1. Benjamin Doerr 1, Michael Gnewuch 2, Nils Hebbinghaus 1, Frank Neumann 1 1 Max-Planck-Institut für Informatik Saarbrücken A Rigorous View on Neutrality 2 Christian-Albrechts-Universität zu Kiel

2. Benjamin Doerr Neutrality  Observation: Different genotypes have the same phenotype, and hence, identical fitness. [Kimura. Evolutionary rate at the molecular level. Nature 217 (1968).]  Consequence: –Selection cannot distinguish between them. –But: Variation may produce different (also differently fit) offsprings.  Question: Effect on evolutionary computation (EC)?

3. Benjamin Doerr Neutrality in EC: Observed Behavior  Positive results: –reduces premature convergence –maintains genetic diversity –eases leaving local optima –absorbs destructive mutation  Negative results: –enlarges search space –population evolves slower  Summary: No consistent findings

4. Benjamin Doerr How can results differ that much?  Previous works... –regard different problems –regard different implementations of neutrality –regard complicated settings  difficult to attribute effects to neutrality  difficult to explain why neutrality should have the observed effect –are purely experimental: Difficult to derive general results.  Our aim: Analyze a simple setting with mathematical methods.

5. Benjamin Doerr A First Step to Understanding Neutrality  Galvan-Lopez, Poli (GECCO&PPSN 2006): –Simple pseudo-boolean functions:  OneMax: Unimodal  Trap-function: Deceptive fitness landscape –Add one extra bit to individuals to indicate neutrality. All neutral individuals have the same fitness f neutral. –Experimental investigation:  Genome length 10 or 14 (+ 1 for neutrality)  Population size 80  100 generations  Mutation rate per bit: 0.02 –One finding: Neutrality may help for trap-function, but slows down optimizing OneMax.

6. Benjamin Doerr A Rigorous Analysis of Neutrality  Our work: –same concept of neutrality –mathematical investigation:  arbitrary genome length n (+ 1 for neutrality)  Population size 1, i.e., (1+1)-EA  Prove bounds for the time needed to find the optimum –Results:  Both for OneMax and the trap-function, neutrality has no significant effect for larger genome lengths  Neutrality can reduce the run-time from exponential to polynomial for the peak-function  Results carry over to larger populations

7. Benjamin Doerr Our Result for the Trap-Function  Trap function:  Result: –For all values of f neutral (and without neutrality), the runtime of the (1+1)-EA is at least 2 Ω(n) with probability 1 – 2 -Ω(n). –Consequence: This function cannot be optimized, regardless of neutrality. –Same would hold for larger populations. number of ones in the bit-string fitness n/2n

8. Benjamin Doerr Our Result for the Peak-Function  Peak-function:  Result: –Without neutrality, the run-time is n Ω(n) with prob. 1-o(1) –If f neutral > n/2, then the run-time is O(n 2 log(n)) –Consequence: Neutrality can reduce the deceptive influence of the Peak-function. number of ones in the bit-string fitness n/2n + leading ones n/2

9. Benjamin Doerr Summary and Conclusion  Results: Rigorous analysis for all genome-lengths –Neutrality has no significant impact on optimizing OneMax and the trap-function –There are functions where neutrality reduces the optimization time from exponential to polynomial  Conclusion: –Neutrality can be beneficial, but is no sure-win concept. –Most likely to be useful if it both  reduces deception and  produces only plateaus that can be left easily –True understanding of neutrality still missing. Thanks!