A comparative analysis of selection schemes used in genetic algorithms David E. Goldberg Kalyanmoy Deb

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

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

What selection strategies? Proportionate reproduction Ranking selection Tournament selection Genitor (“steady state”) selection Genetic Algorithms

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

Proportionate Reproduction Probability of selection: prob(i,t) = f(i)/∑m(j,t) f(j) Various methods for implementation: Roulette wheel Stochastic remainder Stochastic universal Genetic Algorithms

How many in next generation? m(i,t+1) = m(i,t) * n * prob(i,t) m(i,t+1) = m(i,t) * f(i)/f(avg,t) Proportion(i,t+1) = Proportion(i,t) * f(i)/f(avg,t) Genetic Algorithms

Graph of Eqn, implementation Genetic Algorithms Convergence behavior

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

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

Takeover time for f(x) = x^c Solving for t and approximating Setting x = 1, we get proportion of best individual Setting P = n-1/n, we calculate when population contains n-1 best individuals Thus the takeover time for a polynomially distributed objective function is O(nlogn) Genetic Algorithms

Takeover time for f(x) = e^cx The takeover time for a polynomially and exponentially distributed objective function is O(nlogn) Genetic Algorithms

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 Genetic Algorithms

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) Genetic Algorithms

Tournament Selection Binary N-ary Randomly choose N individuals from population Select best for further processing Genetic Algorithms

Binary Tournament Tournament size = 3 Genetic Algorithms

Tournaments Genetic Algorithms

Genitor Choose an offspring based on ranking Replace worst individual in population Genetic Algorithms

Genitor Genetic Algorithms

Growth Comparison Genetic Algorithms

Takeover time comparison Genetic Algorithms

Time complexity Genetic Algorithms

Conclusions The paper provides a framework for comparing selection operators Compares selection “pressure” for each type of selection Introduces the concept of takeover time to help us understand the exploration/exploitation tradeoff Provides takeover time estimates for different types of selection Implications for genetic search The models provide us with theory necessary to compare selection methods and Balance growth ratio (quick convergence) with higher crossover/mutation (more exploration) Genetic Algorithms