This is a paper from Melanie Mitchell, Stephanie Forrest, John H. Holland

1. The Royal Road for Genetic Algorithms: Fitness Landscape and GA Performance Melanie Mitchell University of Michigan Stephanie Forrest University of New Mexico John H. Holland University of Michigan Presented By Md Mishfaq Ahmed September 22nd , 2011

2. Effect of fitness landscape on GA performance Three key properties of fitness landscape -- Deception Low order schemas leading away from higher order optimum instance Sampling Error Functions with high variance in the fitness of a correct low order schema leading to error Number of Local Optima in the landscape Affects the ease of adaptation under mutation and crossover

4. Parameterizable Landscape feature: Hierarchical Structure of schemas and Stepping stone Royal Road function is used to model the hierarchical structure F(x) = ∑ csσs(x) ; s ϵ S where: S = {s1,s2,s3 ….. s15}; a set of schemas The term “Royal road” is used to emphasize: GA with xover can easily follow the path laid out to reach global optimum Algorithms like Hillclimbing that relies on single-bit mutation cannot easily find high values in royal road family of functions

5. Parameterizable Landscape features: Royal Road function

6. Landscape feature: Isolated High-Fitness Regions Consider the fitness function: The optimum is 1111 with fitness 5-16+5-16+31= 9. Average fitness for u(**11) is 5-16(0.5)+5(0.25)-16(0.25)+31(0.25) = 2 The average fitnesses for five schemas are given on left.

7. Landscape feature: Isolated High-Fitness Regions Hillclimbing : Reach the largest areas of intermediate fitness (**11 and 11**) but will be slow in crossing the “deserts” (*111 and 111*) to reach the global optimum (1111) GA: Once instances of **11 and 11** are present, the “deserts” can be crossed quickly via crossover among those instances to produce 1111

8. Landscape feature: Multiple Conflicting solutions Consider a function with two equal peaks: F(x) = (x-(1/2))2 two conflicting optima: 0 and 1 Conventional GA: Initially sample both peaks but eventually converges on one by exploiting random fluctuations in the sampling process Crossover may cause useless hybrids by crossing good solutions from conflicting peaks (ex: 0000 and 1111)

9. GA performance on Royal Road functions: Experimental setup Royal road function (of figure 1) is used GA performance is compared with stochastic hillclimbing l = 64 . Individuals of GA population are bit strings of length 64 Population size is always fixed at n = 128 GA was allowed to continue until global optimum is found GA used is single-point xover with xover rate 0.7 per pair of parents and mutation rate 0.005

10. Results: Effects of crossover on GA performance 1 GA with Xover 2 GA without Xover 3 HillClimbing Chart 1: summary of results on the royal road function for GA with and without crossover, and for Hillclimbing Each result summerizes 50 runs. For hillclimbing the optimum was not reached after 2000 generations, the best solution had a fitness of 38% of the optimum

11. Results: Effects of crossover on GA performance 1 Order 8 2 Order 16 3 Order 32 4 Order 64 Chart 2: The average generation of first appearance of a schema of each order for the Royal Road function, averaged over 50 runs for GA with and without crossover

12. Results: Effects of crossover on GA performance What is the bottleneck in the discovery process of higher order schema with higher fitness? Time for instances of the components schema to appear in the population? OR The waiting time for the crossover to take place? To answer this question, Mean time to combine (MTTC) is measured

13. Results: Effects of crossover on GA performance 1 Order 8 2 Order 16 3 Order 32 Chart 3: MTTC1 is the average difference in generations between the first appearance of two component schemas of given order and the appearance of the schema that is the combination of the two. MTTC2 is the same data with the first appearance is only taken into account if that lasts for at least 10 population. ( figure in brackets shows number of cases)

14. Effects of Intermediate levels on GA performance Does GA perform better when intermediate levels schemas are used as stepping stone? To answer this question Royal road function of figure 1 is modified to have no intermediate (no order 16 or order 32) schema Only order 8 schemas are assigned fitness coefficient c = 8 Global optimum is still at x = 111…1 (string of 64 1s) But F(x) = 8*8=64

15. Effects of Intermediate levels on GA performance 1 GA with Xover with intermediate levels 2 GA with Xover without intermediate levels 3 HillClimbing without intermediate levels Chart 4: Summary of results for the original royal road function (with intermediate levels) for GA with Xover and for the modified function (no intermediate levels) for GA with Xover and Hillclimbing

16. Conclusion The paper proposes three features of fitness landscapes that are relevant to the GA performance Royal road functions isolate one important aspect of fitness landscapes: hierarchies of schemas Crossover boosts performance for straight schema hierarchy with single global optimum Adding intermediate level schemas have detrimental effect in performance

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