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This paper analyzes the relative advantages between crossover and mutation on a class of deterministic and stochastic additively separable problems with substructures of non-uniform salience. This study assumes that the recombination and mutation operators have the knowledge of the building blocks (BBs) and effectively exchange or search among competing BBs. Facetwise models of convergence time and population sizing have been used to determine the scalability of each algorithm. The analysis shows that for deterministic exponentially-scaled additively separable, problems, the BB-wise mutation is more efficient than crossover yielding a speedup of Θ(l logl), where l is the problem size. For the noisy exponentially-scaled problems, the outcome depends on whether scaling on noise is dominant. When scaling dominates, mutation is more efficient than crossover yielding a speedup of Θ(l logl). On the other hand, when noise dominates, crossover is more efficient than mutation yielding a speedup of Θ(l).
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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 Laboratory2Materials Computation Center
University of Illinois at Urbana-Champaign, Urbana IL 61801http://www.illigal.uiuc.edu
[email protected], [email protected]
Supported by AFOSR FA9550-06-1-0096 and NSF DMR 03-25939.
MotivationGreat 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 literatureEmphasis 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]
Outline
Related work
Assumption of known or discovered linkage
Objective
Algorithm Description
Scalability analysis: Crossover vs. MutationKnown or discovered linkageExponentially scaled additively-separable problem with and without Gaussian noise
Summary and Conclusions
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 difficultyLinkage and neighborhood information is critical
Known or Discovered Linkage
Assumption of known or induced linkageCan use linkage-learning techniques
Linkage information is critical for selectorecombinative GA success
Provide the same information for mutationMutation searches in the building-block subspace
Pelikan, Ph.D. Thesis, 2002
Exponential Polynomial Scalability
Algorithm Description
Selectorecombinative genetic algorithmPopulation of size nBinary tournament selectionUniform building-block-wise crossover
Exchange BBs with probability 0.5
Selectomutative genetic algorithmStart with a random individualEnumerative BB-wise mutation
Consider BB partitions– Arbitrary left-to-right order
Choose the best schemata– Among the 2k possible ones
BBs #1 and #3 exchanged
Crossover Versus Mutation: Uniform Scaling
Deterministic fitness: Mutation is more efficient
Noisy fitness: Recombination is more efficient
[Sastry & Goldberg, 2004]
Objective
Crossover and mutation both have access to same neighborhood information
Known or discovered linkageRecombination exchanges building blocksMutation searches for the best BB in each partition
Compare scalability of crossover and mutationAdditively separable problems with exponentially-scaled BBs
With and without additive Gaussian noise
Where do they excel?
Derive, verify, and use facetwise modelsConvergence time and population sizing
Scaling and Noise Cover Most Problems
Adversarial problem design [Goldberg, 2002]
Noisy BinInt
P
Fluctuating
Deception NoiseScaling R
Convergence Time for Crossover: Deterministic Fitness Functions
Selection-Intensity based model [Rudnick, 1992; Thierens et al, 1998]
Derived for the BinInt problemApplicable to additively-separable problems
Selection Intensity
Problem size (m·k )
Population Sizing for Crossover:Deterministic Fitness Functions
Domino convergence [Rudnick, 1992]
BB convergence in order of salienceDrift bound dictates population sizing
Drift time [Goldberg and Segrest, 1987]
Size the population such that:
Population size:
...
time
Pro
porti
on Mostsalient
Leastsalient
Scalability Analysis of Crossover & Mutation: Deterministic Fitness Functions
Selectorecombinative GAPopulation size:
Convergence time:
Number of function evaluations:
Selectomutative GAInitial solution is evaluated once2k –1 evaluations in each of m partitions
Crossover vs. Mutation: Deterministic Fitness Functions
Speed-Up: Scalability ratio of mutation to that of crossover
Convergence Time for Crossover: Noisy Fitness Functions
Additive Gaussian noise with variance σ2N
Set proportional to maximum fitness variance
Scaling dominated:
Noise dominated:
Population Sizing for Crossover:Noisy Fitness Functions
Scaling dominated:
Noise dominated:
Scalability Analysis of Mutation: Noisy Fitness Functions
Fitness should be sampled to average out noiseWhat 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
Scalability Analysis of Crossover & Mutation: Noisy Fitness Functions
Selectorecombinative GA
Selectomutative GAFitness of each individual is sampled ns times2k –1 evaluations in each of m partitions
Crossover vs. Mutation: Noisy BinInt
Speed-Up: Scalability ratio of crossover to that of mutation
Summary
Deterministic fitness: Mutation is more efficient
Noisy fitness: Recombination is more efficient in noise dominated regime
Conclusions
Good neighborhood information is essentialQuadratic scalability of crossover and mutationExponential scalability of simple crossover [Thierens & Goldberg, 1994]
ekmk scalability of simple mutation [Mühlenbein, 1991]
Leads to a theory of time continuationKey facet of efficiency enhancement
Leads to principled design and development of adaptive time continuation operators
Promise of yielding supermultiplicative speedups
