Genetic Algorithms 10/10/2004 1Computer Science

A comparative analysis of selection schemes

used in genetic algorithms

David E. GoldbergKalyanmoy Deb

Genetic Algorithms 10/10/2004 2Computer Science

What is the paper about?

• Defines and compare four selection schemes

• Presents a technique for comparisons:– Produce a difference/differential equation

modeling the selection scheme– Test computer implementation against diff.

equation model

• Defines criteria for comparison:– Convergence time– Schema growth ratios

• Conclusions: practical applications of analysis

Genetic Algorithms 10/10/2004 3Computer Science

Where are we now?

• Many papers claim the superiority of this or that selection scheme

• But many of these claims are based on limited (and uncontrolled experiments).

• Little analysis has been done

• This paper attempts to provide the needed analysis

Genetic Algorithms 10/10/2004 4Computer Science

What selection strategies?

• Proportionate reproduction• Ranking selection• Tournament selection• Genitor (“steady state”) selection

Genetic Algorithms 10/10/2004 5Computer Science

Birth, life, and death

• m(i, t+1) = m(i, t) + m(i, t, b) – m(i,t,d)

• Ex: in non-overlapping population models:– m(i,t+1) = m(i,t,b) ; m(i,t,d) =

m(i,t)

• We can also do proportions:– P(i,t+1) = P(i,t) + P(i,t,b) – P(i,t,d)

Genetic Algorithms 10/10/2004 6Computer Science

Proportionate Reproduction

• Probability of selection:– p(i,t) = f(i)/∑m(j,t) f(j)

• Various methods for implementation:– Roulette wheel– Stochastic remainder– Stochastic universal

Genetic Algorithms 10/10/2004 7Computer Science

How many in next generation?

• m(i,t+1) = m(i,t) * n * p(i,t)• m(i,t+1) = m(i,t) * f(i)/f(avg,t)

• P(i,t+1) = P(i,t) * f(i)/f(avg,t)

Genetic Algorithms 10/10/2004 8Computer Science

Solution to diff eq

• m(i,t+1) = m(i,t) * f(i) divide by population size

– P(i,t+1) = f(i) m(i,t)/∑m(j,t+1)– P(i,t+1) = f(i) m(i,t)/∑f(j)m(j,t)– P(i,t+1) = f(i) P(i,t)/∑f(j)P(j,t)

• Note: m(i,t) = m(i,0) f(i)^t

– P(i,t) = f(i)t P(i,0)/∑f(j)t P(j,0)

Genetic Algorithms 10/10/2004 9Computer Science

Graph of Eqn, implementation

Convergence behavior

Genetic Algorithms 10/10/2004 10Computer Science

Takeover time

• How many individuals between specified values of x in objective function f(x)?

• Let p0(x) be uniform, integral 1• Consider f(x) = xc and f(x) = ecx

• Limits x and x – 1/n

Genetic Algorithms 10/10/2004 11Computer Science

Behavior of f(x) = x^c

Integrate with limits x & x – 1/n

x = 1 is best, x = 0 is worst individual

Compare theory and experiment for f(x) = x

Genetic Algorithms 10/10/2004 12Computer Science

Takeover time for f(x) = x^c

Setting x = 1, we get proportion of best individual

Setting P = n-1/n, we calculate when population contains n-1 best individuals

Solving for t and approximating

Thus the takeover time for a polynomially distributed objective function is O(nlogn)

Genetic Algorithms 10/10/2004 13Computer Science

Takeover time for f(x) = e^cx

The takeover time for a polynomially and exponentially distributed objective function is O(nlogn)

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Time complexity of Proportionate Reproduction

• Roulette Wheel – O(n2) or O(nlogn) with binary search

• Stochastic remainder selection– floor(f(i)/favg) number of copies– Remainder = flip(fractional(f(i)/favg))– O(n) without replacement or O(n2)

with

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Ranking

• Sort from best to worst• Create a transformation function

called an assignment function that converts a rank to an equivalent “fitness” – assignFunction(rank)

• Proportionate reproduction on assignFunction(rank)

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Tournament Selection

• Binary• N-ary

• Randomly choose N individuals from population

• Select best for further processing

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Binary Tournament

• Tournament size = 3

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Tournaments

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Genitor

• Choose an offspring based on ranking

• Replace worst individual in population

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Genitor

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Growth Comparison

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Takeover time comparison

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Time complexity

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Conclusions

• The paper provides a framework for comparing selection operators

• Implications for genetic search– The models provide us with theory

necessary to compare selection methods and

– Balance growth ration (quick convergence) with higher crossover/mutation (more exploration)