 4            3127 . 0 0127 . 0 3 . 0 5 5 5         
Learning curve for operation  Exclusive-OR
Final results of three-layer network learning
Network represented by McCulloch-Pitts model for solving the  Exclusive-OR  operation
( a ) Decision boundary constructed by hidden neuron 3; ( b ) Decision boundary constructed by hidden neuron 4;  ( c ) Decision boundaries constructed by the complete three-layer network Decision boundaries
Accelerated learning in multilayer neural networks ,[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
Learning with momentum for operation  Exclusive-OR
Learning with adaptive learning rate To accelerate the convergence and yet avoid the  danger of instability, we can apply two heuristics: Heuristic 1 If the change of the sum of squared errors has the same algebraic sign for several consequent epochs, then the learning rate parameter,   , should be increased. Heuristic 2 If the algebraic sign of the change of the sum of squared errors alternates for several consequent epochs, then the learning rate parameter,   , should be decreased.
[object Object],[object Object],[object Object]
Learning with adaptive learning rate
Learning with momentum and adaptive learning rate
[object Object],The Hopfield Network
[object Object],[object Object]
[object Object]
Single-layer  n -neuron Hopfield network
[object Object]
[object Object],[object Object],[object Object]
[object Object],[object Object]
Possible states for the three-neuron Hopfield network
[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
[object Object]
[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],Storage capacity of the Hopfield network
[object Object],[object Object],Bidirectional associative memory (BAM)
[object Object],[object Object]
BAM operation
The basic idea behind the BAM is to store pattern pairs so that when  n -dimensional vector  X  from set  A  is presented as input, the BAM recalls  m -dimensional vector  Y  from set  B , but when  Y  is presented as input, the BAM recalls  X .
[object Object],[object Object]
[object Object],[object Object],[object Object],Stability and storage capacity of the BAM

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lecture07.ppt

  • 1.
  • 2. Introduction, or how the brain works Machine learning involves adaptive mechanisms that enable computers to learn from experience, learn by example and learn by analogy. Learning capabilities can improve the performance of an intelligent system over time. The most popular approaches to machine learning are artificial neural networks and genetic algorithms . This lecture is dedicated to neural networks.
  • 3.
  • 4.
  • 6.
  • 7.
  • 8. Architecture of a typical artificial neural network
  • 9. Analogy between biological and artificial neural networks
  • 10. The neuron as a simple computing element Diagram of a neuron
  • 11.
  • 13.
  • 15.
  • 16.
  • 17. Linear separability in the perceptrons
  • 18. This is done by making small adjustments in the weights to reduce the difference between the actual and desired outputs of the perceptron. The initial weights are randomly assigned, usually in the range [  0.5, 0.5], and then updated to obtain the output consistent with the training examples. How does the perceptron learn its classification tasks?
  • 19.
  • 20. The perceptron learning rule where p = 1, 2, 3, . . .  is the learning rate , a positive constant less than unity. The perceptron learning rule was first proposed by Rosenblatt in 1960. Using this rule we can derive the perceptron training algorithm for classification tasks.
  • 21. Step 1 : Initialisation Set initial weights w 1 , w 2 ,…, w n and threshold  to random numbers in the range [  0.5, 0.5]. If the error, e ( p ), is positive, we need to increase perceptron output Y ( p ), but if it is negative, we need to decrease Y ( p ). Perceptron’s training algorithm
  • 22. Step 2 : Activation Activate the perceptron by applying inputs x 1 ( p ), x 2 ( p ),…, x n ( p ) and desired output Y d ( p ). Calculate the actual output at iteration p = 1 where n is the number of the perceptron inputs, and step is a step activation function. Perceptron’s training algorithm (continued)
  • 23. Step 3 : Weight training Update the weights of the perceptron where  w i ( p ) is the weight correction at iteration p . The weight correction is computed by the delta rule : Step 4 : Iteration Increase iteration p by one, go back to Step 2 and repeat the process until convergence. Perceptron’s training algorithm (continued)
  • 24. Example of perceptron learning: the logical operation AND
  • 25. Two-dimensional plots of basic logical operations A perceptron can learn the operations AND and OR , but not Exclusive-OR .
  • 26.
  • 27. Multilayer perceptron with two hidden layers
  • 28.
  • 29.
  • 30.
  • 32. Step 1 : Initialisation Set all the weights and threshold levels of the network to random numbers uniformly distributed inside a small range: where F i is the total number of inputs of neuron i in the network. The weight initialisation is done on a neuron-by-neuron basis. The back-propagation training algorithm
  • 33. Step 2 : Activation Activate the back-propagation neural network by applying inputs x 1 ( p ), x 2 ( p ),…, x n ( p ) and desired outputs y d ,1 ( p ), y d ,2 ( p ),…, y d , n ( p ). ( a ) Calculate the actual outputs of the neurons in the hidden layer: where n is the number of inputs of neuron j in the hidden layer, and sigmoid is the sigmoid activation function.
  • 34. ( b ) Calculate the actual outputs of the neurons in the output layer: where m is the number of inputs of neuron k in the output layer. Step 2 : Activation (continued)
  • 35. Step 3 : Weight training Update the weights in the back-propagation network propagating backward the errors associated with output neurons. ( a ) Calculate the error gradient for the neurons in the output layer: where Calculate the weight corrections: Update the weights at the output neurons:
  • 36. ( b ) Calculate the error gradient for the neurons in the hidden layer: Calculate the weight corrections: Update the weights at the hidden neurons: Step 3 : Weight training (continued)
  • 37. Step 4 : Iteration Increase iteration p by one, go back to Step 2 and repeat the process until the selected error criterion is satisfied. As an example, we may consider the three-layer back-propagation network. Suppose that the network is required to perform logical operation Exclusive-OR . Recall that a single-layer perceptron could not do this operation. Now we will apply the three-layer net.
  • 38. Three-layer network for solving the Exclusive-OR operation
  • 39.
  • 40.
  • 41.
  • 42.
  • 43.
  • 44. Learning curve for operation Exclusive-OR
  • 45. Final results of three-layer network learning
  • 46. Network represented by McCulloch-Pitts model for solving the Exclusive-OR operation
  • 47. ( a ) Decision boundary constructed by hidden neuron 3; ( b ) Decision boundary constructed by hidden neuron 4; ( c ) Decision boundaries constructed by the complete three-layer network Decision boundaries
  • 48.
  • 49.
  • 50. Learning with momentum for operation Exclusive-OR
  • 51. Learning with adaptive learning rate To accelerate the convergence and yet avoid the danger of instability, we can apply two heuristics: Heuristic 1 If the change of the sum of squared errors has the same algebraic sign for several consequent epochs, then the learning rate parameter,  , should be increased. Heuristic 2 If the algebraic sign of the change of the sum of squared errors alternates for several consequent epochs, then the learning rate parameter,  , should be decreased.
  • 52.
  • 53. Learning with adaptive learning rate
  • 54. Learning with momentum and adaptive learning rate
  • 55.
  • 56.
  • 57.
  • 58. Single-layer n -neuron Hopfield network
  • 59.
  • 60.
  • 61.
  • 62. Possible states for the three-neuron Hopfield network
  • 63.
  • 64.
  • 65.
  • 66.
  • 67.
  • 68.
  • 69.
  • 71. The basic idea behind the BAM is to store pattern pairs so that when n -dimensional vector X from set A is presented as input, the BAM recalls m -dimensional vector Y from set B , but when Y is presented as input, the BAM recalls X .
  • 72.
  • 73.