1. Soft computing Lecture 7 Multi-Layer perceptrons

2. Why hidden layer is needed

4. Problem of XOR for simple perceptron X2X2 X1X1 (0,1) (0,0) (1,0) (1,1) Class 1 Class 2 In this case it is not possible to draw descriminant line

7. Minimization of error

9. Main algorithm of training

10. Kinds of sigmoid used in perceptrons Exponential Rational Hyperbolic tangent

11. Formulas for error back propagation algorithm Modification of weights of synapses of j th neuron connected with i th ones, x j – state of j th neuron (output) For output layer For hidden layers k – number of neuron in next layer connected with j th neuron (1) (2) (3)

12. (2), (1) (1) (3), (1) (1)

13. Example of implementation TNN=Class(TObject) public State:integer; N,NR,NOut,NH:integer; a:real; Step:real; NL:integer; // ъюы-тю шЄхЁрЎшщ яЁш юсєўхэшш S1:array[1..10000] of integer; S2:array[1..200] of real; S3:array[1..5] of real; G3:array[1..5] of real; LX,LY:array[1..10000] of integer; W1:array[1..10000,1..200] of real; W2:array[1..200,1..5] of real; W1n:array[1..10000,1..200] of real; W2n:array[1..200,1..5] of real; SymOut:array[1..5] of string[32]; procedure FormStr; procedure Learn; procedure Work; procedure Neuron(i,j:integer); end;

14. Procedure of simulation of neuron; procedure TNN.Neuron(i,j:integer); var k:integer; Sum:real; begin case i of 1: begin if Form1.PaintBox1.Canvas.Pixels[LX[j],LY[j]]= clRed then S1[j]:=1 else S1[j]:=0; end; 2: begin Sum:=0.0; for k:=1 to NR do Sum:=Sum + S1[k]*W1[k,j]; if Sum> 0 then S2[j]:=Sum/(abs(Sum)+Net.a) else S2[j]:=0; end; 3: begin Sum:=0.0; for k:=1 to NH do Sum:=Sum + S2[k]*W2[k,j]; if Sum> 0 then S3[j]:=Sum/(abs(Sum)+Net.a) else S3[j]:=0; end;

15. Fragment of procedure of learning For i:=1 to NR do for j:=1 to NH do begin S:=0; for k:=1 to NOut do begin if (S3[k]>0) and (S3[k]<1) then D:=S3[k]*(1-S3[k]) else D:=1; W2n[j,k]:=W2[j,k]+Step*S2[j]*(G3[k]-S3[k])*D; S:=S+D*(G3[k]-S3[k])*W2[j,k] end; if (S2[j]>0) and (S2[j]<1) then D:=S2[j]*(1-S2[j]) else D:=1; S:=S*D; W1n[i,j]:=W1[i,j]+Step*S*S1[i]; end;

16. Generalization

17. Some of the test data are now misclassified. The problem is that the network, with two hidden units, now has too much freedom and has fitted a decision surface to the training data which follows its intricacies in pattern space without extracting the underlying trends.

18. Overfitting

19. Local minima

20. Two tasks solved by MLP Classification (recognition) –Usually binary outputs Regression (approximation) –Analog outputs

21. Theorem of Kolmogorov “Any continuous function from input to output can be implemented in a three- layer net, given sufficient number of hidden units n H, proper nonlinearities, and weights.”

22. Advantages and disadvantages of MLP with back propagation Advantages: –Guarantee of possibility of solving of tasks Disadvantages: –Low speed of learning –Possibility of overfitting –Impossible to relearning –Selection of structure needed for solving of concrete task is unknown

23. Increase of speed of learning Preliminary processing of features before getting to inputs of percepton Dynamical step of learning (in begin one is large, than one is decreasing) Using of second derivative in formulas for modification of weights Using hardware implementation

24. Fight against of overfitting Don’t select too small error for learning or too large number of iteration

25. Choice of structure Using of constructive learning algorithms –Deleting of nodes (neurons) and links corresponding to one –Appending new neurons if it is needed Using of genetic algorithms for selection of suboptimal structure

26. Impossible to relearning Using of constructive learning algorithms –Deleting of nodes (neurons) and links corresponding to one –Appending new neurons if it is needed