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Chapter 1 - Write Your First Genetic Algorithm

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Scott Chu
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Chapter 1 - Write Your First Genetic Algorithm

Introduction

Optimization is the practice of making the best possible decisions in a situation. You can think of optimization as the search for the best.

Optimization algorithms are techniques for solve optimization problems. Problems where your goal is to find the best of something.

Understanding Genetic Algorithms

Genetic algorithms are a class of optimization algorithms based on evolution and natural section. they use strategies loosely based on genetics and biology to produce optimal (the best or near-optimal) solutions to complicated problems.

At the core, optimization problems are search problems. Search problems require you to navigate an area, like a maze, to find an objective, like the end of the maze. Optimization problems are basically the same thing, only there are multiple possible solutions.

Two basic approaches are used for search problems:

  • brute-force search
  • informed search

Informed search relies on a search strategy. You make smart decisions based on the available information.

In brute-force search, you iterate over every possible solution linearly. Brute-force searches use no knowledge of the search area to make decisions.

The key to informed search and thus optimization techniques, like genetic algorithms, lies in how they balance exploration versus exploitation.

Use Crossover to Exploit

In the scenario: You find yourself lost in the woods without a map or compass. how would you navigate out of the woods?

For brute-force strategy, walk in circles around every tree, hoping you make it back to civilization before you get too tired.

For informed strategy, to exploit or take advantage of the information available to you.

Example:

  • Perhaps you know that the nearest town is north.
  • Find out north based off on the position of the sun.

Crossover is how genetic algorithms exploit in search. Crossover is the process of creating new child solutions from parent solutions.

the iea is that the strongest solutions have characteristics that make them strong. These characteristics are call schemas. Schemas are building blocks of fit solutions.

Use Mutation to explore

To explore in search is to try new, random paths to see if they produce a better outcome. This concept of getting stuck in the same place in the search space is parallel to a common pitfall in optimization problems known as premature convergence. It’s easy for genetic algorithms to get stuck in one party of a search space because some solutions appear to be good enough - even though better solutions exist.

Mutation is how genetic algorithms explore. It is not enough to simply keep trying to build new solutions from previous ones, which is essentially the same as trying the same path over and over again. Mutation introduces randomness into the algorithms. The goal is to slightly alter some aspect of the previous solutions to create newer solutions, which may lead to newer, better paths.

The effectiveness of the genetic algorithms largely replies on how you balance exploitation versus exploration. Favoring one over the other has mertis. Often time you don’t know much about a search space, it’s best to favor exploration first and then slowly shift toward exploiting the information you already know.

Introducing the One-Max Problem

Question

What is the maximum sum of a bitstring (a string consisting of only 1s and 0s) of length N?

Answers

The maximum sum of a bitstring of length N is N.

Brute-force search

Needing to search through $2^N$ different solutions. What happens if you want to use this technique for bitstrings of length 40?

$2^{40} = 1,099,511,627,776$

| Genetic algorithms work via transformations on populations of | chromosomes over some number of generations.

Example:

Imagine you’re playing a car game where your goal is to get the highest possible card after some number of turns. You are initially given five cards and you can choose to keep any number of cards at the end of every turn.

  • A single card is a chromosome. It represents one solution to your problem.
  • Your entire hand is the population. It’s a collection of possible solutions.
  • The changes you make to your hand after every turn are transformations.
  • Finally, every turn represents one generation - one transformation of the population.

lifecycle

Initializing the Population

The number of chromosomes in your populations is irrelevant. A larger population means you’re currently looking at a larger area of the search space.

Typically, the more chromosomes you have, the longer it takes to perform transformations on the entire population.

Conversely, the fewer chromosomes you have, the longer it takes to produce a viable solution to the problem and the more susceptible your population is to premature convergence.

For this example problem, a population size of 100 strikes a nice balance between the two.

Understanding Population Size

In a traditional search method like depth-first search or breadth-frist search, you examine one solution at a time and determine where to go next. Even in other informed search algorithms like A* or uniform-cost search, you examine one solution at a time. Genetic algorithms examine many solutions at once, ruling out large areas of the search space after every generation. The population size dictates the size of the area of the search space you’re looking at.

Understanding the Flow of Genetic Algorithms

The process repeats itself. Genetic algorithms are recursive meaning the algorithm repeats itself over and over until it hits a termination point.

import IEx.Helpers
import ExUnit.Assertions
ExUnit.Assertions
select_best = fn population ->
  Enum.max_by(population, &Enum.sum/1)
end

assert select_best.([[1, 2, 3], [1, 2, 4]]) == [1, 2, 4]
true

Evaluating the Population

This function represents Step 2 in the genetic algorithm. This function takes population, evaluates each chromosome based on a fitness function and sorts the population based on each chromosome’s fitness.

Fitness is simply a heuristic that tells you how good or bad a solution is - a fitness function calculates this fitness for you.

In this problem, the fitness of a chromosome is represented by the sum of bitstring.

evaluate = fn population ->
  population
  |> Enum.sort_by(&Enum.sum/1, &>=/2)
end

assert evaluate.([
         [1, 2, 3],
         [2, 3, 4],
         [3, 1, 5],
         [1, 2, 0]
       ]) == [[2, 3, 4], [3, 1, 5], [1, 2, 3], [1, 2, 0]]
true

Selecting Parents

This step of selecting the parents for reproduction is referred to as selection. Selection is the process of picking the parents that will be combined to create new solutions.

The goal of selection is to pick some parents that can easily be combined to create better solutions.

For this problem, the result of the selection function to be formatted nicely for crossover. The selection function should return a list of tuples consisting of a pair of parents to be combined.

selection = fn population ->
  population
  |> Enum.chunk_every(2)
  |> Enum.map(&List.to_tuple(&1))
end

assert selection.([
         [2, 3, 4],
         [3, 1, 5],
         [1, 2, 3],
         [1, 2, 0]
       ]) == [{[2, 3, 4], [3, 1, 5]}, {[1, 2, 3], [1, 2, 0]}]
true

Crossover

Crossover is analogous to reproduction. It’s genetic operator that takes two or more parent chromosomes and produces two or more child chromosomes.

In this problem, every pair of parents are splitted at the random point selected. the chromosomes are then recombined with the tails swapped. This is known as single-point crossover and is one of the simplest crossover method used.

crossover = fn population, size ->
  population
  |> Enum.reduce([], fn {p1, p2}, acc ->
    cx_point = :rand.uniform(size)
    {h1, t1} = Enum.split(p1, cx_point)
    {h2, t2} = Enum.split(p2, cx_point)
    [h1 ++ t2, h2 ++ t1 | acc]
  end)
end

crossover.(
  [
    {[2, 3, 4], [3, 1, 5]},
    {[1, 2, 3], [1, 2, 0]}
  ],
  3
)
[[1, 2, 0], [1, 2, 3], [2, 3, 5], [3, 1, 4]]

Adding Mutation

Despite initializing your population to a seemingly random distribution, eventually the parents got too genetically similar to make any improvements during crossover.

This illustrates the importance of including the mutation step in your algorithm and how vital exploration is to informed search techniques.

mutate = fn
  chromosome, chance, threshold when chance < threshold ->
    Enum.shuffle(chromosome)

  chromosome, _, _ ->
    chromosome
end

mutation = fn population, threshold ->
  population
  |> Enum.map(&amp;mutate.(&amp;1, :rand.uniform(), threshold))
end
#Function<43.65746770/2 in :erl_eval.expr/5>
inform = fn vector ->
  value =
    vector
    |> Enum.sum()
    |> Integer.to_string()

  message = "Current Best: " <> value <> "\n"
  IO.write(message)
end

algorithm = fn population, algorithm ->
  best = select_best.(population)

  inform.(best)

  if Enum.sum(best) === 1000 do
    best
  else
    population
    |> evaluate.()
    |> selection.()
    |> crossover.(1000)
    |> mutation.(0.5)
    |> algorithm.(algorithm)
  end
end
#Function<43.65746770/2 in :erl_eval.expr/5>

Run the Algorithm

population = fn chromosome_size, population_size ->
  for _ <- 1..population_size, do: for(_ <- 1..chromosome_size, do: Enum.random(0..1))
end
#Function<43.65746770/2 in :erl_eval.expr/5>
solution = algorithm.(population.(1000, 100), algorithm)

IO.write("\nAnswer is \n")
Current Best: 534
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Current Best: 891
Current Best: 886
Current Best: 896
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Current Best: 892
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Current Best: 887
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Current Best: 887
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Current Best: 890
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Current Best: 897
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Current Best: 901
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Current Best: 890
Current Best: 890
Current Best: 892
Current Best: 890
Current Best: 891
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Current Best: 891
Current Best: 889
Current Best: 889
Current Best: 889
Current Best: 894
Current Best: 895
Current Best: 892
Current Best: 896
Current Best: 888
Current Best: 887
Current Best: 887
Current Best: 891
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Current Best: 890
Current Best: 895
Current Best: 896
Current Best: 902
Current Best: 891
Current Best: 888
Current Best: 889
Current Best: 893
Current Best: 892
Current Best: 890
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Current Best: 890
Current Best: 897
Current Best: 895
Current Best: 892
Current Best: 890
Current Best: 890
Current Best: 888
Current Best: 891
Current Best: 896
Current Best: 892
Current Best: 892
Current Best: 889
Current Best: 892
Current Best: 894
Current Best: 897
Current Best: 890
Current Best: 895
Current Best: 891
Current Best: 892
Current Best: 889
Current Best: 891
Current Best: 895
Current Best: 891
Current Best: 901
Current Best: 900
Current Best: 903
Current Best: 906
Current Best: 897
Current Best: 894
Current Best: 901
Current Best: 901
Current Best: 899
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Current Best: 899
Current Best: 891
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Current Best: 890
Current Best: 892
Current Best: 898
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Current Best: 898
Current Best: 896
Current Best: 900
Current Best: 899
Current Best: 895
Current Best: 903
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Current Best: 899
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Current Best: 906
Current Best: 901
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Current Best: 899
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Current Best: 899
Current Best: 901
Current Best: 897
Current Best: 905
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Current Best: 896
Current Best: 901
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Current Best: 897
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Current Best: 900
Current Best: 903
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Current Best: 898
Current Best: 897
Current Best: 898
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Current Best: 897
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Current Best: 906
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Current Best: 901
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Current Best: 900
Current Best: 900
Current Best: 903
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Current Best: 902
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Current Best: 902
Current Best: 900
Current Best: 899
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Current Best: 903
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Current Best: 901
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Current Best: 898
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Current Best: 899
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Current Best: 902
Current Best: 898
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Current Best: 911
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Current Best: 907
Current Best: 907
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Current Best: 916
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Current Best: 914
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Current Best: 916
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Current Best: 919
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Current Best: 917
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Current Best: 937
Current Best: 921
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Current Best: 918
Current Best: 921
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Current Best: 923
Current Best: 918
Current Best: 919
Current Best: 918
Current Best: 919
Current Best: 919
Current Best: 918
Current Best: 918
Current Best: 919
Current Best: 923
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Current Best: 924
Current Best: 930
Current Best: 928
Current Best: 925
Current Best: 921
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Current Best: 918
Current Best: 918
Current Best: 918
Current Best: 926
Current Best: 918
Current Best: 920
Current Best: 924
Current Best: 927
Current Best: 931
Current Best: 929
Current Best: 922
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Current Best: 926
Current Best: 935
Current Best: 929
Current Best: 936
Current Best: 930
Current Best: 938
Current Best: 931
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Current Best: 924
Current Best: 926
Current Best: 935
Current Best: 928
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Current Best: 929
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Current Best: 927
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Current Best: 926
Current Best: 928
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Current Best: 926
Current Best: 926
Current Best: 929
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Current Best: 930
Current Best: 926
Current Best: 935
Current Best: 927
Current Best: 927
Current Best: 929
Current Best: 925
Current Best: 927
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Current Best: 925
Current Best: 926
Current Best: 934
Current Best: 927
Current Best: 926
Current Best: 924
Current Best: 926
Current Best: 925
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Current Best: 924
Current Best: 929
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Current Best: 927
Current Best: 933
Current Best: 932
Current Best: 935
Current Best: 927
Current Best: 929
Current Best: 925
Current Best: 927
Current Best: 933
Current Best: 930
Current Best: 927
Current Best: 924
Current Best: 926
Current Best: 931
Current Best: 925
Current Best: 925
Current Best: 924
Current Best: 924
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Current Best: 928
Current Best: 926
Current Best: 925
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Current Best: 924
Current Best: 926
Current Best: 928
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Current Best: 924
Current Best: 927
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Current Best: 932
Current Best: 924
Current Best: 929
Current Best: 927
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Current Best: 927
Current Best: 928
Current Best: 926
Current Best: 934
Current Best: 938
Current Best: 927
Current Best: 928
Current Best: 929
Current Best: 933
Current Best: 927
Current Best: 926
Current Best: 924
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Current Best: 930
Current Best: 930
Current Best: 926
Current Best: 929
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Current Best: 927
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Current Best: 929
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Current Best: 933
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Current Best: 931
Current Best: 927
Current Best: 926
Current Best: 933
Current Best: 937
Current Best: 937
Current Best: 930
Current Best: 930
Current Best: 936
Current Best: 933
Current Best: 931
Current Best: 931
Current Best: 929
Current Best: 940
Current Best: 937
Current Best: 937
Current Best: 934
Current Best: 934
Current Best: 941
Current Best: 931
Current Best: 932
Current Best: 940
Current Best: 940
Current Best: 936
Current Best: 943
Current Best: 937
Current Best: 934
Current Best: 933
Current Best: 935
Current Best: 941
Current Best: 932
Current Best: 938
Current Best: 937
Current Best: 938
Current Best: 935
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Current Best: 938
Current Best: 936
Current Best: 933
Current Best: 938
Current Best: 932
Current Best: 935
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Current Best: 933
Current Best: 932
Current Best: 932
Current Best: 935
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Current Best: 935
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Current Best: 938
Current Best: 938
Current Best: 938
Current Best: 940
Current Best: 938
Current Best: 936
Current Best: 936
Current Best: 935
Current Best: 945
Current Best: 939
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Current Best: 947
Current Best: 940
Current Best: 934
Current Best: 936
Current Best: 943
Current Best: 939
Current Best: 947
Current Best: 942
Current Best: 941
Current Best: 940
Current Best: 938
Current Best: 941
Current Best: 936
Current Best: 934
Current Best: 934
Current Best: 941
Current Best: 947
Current Best: 943
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Current Best: 938
Current Best: 942
Current Best: 942
Current Best: 941
Current Best: 939
Current Best: 944
Current Best: 945
Current Best: 940
Current Best: 947
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Current Best: 946
Current Best: 947
Current Best: 948
Current Best: 947
Current Best: 944
Current Best: 936
Current Best: 938
Current Best: 942
Current Best: 944
Current Best: 938
Current Best: 946
Current Best: 943
Current Best: 942
Current Best: 939
Current Best: 939
Current Best: 936
Current Best: 942
Current Best: 940
Current Best: 942
Current Best: 949
Current Best: 937
Current Best: 947
Current Best: 939
Current Best: 949
Current Best: 942
Current Best: 941
Current Best: 949
Current Best: 950
Current Best: 950
Current Best: 946
Current Best: 951
Current Best: 946
Current Best: 954
Current Best: 945
Current Best: 948
Current Best: 946
Current Best: 945
Current Best: 945
Current Best: 947
Current Best: 945
Current Best: 946
Current Best: 945
Current Best: 946
Current Best: 947
Current Best: 948
Current Best: 949
Current Best: 956
Current Best: 951
Current Best: 957
Current Best: 945
Current Best: 957
Current Best: 952
Current Best: 947
Current Best: 953
Current Best: 948
Current Best: 953
Current Best: 954
Current Best: 946
Current Best: 945
Current Best: 948
Current Best: 949
Current Best: 950
Current Best: 948
Current Best: 946
Current Best: 954
Current Best: 952
Current Best: 952
Current Best: 950
Current Best: 949
Current Best: 946
Current Best: 947
Current Best: 947
Current Best: 948
Current Best: 946
Current Best: 945
Current Best: 950
Current Best: 945
Current Best: 952
Current Best: 948
Current Best: 947
Current Best: 945
Current Best: 950
Current Best: 947
Current Best: 951
Current Best: 954
Current Best: 951
Current Best: 950
Current Best: 954
Current Best: 950
Current Best: 951
Current Best: 951
Current Best: 954
Current Best: 960
Current Best: 958
Current Best: 957
Current Best: 959
Current Best: 949
Current Best: 952
Current Best: 964
Current Best: 963
Current Best: 962
Current Best: 953
Current Best: 957
Current Best: 950
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Current Best: 950
Current Best: 950
Current Best: 950
Current Best: 954
Current Best: 955
Current Best: 959
Current Best: 955
Current Best: 959
Current Best: 953
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Current Best: 953
Current Best: 958
Current Best: 955
Current Best: 953
Current Best: 953
Current Best: 953
Current Best: 949
Current Best: 950
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Current Best: 951
Current Best: 951
Current Best: 951
Current Best: 949
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Current Best: 950
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Current Best: 954
Current Best: 954
Current Best: 954
Current Best: 951
Current Best: 951
Current Best: 950
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Current Best: 953
Current Best: 952
Current Best: 950
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Current Best: 951
Current Best: 953
Current Best: 954
Current Best: 950
Current Best: 950
Current Best: 953
Current Best: 949
Current Best: 950
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Current Best: 950
Current Best: 951
Current Best: 950
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Current Best: 950
Current Best: 950
Current Best: 951
Current Best: 956
Current Best: 949
Current Best: 949
Current Best: 952
Current Best: 954
Current Best: 954
Current Best: 951
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Current Best: 950
Current Best: 950
Current Best: 954
Current Best: 951
Current Best: 953
Current Best: 950
Current Best: 950
Current Best: 951
Current Best: 955
Current Best: 953
Current Best: 953
Current Best: 953
Current Best: 957
Current Best: 953
Current Best: 957
Current Best: 956
Current Best: 951
Current Best: 955
Current Best: 964
Current Best: 957
Current Best: 953
Current Best: 953
Current Best: 953
Current Best: 954
Current Best: 953
Current Best: 954
Current Best: 959
Current Best: 957
Current Best: 961
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Current Best: 958
Current Best: 958
Current Best: 958
Current Best: 957
Current Best: 959
Current Best: 954
Current Best: 955
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Current Best: 956
Current Best: 953
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Current Best: 954
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Current Best: 957
Current Best: 955
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Current Best: 953
Current Best: 957
Current Best: 957
Current Best: 956
Current Best: 954
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Current Best: 955
Current Best: 957
Current Best: 954
Current Best: 959
Current Best: 955
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Current Best: 956
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Current Best: 956
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Current Best: 955
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Current Best: 956
Current Best: 958
Current Best: 960
Current Best: 956
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Current Best: 963
Current Best: 962
Current Best: 955
Current Best: 957
Current Best: 959
Current Best: 956
Current Best: 954
Current Best: 958
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Current Best: 956
Current Best: 955
Current Best: 962
Current Best: 954
Current Best: 959
Current Best: 958
Current Best: 957
Current Best: 956
Current Best: 958
Current Best: 955
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Current Best: 958
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Current Best: 955
Current Best: 957
Current Best: 960
Current Best: 956
Current Best: 954
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Current Best: 956
Current Best: 960
Current Best: 961
Current Best: 954
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Current Best: 956
Current Best: 961
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Current Best: 959
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Current Best: 963
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Current Best: 963
Current Best: 964
Current Best: 959
Current Best: 962
Current Best: 958
Current Best: 957
Current Best: 963
Current Best: 962
Current Best: 958
Current Best: 959
Current Best: 962
Current Best: 958
Current Best: 960
Current Best: 957
Current Best: 960
Current Best: 960
Current Best: 961
Current Best: 962
Current Best: 964
Current Best: 964
Current Best: 959
Current Best: 963
Current Best: 957
Current Best: 959
Current Best: 959
Current Best: 959
Current Best: 959
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Current Best: 961
Current Best: 958
Current Best: 959
Current Best: 963
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Current Best: 960
Current Best: 957
Current Best: 960
Current Best: 963
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Current Best: 959
Current Best: 958
Current Best: 961
Current Best: 962
Current Best: 959
Current Best: 961
Current Best: 962
Current Best: 961
Current Best: 963
Current Best: 960
Current Best: 960
Current Best: 958
Current Best: 961
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Current Best: 959
Current Best: 959
Current Best: 963
Current Best: 961
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Current Best: 959
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Current Best: 960
Current Best: 959
Current Best: 961
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Current Best: 962
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Current Best: 964
Current Best: 959
Current Best: 960
Current Best: 958
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Current Best: 962
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Current Best: 961
Current Best: 964
Current Best: 962
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Current Best: 962
Current Best: 963
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Current Best: 964
Current Best: 963
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Current Best: 964
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Current Best: 964
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Current Best: 961
Current Best: 964
Current Best: 963
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Current Best: 966
Current Best: 968
Current Best: 963
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Current Best: 963
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Current Best: 963
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Current Best: 964
Current Best: 965
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Current Best: 964
Current Best: 962
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Current Best: 963
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Current Best: 961
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Current Best: 963
Current Best: 964
Current Best: 962
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Current Best: 963
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Current Best: 966
Current Best: 962
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Current Best: 964
Current Best: 963
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Current Best: 964
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Current Best: 964
Current Best: 961
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Current Best: 965
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Current Best: 964
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Current Best: 964
Current Best: 968
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Current Best: 962
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Current Best: 964
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Current Best: 964
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Current Best: 968
Current Best: 970
Current Best: 965
Current Best: 968
Current Best: 966
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Current Best: 972
Current Best: 968
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Current Best: 998
Current Best: 997
Current Best: 996
Current Best: 995
Current Best: 996
Current Best: 997
Current Best: 995
Current Best: 996
Current Best: 997
Current Best: 997
Current Best: 996
Current Best: 997
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 997
Current Best: 998
Current Best: 998
Current Best: 996
Current Best: 995
Current Best: 996
Current Best: 995
Current Best: 997
Current Best: 995
Current Best: 995
Current Best: 995
Current Best: 996
Current Best: 997
Current Best: 996
Current Best: 995
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 995
Current Best: 995
Current Best: 996
Current Best: 997
Current Best: 996
Current Best: 997
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 997
Current Best: 999
Current Best: 997
Current Best: 995
Current Best: 997
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 995
Current Best: 997
Current Best: 995
Current Best: 996
Current Best: 997
Current Best: 997
Current Best: 997
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 995
Current Best: 995
Current Best: 995
Current Best: 996
Current Best: 997
Current Best: 997
Current Best: 997
Current Best: 997
Current Best: 997
Current Best: 999
Current Best: 995
Current Best: 996
Current Best: 995
Current Best: 995
Current Best: 999
Current Best: 997
Current Best: 997
Current Best: 999
Current Best: 999
Current Best: 996
Current Best: 998
Current Best: 998
Current Best: 996
Current Best: 996
Current Best: 995
Current Best: 996
Current Best: 996
Current Best: 998
Current Best: 997
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 997
Current Best: 997
Current Best: 998
Current Best: 996
Current Best: 995
Current Best: 995
Current Best: 995
Current Best: 997
Current Best: 999
Current Best: 996
Current Best: 998
Current Best: 997
Current Best: 995
Current Best: 996
Current Best: 996
Current Best: 995
Current Best: 995
Current Best: 995
Current Best: 995
Current Best: 998
Current Best: 997
Current Best: 997
Current Best: 997
Current Best: 998
Current Best: 998
Current Best: 997
Current Best: 995
Current Best: 996
Current Best: 996
Current Best: 997
Current Best: 997
Current Best: 996
Current Best: 997
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 996
Current Best: 997
Current Best: 996
Current Best: 998
Current Best: 997
Current Best: 1000

Answer is 
:ok

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