Evolutionary Computation

1
Evolutionary Computation

• Evolving Neural Network Topologies

2
Project Problem

• There is a class of problems that are not
  linearly separable
• The XOR function is a member of this class
• The BACKPROPAGATION algorithm can express a
  variety of these non-linear decision surfaces

3
Project Problem

• Non-linear decision surfaces cant be learned
  with a single perceptron
• Backprop uses a multi-layer approach where there
  is a number of input units, a number of hidden
  units, and the corresponding output units

4
Parametric Optimization

• Parametric Optimization was the goal of this
  project
• The parameter to be optimized was the number of
  hidden units in the hidden layer of a backprop
  network used to learn the output for the XOR
  benchmark function

5
Tool Boxes

• A neural network tool box developed by Herve Abdi
  (available from Matlab Central) was used for the
  backprop application
• The Genetic Algorithm for Function Optimization
  (GAOT) was used for the GA application

6
Graphical Illustrations

• The next plot shows the randomness associated
  with successive runs of the backprop algorithm
• xx mytestbpg(1,5000,0.25)
• Where 1 is the number of hidden units, 5000 is
  the number of training iterations, and 0.25 is
  the learning rate

7
Graphical Illustrations
8
Graphical Illustrations

• The error from the above plot (and the following)
  is calculated as follows

9
Graphical Illustrations

• E(n) is the error at time epoch n
• Ti(n) 0 11 0 is the output of the target
  function with input sets (0,0), (0,1), (1,0),
  and (1,1) at time epoch n
• Oi(n) o1 o2 o3 o4 is the output of the
  backprop network at time epoch n
• Notice that the training examples will cover the
  entire state space of this function

10
Graphical Illustrations

• The next few plots will show the effects of the
  rate of error convergence for different numbers
  of hidden units

11
Graphical Illustrations-1 hidden
12
Graphical Illustrations-1,2 hidden
13
Graphical Illustrations-1,2,3,5 hidden
14
Graphical Illustrations-1,2,3,5,50 hidden
15
Parametric Optimization

• The GA supplied by the GAOT tool box was used to
  optimize the number of hidden units needed for
  the XOR backprop benchmark
• A real-valued (floating point representation
  instead of binary value representation) was used
  in conjunction with the selection, mutation, and
  cross-over operators

16
Parametric Optimization

• The fitness (evaluation) function is the driving
  factor for the GA in the GAOT toolbox
• The fitness function is specific to the problem
  at hand

17
Fitness Function
18
Parametric Optimization

• Authors of the NEAT paper give the following
  results for their NeuroEvolutionary
  implementation
• Optimum number of hidden nodes (average value)
  2.35
• Average number of generations 32

19
Parametric Optimization

• Results of my experimentation
• Optimum number of hidden units (average value)
  2.9
• Convergence of this value after approximately 17
  generations

20
Parametric Optimization

• GA parameters
• Population size of 20
• Maximum number of generations 50
• Backprop fixed parameters 5000 training
  iterations learning rate of 0.25
• Note Approximately 20 minutes per run of the GA
  with these parameter values. Running on Matlab
  with 768MB of RAM at 2.2 GHz

21
Initial Population Values
22
Final Population Values
23
Paths of Best and Average Solution for a Run of
the GA
24
Target and Backprop Output
25
Conclusion

• Interesting experimental results. Also agrees
  with other researchers results
• GA is a good tool to use for parameter
  optimization. However, the results depend on a
  good fitness function for the task at hand