Effect of Spatial Locality on An Evolutionary Algorithm for Multimodal Optimization

1. Effect of Spatial Locality on An Evolutionary Algorithmfor Multimodal Optimization EvoNum 2010 Ka-Chun Wong, Kwong-Sak Leung, and Man-Hon Wong Department of Computer Science & Engineering The Chinese University of Hong Kong, HKSAR, China {kcwong, ksleung, mhwong}@cse.cuhk.edu.hk

2. Background • Differential Evolution (DE) • Proposed by Price and Storn in 1995 • Considered as one of the most powerful evolutionary algorithms for real number function optimization nowadays • "Differential Evolution -- a simple and efficient adaptive scheme for global optimization over continuous spaces, 1995”

3. Background • DE’s Main Idea: (DE/rand/1) • Generate trial vectors (v) using the following formula: • It elegantly replaces the two operations: • Crossover • Mutation • Less parameters to be tuned • Self-organizing ability

4. Background http://ocw.mit.edu/NR/rdonlyres/Sloan-School-of-Management/15-099Fall2003/A40397B9-E8FB-4B45-A41B-D1F69218901F/0/ses2_storn_price.pdf

5. Motivation • Given an optimization problem, traditional optimization algorithms can be applied to obtain a optimum. • However, in the real world, we are often interested in not only a single optimum, but also other possible global and local optima.

6. Problem Definition • Given a function, an algorithm should work out all optimal points in a single run. Six-hump Camel Back Function (http://www.it.lut.fi)

7. Previous works • AEGA (Leung et al. 2003) • SCGA (Li et al. 2002) • Crowding (Kenneth De Jong 1975) • Fitness Sharing (Goldberg et al. 1989) • CrowdingDE (R. Thomsen 2004) • SDE (Xiaodong Li 2005) • Repeated iterations (Beasley et al. 1993) • ……

8. CrowdingDE • Proposed by R. Thomsen in CEC2004 • Main Idea: • Incorporate Crowding technique into Differential Evolution (DE) for multimodal optimization • An individual can only replace the most similar individual

9. CrowdingDE Crowding (Crowding Factor = whole population)

10. Proposed Method • CrowdingDE-L (CrowdingDE using Spatial Locality) • Improve the accuracy • A case study for incorporating “The Principle of Locality” into CrowdingDE ^Peter J. DenningThe locality principle, 2005. The story of the computing fundamental principle of locality of reference.

11. Individual Optimum Proposed Method • Observation: • During a run, individuals around different optima tend to exhibit different convergence rates. • Close individuals (within the same niche) tend to have similar: • Step-size for improvement • Crossover between them is good

12. Proposed Method • Apply spatial locality : • Given an parent individual, favor the close individuals to be selected for trial vector (offspring) generation • Transform the distances between the parent and the candidate individuals to the proportion to be selected. • Use the proportion to form a roulette-wheel to select candidate individuals for trial vector generation

13. Proposed Method • Previous Idea: • Randomly selects candidate individuals for trial vector generation • Proposed Idea: • Apply spatial locality to select candidate individuals for trial vector generation

14. Individual Optimum Proposed Method • Apply spatial locality : • Given an parent individual, favor the close individuals to be selected for trial vector (offspring) generation

15. Proposed Method New New Old

16. Proposed Method • Transformation functions • Transform distance to proportion for selection

17. Experiments • All algorithms were run up to maximum 40000 fitness function evaluations. The performance measurements are obtained by taking the average and standard deviation of 50 runs.

18. Experiments • Performance measurements

19. Experiments

20. Experiments

21. Further Experiments • We conducted further experiments on the number of successful trial vector generation • A successful trial vector generation is defined: • The generation of a trial vector, which can replace an individual in the parent population.

22. Further Experiments • The proposed method (red colour) does improve the selection of candidate individuals for trial vector generation, comparing to the original method (green colour)

23. Critical Thinking • Pros: • Simple and easy to implement • Less parameters to be tuned • Only one DE parameter needs to be set • Cons: • Computationally expensive • Crowding Factor is set to the population size • O(N^2)

24. Conclusion • The experimental results should not be taken to mean that the proposed method (CrowdingDE-L) is “better” than other evolutionary algorithms tested for multimodal optimization. Such a conclusion is oversimplified. • However, it shows that the proposed method does improve CrowdingDE for generating trial vectors. • A case study for integrating locality principle into evolutionary algorithm • The numerical results can also be viewed as a valuable resource for comparing the state-of-the-art algorithms for multimodal optimization.

25. Future Works • Temporal Locality • With the success of spatial locality in this paper, other local techniques involving the principle of locality could be further explored and verified. For instances, besides space, temporal locality can be integrated into evolutionary algorithms. Say, individuals with the same age could be given higher priority for crossovers. Mutation step size could also be linked to the previous step sizes. • Different distance metrics & transformation functions • Different distance metrics could be adopted in calculating the locality. For instances, although Euclidean distance is adopted in this paper, it can be further generalized to p-norm distance (or Minkowski distance).

26. Q & A