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Selection and Recombination

Selection and Recombination. Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello Department of Mathematics and Computer Science University of Catania. Selection and Recombination. 1. Review of the previous lecture (a) How a simple evolutionary algorithm works

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Selection and Recombination

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  1. Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello Department of Mathematics and Computer Science University of Catania

  2. Selection and Recombination • 1. Review of the previous lecture • (a) How a simple evolutionary algorithm works • (b) Some crossover and mutation operators • 2. Selection and reproduction • 3. Recombination • 4. A sample algorithm Selection and Recombination - Lecture 4

  3. Selection and Reproduction • Selection is not normally regarded as a search operator although it does in search significantly. • Selection can be used either before or after search operators. • When selection is used before search operators, the process of choosing the next generation from the union of all parents and offspring is sometimes called reproduction. • The generational gap of an EA refers to the overlap (i.e., individuals that did not go through any search operators) between the old and new generations. • The two extremes are generational EAs and steady-state EAs. • Elitism = copying the best individual to the next generation. Selection and Recombination - Lecture 4

  4. Fitness Proportional Selection • Also known as roulette wheel selection. • Use raw fitness in computing selection probabilities. Does not allow negative fitness values. • Weakness: Domination of super individuals in early generations and slow convergence in later generations. • Fitness scaling has often been used in early days to combat the two problems. Selection and Recombination - Lecture 4

  5. Fitness Scaling • Simple scaling • The ith individual's fitness is defined as: fscaled (t) = foriginal (t) - f worst (t); • where t is the generation number and f worst (t) the fitness of the worst individual so far. • Sigma scaling • The ith individual's fitness is defined as: • where c is a constant, e.g., 2, f*(t) is the average fitness in the current population, and f (t) is the standard deviation of the fitness in the current population. Selection and Recombination - Lecture 4

  6. Power scaling • The ith individual's fitness is defined as: fscaled(t) = (foriginal(t)) k where k > 0. • Exponential scaling • The ith individual's fitness is defined as: fscaled (t) = exp( foriginal (t)/T ) where T > 0 is the temperature, approaching zero. Selection and Recombination - Lecture 4

  7. Ranking • Linear ranking • Assume that the population size is , rank 0 indicates the worst individual and rank -1 the best. The probability of selecting the ith individual is: • Where  () indicates the expected number of offspring produced by the worst (best) individuals. Note that: • Hence  + = 2. So 1  2 and  = 2- . • Power ranking • C below is a normalising factor. 0<  < . Selection and Recombination - Lecture 4

  8. Geometric ranking: • Exponential ranking: • Xin Yao ranking: Selection and Recombination - Lecture 4

  9. Recombination for Real-valued Representation • Discrete Recombination • does not change actual (gene) values. Very similar to the crossover operators on binary strings. • Intermediate Recombination • does change actual (gene) values. Usually based on some kind of average/mixture among multiple parents. Selection and Recombination - Lecture 4

  10. Discrete Recombination • Multi-point Recombination • Similar to that for the binary representation. • Global Discrete Recombination • Similar to uniform crossover for the binary representation. • Geometric explanation. Selection and Recombination - Lecture 4

  11. Intermediate Recombination • With two parents • Given x1 and x2: • With more parents • Givenx1 and x2: where Selection and Recombination - Lecture 4

  12. Other Recombination • Heuristic Recombination • Assume x2 is no worse than x1: where u is a uniformly distributed random number in [0,1]. • Simplex Recombination • Randomly select a group (> 2) of parents. Assume x1 is the best individual and x2 the worst in the group. Compute the centroid, c, of the group without x2 . • Geometric Recombination • Can be generalised to multiple parents. Selection and Recombination - Lecture 4

  13. Quadratic Recombination • Let xij be the j-th component of the vectors xi, i 1,2,3, j 1,…,n, where n is the dimensionality. We approximate the position of P4 using the quadratic interpolation method as follow. • One offspring is generated from three parents. Selection and Recombination - Lecture 4

  14. What Does Quadratic Recombination Mean Note that we are minimising “fitness” here Selection and Recombination - Lecture 4

  15. A Hybrid EA With Local Search • 1. Initialize  individuals at random. • 2. Perform local search from each individual. • 3. REPEAT • (a) Generate 3 points P1 ,P2 , P3 by global discrete recombination. • (b) Perform a quadratic approximation using P1 ,P2 , P3 to produce a point P4 . • (c) Perform a local search from P4 and update P4 with the search result. (Sounds Lamarkian.) • (d) Place P1 ,P2 , P3 , P4 into the population and do a ( + 4) truncation selection. • 4. UNTIL termination criteria are met. Selection and Recombination - Lecture 4

  16. Local Search With Random Memorising • Store best solutions in a memory. • Retrieve a random one (old best) when a new best solution is found. • Search along the direction of oldnew, i.e., the direction of: Selection and Recombination - Lecture 4

  17. Experimental Studies • 18 multimodal benchmark functions. • Population size 30 • Maximum function evaluation 500,000. • 50 independent runs for each function. Selection and Recombination - Lecture 4

  18. Benchmark Functions (f8 - f25 ) Selection and Recombination - Lecture 4

  19. Results on f8-f13 Selection and Recombination - Lecture 4

  20. Results on f8-f13 with population size 50 and 60 Selection and Recombination - Lecture 4

  21. Results on f14-f25 Selection and Recombination - Lecture 4

  22. Results on f16-f23 with population size 10 Selection and Recombination - Lecture 4

  23. Summary • Different problems require different search operators and selection schemes. There is no universally best one. • Using real vectors is usually more appropriate than binary strings for function optimisation. • Many search operators are heuristics-based. Domain knowledge can often be incorporated into search operators and representation. Selection and Recombination - Lecture 4

  24. References • T. Bäck, D. B. Fogel, and Z. Michalewicz (eds.), Handbook of Evolutionary Computation, IOP Publ. Co. & Oxford University Press, 1997. Part C. (in the School library) • K.-H. Liang, X. Yao and C. S. Newton, “Combining landscape approximation and local search in global optimization" Proc. of the 1999 Congress on Evolutionary Computation, Vol. 2, IEEE Press, Piscataway, NJ, USA, pp.1514-1520, July 1999. Selection and Recombination - Lecture 4

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