Let's get ready to rumble redux: Crossover versus mutation head to head on exponentially scaled problems

Slide 1: Let’s Get Ready to Rumble Redux: Crossover vs. Mutation Head to Head on Exponentially-Scaled Problems Kumara Sastry1,2 and David E. Goldberg1 1Illinois Genetic Algorithms Laboratory 2Materials Computation Center University of Illinois at Urbana-Champaign, Urbana IL 61801 http://www.illigal.uiuc.edu ksastry@uiuc.edu, deg@uiuc.edu Supported by AFOSR FA9550-06-1-0096 and NSF DMR 03-25939.

Slide 2: Motivation Great debate between crossover and mutation When mutation works, it’s lightning quick When crossover works, it tackles more complex problems Compare crossover and mutation where both operators have access to same neighborhood information Local search literature Emphasis on good neighborhood operators [Barnes et al, 2003; Watson, 2003; Hansen et al, 2001] Need for automatic induction of neighborhoods Leads to adaptive time continuation operator [Lima et al 2005, 2006, 2007]

Slide 3: Outline Related work Assumption of known or discovered linkage Objective Algorithm Description Scalability analysis: Crossover vs. Mutation Known or discovered linkage Exponentially scaled additively-separable problem with and without Gaussian noise Summary and Conclusions

Slide 4: Background Emprical studies comparing crossover and mutation Scalability of GAs and mutation-based hillclimber [Mühlenbein, 1991 & 1992; Mitchell, Holland, and Forrest, 1994; Baum, Boneh, and Garett, 2001; Dorste, 2002; Garnier, 1999; Jansen and Wegener, 2002, 2005] Single GA run with large population vs. multiple GA runs with small population at fixed computational cost [Goldberg, 1999; Srivastava & Goldberg, 2001; Srivastava, 2002; Cantú-Paz & Goldberg, 2003; Luke, 2001; Fuchs, 1999] Used fixed operators that don’t adapt linkage Did not consider problems of bounded difficulty Linkage and neighborhood information is critical

Slide 5: Known or Discovered Linkage Assumption of known or induced linkage Can use linkage-learning techniques Linkage information is critical for selectorecombinative GA success Exponential Polynomial Scalability Pelikan, Ph.D. Thesis, 2002 Provide the same information for mutation Mutation searches in the building-block subspace

Slide 6: Algorithm Description Selectorecombinative genetic algorithm Population of size n Binary tournament selection Uniform building-block-wise crossover BBs #1 and #3 exchanged Exchange BBs with probability 0.5 Selectomutative genetic algorithm Start with a random individual Enumerative BB-wise mutation Consider BB partitions – Arbitrary left-to-right order Choose the best schemata – Among the 2k possible ones

Slide 7: Crossover Versus Mutation: Uniform Scaling Deterministic fitness: Noisy fitness: Recombination Mutation is more efficient is more efficient [Sastry & Goldberg, 2004]

Slide 8: Objective Crossover and mutation both have access to same neighborhood information Known or discovered linkage Recombination exchanges building blocks Mutation searches for the best BB in each partition Compare scalability of crossover and mutation Additively separable problems with exponentially-scaled BBs With and without additive Gaussian noise Where do they excel? Derive, verify, and use facetwise models Convergence time and population sizing

Slide 9: Scaling and Noise Cover Most Problems Adversarial problem design [Goldberg, 2002] Fluctuating R P Noise Deception Scaling Noisy BinInt

Slide 10: Convergence Time for Crossover: Deterministic Fitness Functions Selection-Intensity based model [Rudnick, 1992; Thierens et al, 1998] Derived for the BinInt problem Applicable to additively-separable problems Selection Intensity Problem size (m·k )

Slide 11: Population Sizing for Crossover: Deterministic Fitness Functions Domino convergence [Rudnick, 1992] Proportion Most Least salient salient BB convergence in order of salience ... Drift bound dictates population sizing Drift time [Goldberg and Segrest, 1987] time Size the population such that: Population size:

Slide 12: Scalability Analysis of Crossover & Mutation: Deterministic Fitness Functions Selectorecombinative GA Population size: Convergence time: Number of function evaluations: Selectomutative GA Initial solution is evaluated once 2k –1 evaluations in each of m partitions

Slide 13: Crossover vs. Mutation: Deterministic Fitness Functions Speed-Up: Scalability ratio of mutation to that of crossover

Slide 14: Convergence Time for Crossover: Noisy Fitness Functions Additive Gaussian noise with variance σ2N Set proportional to maximum fitness variance Scaling dominated: Noise dominated:

Slide 15: Population Sizing for Crossover: Noisy Fitness Functions Scaling dominated: Noise dominated:

Slide 16: Scalability Analysis of Mutation: Noisy Fitness Functions Fitness should be sampled to average out noise What should the sample size, ns, be? BB-wise decision making [Goldberg, Deb, & Clark, 1992] Square of the ordinate of a one-sided Gaussian deviate with specified error probability, α

Slide 17: Scalability Analysis of Crossover & Mutation: Noisy Fitness Functions Selectorecombinative GA Selectomutative GA Fitness of each individual is sampled ns times 2k –1 evaluations in each of m partitions

Slide 18: Crossover vs. Mutation: Noisy BinInt Speed-Up: Scalability ratio of crossover to that of mutation

Slide 19: Summary Deterministic fitness: Noisy fitness: Recombination Mutation is more efficient is more efficient in noise dominated regime

Slide 20: Conclusions Good neighborhood information is essential Quadratic scalability of crossover and mutation Exponential scalability of simple crossover [Thierens & Goldberg, 1994] ekmk scalability of simple mutation [Mühlenbein, 1991] Leads to a theory of time continuation Key facet of efficiency enhancement Leads to principled design and development of adaptive time continuation operators Promise of yielding supermultiplicative speedups