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- 1. © Negnevitsky, Pearson Education, 2005 1 Lecture 7 Artificial neural networks: Supervised learning Introduction, or how the brain works The neuron as a simple computing element The perceptron Multilayer neural networks Accelerated learning in multilayer neural networks The Hopfield network Bidirectional associative memories (BAM) Summary
- 2. © Negnevitsky, Pearson Education, 2005 2 Accelerated learning in multilayer neural networks A multilayer network learns much faster when the sigmoidal activation function is represented by a hyperbolic tangent: where a and b are constants. Suitable values for a and b are: a = 1.716 and b = 0.667 a e a Y bX h tan 1 2
- 3. © Negnevitsky, Pearson Education, 2005 3 We also can accelerate training by including a momentum term in the delta rule: where b is a positive number (0 £ b < 1) called the momentum constant. Typically, the momentum constant is set to 0.95. This equation is called the generalised delta rule. ) ( ) ( ) 1 ( ) ( p p y p w p w k j jk jk
- 4. © Negnevitsky, Pearson Education, 2005 4 Learning with momentum for operation Exclusive-OR 0 20 40 60 80 100 120 10-4 10-2 100 102 Epoch Training for 126 Epochs 0 100 140 -1 -0.5 0 0.5 1 1.5 Epoch 10-3 101 10-1 20 40 60 80 120 Learning Rate
- 5. © Negnevitsky, Pearson Education, 2005 5 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, a, 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, a, should be decreased.
- 6. © Negnevitsky, Pearson Education, 2005 6 Adapting the learning rate requires some changes in the back-propagation algorithm. If the sum of squared errors at the current epoch exceeds the previous value by more than a predefined ratio (typically 1.04), the learning rate parameter is decreased (typically by multiplying by 0.7) and new weights and thresholds are calculated. If the error is less than the previous one, the learning rate is increased (typically by multiplying by 1.05).
- 7. © Negnevitsky, Pearson Education, 2005 7 Learning with adaptive learning rate 0 10 20 30 40 50 60 70 80 90 100 Epoch Tr ainingfor 103Epochs 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 Epoch 10-4 10-2 100 102 10-3 101 10-1 Sum-Squared Erro Learning Rate
- 8. © Negnevitsky, Pearson Education, 2005 8 Learning with momentum and adaptive learning rate 0 10 20 30 40 50 60 70 80 Epoch Tr ainingfor 85Epochs 0 10 20 30 40 50 60 70 80 90 0 0.5 1 2.5 Epoch 10-4 10-2 100 102 10-3 101 10-1 1.5 2 Sum-Squared Erro Learning Rate
- 9. © Negnevitsky, Pearson Education, 2005 9 The Hopfield Network Neural networks were designed on analogy with the brain. The brain’s memory, however, works by association. For example, we can recognise a familiar face even in an unfamiliar environment within 100-200 ms. We can also recall a complete sensory experience, including sounds and scenes, when we hear only a few bars of music. The brain routinely associates one thing with another.
- 10. © Negnevitsky, Pearson Education, 2005 10 Multilayer neural networks trained with the back- propagation algorithm are used for pattern recognition problems. However, to emulate the human memory’s associative characteristics we need a different type of network: a recurrent neural network. A recurrent neural network has feedback loops from its outputs to its inputs. The presence of such loops has a profound impact on the learning capability of the network.
- 11. © Negnevitsky, Pearson Education, 2005 11 The stability of recurrent networks intrigued several researchers in the 1960s and 1970s. However, none was able to predict which network would be stable, and some researchers were pessimistic about finding a solution at all. The problem was solved only in 1982, when John Hopfield formulated the physical principle of storing information in a dynamically stable network.
- 12. © Negnevitsky, Pearson Education, 2005 12 Single-layer n-neuron Hopfield network xi x1 x2 xn yi y1 y2 yn 1 2 i n I n p u t S i g n a l s O u t p u t S i g n a l s
- 13. © Negnevitsky, Pearson Education, 2005 13 The Hopfield network uses McCulloch and Pitts neurons with the sign activation function as its computing element: X Y X X Y sign if , if , 1 0 if , 1
- 14. © Negnevitsky, Pearson Education, 2005 14 The current state of the Hopfield network is determined by the current outputs of all neurons, y1, y2, . . ., yn. Thus, for a single-layer n-neuron network, the state can be defined by the state vector as: n y y y 2 1 Y
- 15. © Negnevitsky, Pearson Education, 2005 15 In the Hopfield network, synaptic weights between neurons are usually represented in matrix form as follows: where M is the number of states to be memorised by the network, Ym is the n-dimensional binary vector, I is n ´ n identity matrix, and superscript T denotes matrix transposition. I Y Y W M M m T m m 1
- 16. © Negnevitsky, Pearson Education, 2005 16 Possible states for the three-neuron Hopfield network y1 y2 y3 (1, 1,1) ( 1, 1,1) ( 1, 1, 1) (1, 1, 1) (1, 1,1) ( 1,1,1) (1, 1, 1) ( 1,1, 1) 0
- 17. © Negnevitsky, Pearson Education, 2005 17 The stable state-vertex is determined by the weight matrix W, the current input vector X, and the threshold matrix q. If the input vector is partially incorrect or incomplete, the initial state will converge into the stable state-vertex after a few iterations. Suppose, for instance, that our network is required to memorise two opposite states, (1, 1, 1) and (-1, -1, -1). Thus, where Y1 and Y2 are the three-dimensional vectors. or 1 1 1 1 Y 1 1 1 2 Y 1 1 1 1 T Y 1 1 1 2 T Y
- 18. © Negnevitsky, Pearson Education, 2005 18 The 3 ´ 3 identity matrix I is Thus, we can now determine the weight matrix as follows: Next, the network is tested by the sequence of input vectors, X1 and X2, which are equal to the output (or target) vectors Y1 and Y2, respectively. 1 0 0 0 1 0 0 0 1 I 1 0 0 0 1 0 0 0 1 2 1 1 1 1 1 1 1 1 1 1 1 1 W 0 2 2 2 0 2 2 2 0
- 19. © Negnevitsky, Pearson Education, 2005 19 First, we activate the Hopfield network by applying the input vector X. Then, we calculate the actual output vector Y, and finally, we compare the result with the initial input vector X. 1 1 1 0 0 0 1 1 1 0 2 2 2 0 2 2 2 0 1 sign Y 1 1 1 0 0 0 1 1 1 0 2 2 2 0 2 2 2 0 2 sign Y
- 20. © Negnevitsky, Pearson Education, 2005 20 The remaining six states are all unstable. However, stable states (also called fundamental memories) are capable of attracting states that are close to them. The fundamental memory (1, 1, 1) attracts unstable states (-1, 1, 1), (1, -1, 1) and (1, 1, -1). Each of these unstable states represents a single error, compared to the fundamental memory (1, 1, 1). The fundamental memory (-1, -1, -1) attracts unstable states (-1, -1, 1), (-1, 1, -1) and (1, -1, -1). Thus, the Hopfield network can act as an error correction network.
- 21. © Negnevitsky, Pearson Education, 2005 21 Storage capacity of the Hopfield network Storage capacity is or the largest number of fundamental memories that can be stored and retrieved correctly. The maximum number of fundamental memories Mmax that can be stored in the n-neuron recurrent network is limited by n Mma x 0.15
- 22. © Negnevitsky, Pearson Education, 2005 22 Bidirectional associative memory (BAM) The Hopfield network represents an autoassociative type of memory - it can retrieve a corrupted or incomplete memory but cannot associate this memory with another different memory. Human memory is essentially associative. One thing may remind us of another, and that of another, and so on. We use a chain of mental associations to recover a lost memory. If we forget where we left an umbrella, we try to recall where we last had it, what we were doing, and who we were talking to. We attempt to establish a chain of associations, and thereby to restore a lost memory.
- 23. © Negnevitsky, Pearson Education, 2005 23 To associate one memory with another, we need a recurrent neural network capable of accepting an input pattern on one set of neurons and producing a related, but different, output pattern on another set of neurons. Bidirectional associative memory (BAM), first proposed by Bart Kosko, is a heteroassociative network. It associates patterns from one set, set A, to patterns from another set, set B, and vice versa. Like a Hopfield network, the BAM can generalise and also produce correct outputs despite corrupted or incomplete inputs.
- 24. © Negnevitsky, Pearson Education, 2005 24 BAM operation yj(p) y1(p) y2(p) ym(p) 1 2 j m Output layer Input layer xi(p) x1 x2 (p) (p) xn(p) 2 i n 1 xi(p+1) x1(p+1) x2(p+1) xn(p+1) yj(p) y1(p) y2(p) ym(p) 1 2 j m Output layer Input layer 2 i n 1 (a) Forward direction. (b) Backward direction.
- 25. © Negnevitsky, Pearson Education, 2005 25 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.
- 26. © Negnevitsky, Pearson Education, 2005 26 To develop the BAM, we need to create a correlation matrix for each pattern pair we want to store. The correlation matrix is the matrix product of the input vector X, and the transpose of the output vector YT . The BAM weight matrix is the sum of all correlation matrices, that is, where M is the number of pattern pairs to be stored in the BAM. T m M m m Y X W 1
- 27. © Negnevitsky, Pearson Education, 2005 27 Stability and storage capacity of the BAM The BAM is unconditionally stable. This means that any set of associations can be learned without risk of instability. The maximum number of associations to be stored in the BAM should not exceed the number of neurons in the smaller layer. The more serious problem with the BAM is incorrect convergence. The BAM may not always produce the closest association. In fact, a stable association may be only slightly related to the initial input vector.