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Understanding Diversity and Recombination in Simple Evolutionary Algorithms

Dirk Sudholt

University of Sheffield

This project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 618091 (SAGE). Dirk Sudholt Understanding Diversity and Recombination in Simple Evolutionary Algorithms 1 / 8

Evolutionary Operators

Consider bit strings of n bits. Standard bit mutation Flip each bit independently with a fixed probability 1/n.

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Evolutionary Operators

Consider bit strings of n bits. Standard bit mutation Flip each bit independently with a fixed probability 1/n. Uniform crossover 1 1 1 1 1 1 1 1 1 1

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Where crossover provably helps

Constructed example functions Jumpk [Jansen and Wegener, 2002] [K¨

• tzing, Sudholt, and Theile, 2011]

Real Royal Road functions [Jansen & Wegener, 2005] [Storch & Wegener, 2004] H-IFF [Dietzfelbinger, Naudts, van Hoyweghen, and Wegener, 2003] building-block Real Royal Roads [Watson and Jansen, 2007]

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Where crossover provably helps

Constructed example functions Jumpk [Jansen and Wegener, 2002] [K¨

• tzing, Sudholt, and Theile, 2011]

Real Royal Road functions [Jansen & Wegener, 2005] [Storch & Wegener, 2004] H-IFF [Dietzfelbinger, Naudts, van Hoyweghen, and Wegener, 2003] building-block Real Royal Roads [Watson and Jansen, 2007] Ising model / vertex colouring ring graphs [Fischer and Wegener, 2005] binary trees [Sudholt, 2005]

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Where crossover provably helps

Constructed example functions Jumpk [Jansen and Wegener, 2002] [K¨

• tzing, Sudholt, and Theile, 2011]

Real Royal Road functions [Jansen & Wegener, 2005] [Storch & Wegener, 2004] H-IFF [Dietzfelbinger, Naudts, van Hoyweghen, and Wegener, 2003] building-block Real Royal Roads [Watson and Jansen, 2007] Ising model / vertex colouring ring graphs [Fischer and Wegener, 2005] binary trees [Sudholt, 2005] All-pairs shortest paths evolutionary algorithms [Doerr, Happ, and Klein, 2008], [Doerr and Theile, 2009],

[Doerr, Johannsen, K¨

• tzing, Neumann, and Theile, 2010]

ant colony optimization [Sudholt and Thyssen, 2011] multi-objective shortest paths [Neumann and Theile, 2010]

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Scheme of a (µ+1) Genetic Algorithm

Initialize population P of size µ ∈ N uniformly at random. while not stopping do Select x1, x2 ∈ P uniformly at random. Let y := uniform crossover(x1, x2). Flip each bit in y independently with probability 1/n. Let P contain the µ best individuals from P ∪ {y}. end

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Scheme of a (µ+1) Genetic Algorithm

Initialize population P of size µ ∈ N uniformly at random. while not stopping do Select x1, x2 ∈ P uniformly at random. Let y := uniform crossover(x1, x2). Flip each bit in y independently with probability 1/n. Let P contain the µ best individuals from P ∪ {y}. end Let’s consider this GA on OneMax(x) := n

i=1 xi.

Aka additive model, Mount Fuji, Hamming distance problem, Onesmax, . . .

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Experiments: Benefit of Uniform Crossover

EA (no crossover) versus GA (uniform crossover)

500 1,000 1,500 2,000 0.5 1 1.5 2 2.5 ·105 n (100+1) EA (100+1) GA

OneMax

500 1,000 1,500 2,000 1 2 3 ·105 n (100+1) EA (100+1) GA

random weights from [0, 1] First impressions: performance gap seems to grow with µ and n.

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What we know

Expected time of the (1+1) Evolutionary Algorithm is en ln(n) − O(n).

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What we know

Expected time of the (1+1) Evolutionary Algorithm is en ln(n) − O(n). Mutation only is inefficient for large populations No crossover: lower runtime bound of Ω(µn + n log n) [Witt, 2006].

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What we know

Expected time of the (1+1) Evolutionary Algorithm is en ln(n) − O(n). Mutation only is inefficient for large populations No crossover: lower runtime bound of Ω(µn + n log n) [Witt, 2006]. Constant-factor speedup Small populations (µ = o((log n)/ log log n))) make GAs twice as fast as the best mutation-only algorithm [Sudholt, 2016].

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What we know

Expected time of the (1+1) Evolutionary Algorithm is en ln(n) − O(n). Mutation only is inefficient for large populations No crossover: lower runtime bound of Ω(µn + n log n) [Witt, 2006]. Constant-factor speedup Small populations (µ = o((log n)/ log log n))) make GAs twice as fast as the best mutation-only algorithm [Sudholt, 2016]. Superconstant speedup (Carola’s talk) Clever non-standard GAs beat the Θ(n log n) bound [Doerr, Doerr, Ebel, 2015].

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What we know

Expected time of the (1+1) Evolutionary Algorithm is en ln(n) − O(n). Mutation only is inefficient for large populations No crossover: lower runtime bound of Ω(µn + n log n) [Witt, 2006]. Constant-factor speedup Small populations (µ = o((log n)/ log log n))) make GAs twice as fast as the best mutation-only algorithm [Sudholt, 2016]. Superconstant speedup (Carola’s talk) Clever non-standard GAs beat the Θ(n log n) bound [Doerr, Doerr, Ebel, 2015]. Current results do not take advantage of initial population diversity.

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What we know

Expected time of the (1+1) Evolutionary Algorithm is en ln(n) − O(n). Mutation only is inefficient for large populations No crossover: lower runtime bound of Ω(µn + n log n) [Witt, 2006]. Constant-factor speedup Small populations (µ = o((log n)/ log log n))) make GAs twice as fast as the best mutation-only algorithm [Sudholt, 2016]. Superconstant speedup (Carola’s talk) Clever non-standard GAs beat the Θ(n log n) bound [Doerr, Doerr, Ebel, 2015]. Current results do not take advantage of initial population diversity. (Exception: paired-crossover EA [Pr¨

ugel-Bennett, Shapiro, Rowe, 2015])

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Approaches

Frequent argument: “If there is no diversity, there is a sequence of mutations paving the way for crossover.”

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Approaches

Frequent argument: “If there is no diversity, there is a sequence of mutations paving the way for crossover.” Our naive perspective (so far): mutation creates diversity.

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Approaches

Frequent argument: “If there is no diversity, there is a sequence of mutations paving the way for crossover.” Our naive perspective (so far): mutation creates diversity. Population Genetics perspective: recombination creates diversity.

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Approaches

Frequent argument: “If there is no diversity, there is a sequence of mutations paving the way for crossover.” Our naive perspective (so far): mutation creates diversity. Population Genetics perspective: recombination creates diversity. Can we get good upper runtime bounds for OneMax by exploiting diversity in the initial population?

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Open questions

What is the performance of the (µ+1) GA on OneMax? Aiming at rigorous results, but other approaches might help get us there. Goal: prove that the expected time for finding the optimum is at most T(n, µ), hopefully smaller than Θ(µn + n log n). When and why is a GA faster than its counterpart without crossover? What is the best population size (possibly depending on n)? Feel free to turn off mutation and only consider selection and

• recombination. Analyse time and success probability.

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