Clever Algorithms: Nature-Inspired Programming Recipes

By Jason Brownlee PhD

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Evolutionary Programming

Evolutionary Programming, EP.

Taxonomy

Evolutionary Programming is a Global Optimization algorithm and is an instance of an Evolutionary Algorithm from the field of Evolutionary Computation. The approach is a sibling of other Evolutionary Algorithms such as the Genetic Algorithm, and Learning Classifier Systems. It is sometimes confused with Genetic Programming given the similarity in name, and more recently it shows a strong functional similarity to Evolution Strategies.

Inspiration

Evolutionary Programming is inspired by the theory of evolution by means of natural selection. Specifically, the technique is inspired by macro-level or the species-level process of evolution (phenotype, hereditary, variation) and is not concerned with the genetic mechanisms of evolution (genome, chromosomes, genes, alleles).

Metaphor

A population of a species reproduce, creating progeny with small phenotypical variation. The progeny and the parents compete based on their suitability to the environment, where the generally more fit members constitute the subsequent generation and are provided with the opportunity to reproduce themselves. This process repeats, improving the adaptive fit between the species and the environment.

Strategy

The objective of the Evolutionary Programming algorithm is to maximize the suitability of a collection of candidate solutions in the context of an objective function from the domain. This objective is pursued by using an adaptive model with surrogates for the processes of evolution, specifically hereditary (reproduction with variation) under competition. The representation used for candidate solutions is directly assessable by a cost or objective function from the domain.

Procedure

Algorithm (below) provides a pseudocode listing of the Evolutionary Programming algorithm for minimizing a cost function.

Input: $Population_{size}$, ProblemSize, BoutSize
Output: $S_{best}$
Population $\leftarrow$ InitializePopulation($Population_{size}$, ProblemSize)
EvaluatePopulation(Population)
$S_{best}$ $\leftarrow$ GetBestSolution(Population)
While ($\neg$StopCondition())
    Children $\leftarrow \emptyset$
    For ($Parent_{i}$ $\in$ Population)
        $Child_{i}$ $\leftarrow$ Mutate($Parent_{i}$)
        Children $\leftarrow$ $Child_{i}$
    End
    EvaluatePopulation(Children)
    $S_{best}$ $\leftarrow$ GetBestSolution(Children, $S_{best}$)
    Union $\leftarrow$ Population + Children
    For ($S_{i}$ $\in$ Union)
        For ($1$ To BoutSize)
            $S_{j}$ $\leftarrow$ RandomSelection(Union)
            If (Cost($S_{i}$) < Cost($S_{j}$))
                $Si_{wins}$ $\leftarrow$ $Si_{wins}$ + 1
            End
        End
    End
    Population $\leftarrow$ SelectBestByWins(Union, $Population_{size}$)
End
Return ($S_{best}$)
Pseudocode for Evolutionary Programming.

Heuristics

  • The representation for candidate solutions should be domain specific, such as real numbers for continuous function optimization.
  • The sample size (bout size) for tournament selection during competition is commonly between 5% and 10% of the population size.
  • Evolutionary Programming traditionally only uses the mutation operator to create new candidate solutions from existing candidate solutions. The crossover operator that is used in some other Evolutionary Algorithms is not employed in Evolutionary Programming.
  • Evolutionary Programming is concerned with the linkage between parent and child candidate solutions and is not concerned with surrogates for genetic mechanisms.
  • Continuous function optimization is a popular application for the approach, where real-valued representations are used with a Gaussian-based mutation operator.
  • The mutation-specific parameters used in the application of the algorithm to continuous function optimization can be adapted in concert with the candidate solutions [Fogel1991a].

Code Listing

Listing (below) provides an example of the Evolutionary Programming algorithm implemented in the Ruby Programming Language. The demonstration problem is an instance of a continuous function optimization that seeks $\min f(x)$ where $f=\sum_{i=1}^n x_{i}^2$, $-5.0\leq x_i \leq 5.0$ and $n=2$. The optimal solution for this basin function is $(v_0,\ldots,v_{n-1})=0.0$. The algorithm is an implementation of Evolutionary Programming based on the classical implementation for continuous function optimization by Fogel et al. [Fogel1991a] with per-variable adaptive variance based on Fogel's description for a self-adaptive variation on page 160 of his 1995 book [Fogel1995].

def objective_function(vector)
  return vector.inject(0.0) {|sum, x| sum +  (x ** 2.0)}
end

def random_vector(minmax)
  return Array.new(minmax.size) do |i|
    minmax[i][0] + ((minmax[i][1] - minmax[i][0]) * rand())
  end
end

def random_gaussian(mean=0.0, stdev=1.0)
  u1 = u2 = w = 0
  begin
    u1 = 2 * rand() - 1
    u2 = 2 * rand() - 1
    w = u1 * u1 + u2 * u2
  end while w >= 1
  w = Math.sqrt((-2.0 * Math.log(w)) / w)
  return mean + (u2 * w) * stdev
end

def mutate(candidate, search_space)
  child = {:vector=>[], :strategy=>[]}
  candidate[:vector].each_with_index do |v_old, i|
    s_old = candidate[:strategy][i]
    v = v_old + s_old * random_gaussian()
    v = search_space[i][0] if v < search_space[i][0]
    v = search_space[i][1] if v > search_space[i][1]
    child[:vector] << v
    child[:strategy] << s_old + random_gaussian() * s_old.abs**0.5
  end
  return child
end

def tournament(candidate, population, bout_size)
  candidate[:wins] = 0
  bout_size.times do |i|
    other = population[rand(population.size)]
    candidate[:wins] += 1 if candidate[:fitness] < other[:fitness]
  end
end

def init_population(minmax, pop_size)
  strategy = Array.new(minmax.size) do |i|
    [0,  (minmax[i][1]-minmax[i][0]) * 0.05]
  end
  pop = Array.new(pop_size, {})
  pop.each_index do |i|
    pop[i][:vector] = random_vector(minmax)
    pop[i][:strategy] = random_vector(strategy)
  end
  pop.each{|c| c[:fitness] = objective_function(c[:vector])}
  return pop
end

def search(max_gens, search_space, pop_size, bout_size)
  population = init_population(search_space, pop_size)
  population.each{|c| c[:fitness] = objective_function(c[:vector])}
  best = population.sort{|x,y| x[:fitness] <=> y[:fitness]}.first
  max_gens.times do |gen|
    children = Array.new(pop_size) {|i| mutate(population[i], search_space)}
    children.each{|c| c[:fitness] = objective_function(c[:vector])}
    children.sort!{|x,y| x[:fitness] <=> y[:fitness]}
    best = children.first if children.first[:fitness] < best[:fitness]
    union = children+population
    union.each{|c| tournament(c, union, bout_size)}
    union.sort!{|x,y| y[:wins] <=> x[:wins]}
    population = union.first(pop_size)
    puts " > gen #{gen}, fitness=#{best[:fitness]}"
  end
  return best
end

if __FILE__ == $0
  # problem configuration
  problem_size = 2
  search_space = Array.new(problem_size) {|i| [-5, +5]}
  # algorithm configuration
  max_gens = 200
  pop_size = 100
  bout_size = 5
  # execute the algorithm
  best = search(max_gens, search_space, pop_size, bout_size)
  puts "done! Solution: f=#{best[:fitness]}, s=#{best[:vector].inspect}"
end
Evolutionary Programming in Ruby

References

Primary Sources

Evolutionary Programming was developed by Lawrence Fogel, outlined in early papers (such as [Fogel1962]) and later became the focus of his PhD dissertation [Fogel1964]. Fogel focused on the use of an evolutionary process for the development of control systems using Finite State Machine (FSM) representations. Fogel's early work on Evolutionary Programming culminated in a book (co-authored with Owens and Walsh) that elaborated the approach, focusing on the evolution of state machines for the prediction of symbols in time series data [Fogel1966].

Learn More

The field of Evolutionary Programming lay relatively dormant for 30 years until it was revived by Fogel's son, David. Early works considered the application of Evolutionary Programming to control systems [Sebald1990], and later function optimization (system identification) culminating in a book on the approach [Fogel1991], and David Fogel's PhD dissertation [Fogel1992]. Lawrence Fogel collaborated in the revival of the technique, including reviews [Fogel1990] [Fogel1994] and extensions on what became the focus of the approach on function optimization [Fogel1991a].

Yao et al. provide a seminal study of Evolutionary Programming proposing an extension and racing it against the classical approach on a large number of test problems [Yao1999]. Finally, Porto provides an excellent contemporary overview of the field and the technique [Porto2000].

Bibliography

[Fogel1962] L. J. Fogel, "Autonomous automata", Industrial Research, 1962.
[Fogel1964] L. J. Fogel, "On the Organization of Intellect", [PhD Thesis] UCLA, 1964.
[Fogel1966] L. J. Fogel and A. J. Owens and M. J. Walsh, "Artificial Intelligence Through Simulated Evolution", Wiley, 1966.
[Fogel1990] L. J. Fogel, "The Future of Evolutionary Programming", in Proceedings of the Conference on Signals, Systems and Computers, 1990.
[Fogel1991] D. B. Fogel, "System Identification Through Simulated Evolution: A Machine Learning Approach to Modeling", Needham Heights, 1991.
[Fogel1991a] D. B. Fogel and L. J. Fogel and J. W. Atmar, "Meta-evolutionary programming", in Proceedings 25th Asilomar Conf. Signals, Systems, and Computers, 1991.
[Fogel1992] D. B. Fogel, "Evolving artificial intelligence", [PhD Thesis] University of California, San Diego, CA, USA, 1992.
[Fogel1994] L. J. Fogel, "Evolutionary Programming in Perspective: the Top-down View", in Computational Intelligence: Imitating Life, pages 135–146, IEEE Press, 1994.
[Fogel1995] D. B. Fogel, "Evolutionary computation: Toward a new philosophy of machine intelligence", IEEE Press, 1995.
[Porto2000] V. W. Porto, "10: Evolutionary Programming", in Evolutionary Computation 1: Basic Algorithms and Operations, pages 89–102, IoP Press, 2000.
[Sebald1990] A. V. Sebald and D. B. Fogel, "Design of SLAYR neural networks using evolutionary programming", in Proceedings of the 24th Asilomar Conference on Signals, Systems and Computers, 1990.
[Yao1999] X. Yao and Y. Liu and G. Lin, "Evolutionary programming made faster", IEEE Transactions on Evolutionary Computation, 1999.
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