1. Artificial Intelligence Techniques Multilayer Perceptrons

2. Overview The multi-layered perceptron Back-propagation Introduction to training Uses

3. Pattern space - linearly separable X2 X1

4. Non-linearly separable problems If a problem is not linearly separable, then it is impossible to divide the pattern space into two regions A network of neurons is needed Until fairly recently, it was not known how to train a multi-layered network

5. Pattern space - non linearly separable X2 X1 Decision surface

6. The multi-layered perceptron (MLP) Input layer Hidden layerOutput layer

7. Complex decision surface The MLP has the ability to emulate any function using one hidden layer with a sigmoid function, and a linear output layer A 3-layered network can therefore produce any complex decision surface However, the number of neurons in the hidden layer cannot be calculated

8. The multi-layered perceptron (MLP) Input layer Hidden layerOutput layer

9. Network architecture All neurons in one layer are connected to all neurons in the next layer The network is a feedforward network, so all data flows from the input to the output The architecture of the network shown is described as 3:4:2 All neurons in the hidden and output layers have a bias connection

10. Input layer Receives all of the inputs Number of neurons equals the number of inputs Does no processing Connects to all the neurons in the hidden layer

11. Hidden layer Could be more than one layer, but theory says that only one layer is necessary The number of neurons is found by experiment Processes the inputs Connects to all neurons in the output layer The output is a sigmoid function

12. Output layer Produces the final outputs Processes the outputs from the hidden layer The number of neurons equals the number of outputs The output could be linear or sigmoid

13. Problems with networks Originally the neurons had a hard- limiter on the output Although an error could be found between the desired output and the actual output, which could be used to adjust the weights in the output layer, there was no way of knowing how to adjust the weights in the hidden layer

14. The invention of back- propagation By introducing a smoothly changing output function, it was possible to calculate an error that could be used to adjust the weights in the hidden layer(s)

15. Output function The sigmoid function 0 0.2 0.4 0.6 0.8 1 1.2 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -0.5 -0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 net y

16. Sigmoid function The sigmoid function goes smoothly from 0 to 1 as net increases The value of y when net=0 is 0.5 When net is negative, y is between 0 and 0.5 When net is positive, y is between 0.5 and 1.0

17. Back-propagation The method of training is called the back-propagation of errors The algorithm is an extension of the delta rule, called the generalised delta rule

18. Generalised delta rule The equation for the generalised delta rule is ΔWi = ηXiδ δ is the defined according to which layer is being considered. For the output layer, δ is y(1-y)(d-y). For the hidden layer δ is a more complex.

19. Pattern recognition Many problems can be described as pattern recognition For example, voice recognition, face recognition, optical character recognition

20. Pattern classification A more precise definition is pattern classification In pattern classification a system is shown examples of a number of objects Each object is given a label or class The task of the system is to correctly classify objects that it hasn’t seen before

21. Example of 2-input data

22. Pattern space

23. Training a network The problem could not be implemented on a single layer - nonlinearly separable A 3 layer MLP was tried with 4 neurons in the hidden layer - which trained The number of neurons in the hidden layer was reduced to 2 and still trained With 1 neuron in the hidden layer it failed to train

24. The weights The weights for the 2 neurons in the hidden layer are -9, 3.6 and 0.1 and 6.1, 2.2 and -7.8 These weights can be shown in the pattern space as two lines The lines divide the space into 4 regions

25. The hidden neurons

26. Training and Testing Starting with a data set, the first step is to divide the data into a training set and a test set Use the training set to adjust the weights until the error is acceptably low Test the network using the test set, and see how many it gets right

27. A better approach Critics of this standard approach have pointed out that training to a low error can sometimes cause “overfitting”, where the network performs well on the training data but poorly on the test data The alternative is to divide the data into three sets, the extra one being the validation set

28. Validation set During training, the training data is used to adjust the weights At each iteration, the test data is also passed through the network and the error recorded but the weights are not adjusted The training stops when the error for the test set starts to increase

29. Stopping criteria error time Stop here Test set Training set

30. Architecture Input layer Hidden layerOutput layer

31. Back-propagation The method of training is called the back-propagation of errors The algorithm is an extension of the delta rule, called the generalised delta rule

32. Generalised delta rule The equation for the generalised delta rule is ΔWi = ηXiδ δ is the defined according to which layer is being considered. For the output layer, δ is y(1-y)(d-y). For the hidden layer δ is a more complex.

33. Hidden Layer We have to deal with the error from the output layer being feedback backwards to the hidden layer. Lets look at example the weight w2(1,2) Which is the weight connecting neuron 1 in the input layer with neuron 2 in the hidden layer.

34. Δw2(1,2)=ηX1(1)δ2(2) Where X1(1) is the output of the neuron 1 in the hidden layer. δ2(2) is the error on the output of neuron 2 in the hidden layer. δ2(2)=X2(2)[1-X2(2)]w3(2,1) δ3(1)

35. δ3(1)= y(1-y)(d-y) =x3(1)[1-x3(1)][d- x3(1)] So we start with the error at the output and use this result to ripple backwards altering the weights.

37. Example Exclusive OR using the network shown earlier: 2:2:1 network Initial weights W2(0,1)=0.862518W2(1,1)=-0.155797 W2(2,1)=0.282885 W2(0,2)=0.834986w2(1,2)=-0.505997w2(2,2)=- 0.864449 W3(0,1)=0.036498w3(1,1)=-0.430437 w3(2,1)=0.48121

38. Feedforward – hidden layer (neuron 1) So if X1(0)=1 (the bias) X1(1)=0 X1(2)=0 The output of weighted sum inside neuron 1 in the hidden layer=0.862518 Then using sigmoid function X2(1)=0.7031864

39. Feedforward – hidden layer (neuron 2) So if X1(0)=1 (the bias) X1(1)=0 X1(2)=0 The output of weighted sum inside neuron 2 in the hidden layer=0.834986 Then using sigmoid function X2(2)=0.6974081

40. Feedforward – output layer So if X2(0)=1 (the bias) X2(1)=0.7031864 X2(2)=0.6974081 The output of weighted sum inside neuron 2 in the hidden layer=0.0694203 Then using sigmoid function X3(1)=0.5173481 Desired output=0

41. δ3(1)=x3(1)[1-x3(1)][d-x3(1)] =-0.1291812 δ2(1)=X2(1)[1-X2(1)]w3(1,1) δ3(1)=0.0116054 δ2(2)=X2(2)[1-X2(2)]w3(2,1) δ3(1)=-0.0131183 Now we can use the delta rule to calculate the change in the weights ΔWi = ηXiδ

42. Examples If we set η=0.5 ΔW2(0,1) = ηX1(0)δ2(1) =0.5 x 1 x 0.0116054 =0.0058027 ΔW3(2,1) = ηX2(1)δ3(1) =0.5 x 0.7031864 x –0.1291812 =-0.04545192

43. What would be the results of the following? ΔW2(2,1) = ηX1(2)δ2(1) ΔW2(2,2) = ηX1(2)δ2(2)

44. ΔW2(2,1) = ηX1(2)δ2(1) =0.5x0x0.0116054 =0 ΔW2(2,2) = ηX1(2)δ2(2) =0.5 x 0 x – 0.131183 =0

45. New weights W2(0,1)=0.868321W2(1,1)=-0.155797 W2(2,1)=0.282885 W2(0,2)=0.828427w2(1,2)=-0.505997w2(2,2)=- 0.864449 W3(0,1)=0.028093w3(1,1)=-0.475856 w3(2,1)=0.436164

46. Conclusions Train using training, test and validation sets An MLP can be used to recognise (classify) complex data It uses supervised learning with back- propagation to adjust the weights It divides the pattern space in the hidden layer

47. Conclusions Extending the delta rule to do back propagation Need to calculate the error at the outputs of neurones in the hidden and output layers δ3(1)=x3(1)[1-x3(1)][d-x3(1)] δ2(1)=X2(1)[1-X2(1)]w3(1,1) δ3(1) δ2(2)=X2(2)[1-X2(2)]w3(2,1) δ3(1)

48. Once you have the error values (δ’s) for the neurones you then use the delta rule to calculate the actual change in the weights. ΔWi = ηXiδ