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Genetic algorithms

- Gentle introduction
- Jim Cohoon and Kimberly Hanks

Genetic Algorithms in a slide

- Premise
- Evolution worked once (it produced us!), it might

work again - Basics
- Pool of solutions
- Mate existing solutions to produce new solutions
- Mutate current solutions for long-term diversity
- Cull population

Originator

- John Holland
- Seminal work
- Adaptation in Natural and Artificial Systems

introduced main GA concepts, 1975

Introduction

- Computing pioneers (especially in AI) looked to

natural systems as guiding metaphors - Evolutionary computation
- Any biologically-motivated computing activity

simulating natural evolution - Genetic Algorithms are one form of this activity
- Original goals
- Formal study of the phenomenon of adaptation
- John Holland
- An optimization tool for engineering problems

Main idea

- Take a population of candidate solutions to a

given problem - Use operators inspired by the mechanisms of

natural genetic variation - Apply selective pressure toward certain

properties - Evolve a more fit solution

Why evolution as a metaphor

- Ability to efficiently guide a search through a

large solution space - Ability to adapt solutions to changing

environments - Emergent behavior is the goal
- The hoped-for emergent behavior is the design of

high-quality solutions to difficult problems and

the ability to adapt these solutions in the face

of a changing environment - Melanie Mitchell, An Introduction to Genetic

Algorithms

Evolutionary terminology

- Abstractions imported from biology
- Chromosomes, Genes, Alleles
- Fitness, Selection
- Crossover, Mutation

GA terminology

- In the spirit but not the letter of biology
- GA chromosomes are strings of genes
- Each gene has a number of alleles i.e., settings
- Each chromosome is an encoding of a solution to a

problem - A population of such chromosomes is operated on

by a GA

Encoding

- A data structure for representing candidate

solutions - Often takes the form of a bit string
- Usually has internal structure i.e., different

parts of the string represent different aspects

of the solution)

Crossover

- Mimics biological recombination
- Ssome portion of genetic material is swapped

between chromosomes - Typically the swapping produces an offspring
- Mechanism for the dissemination of building

blocks (schemas)

Mutation

- Selects a random locus gene location with

some probability and alters the allele at that

locus - The intuitive mechanism for the preservation of

variety in the population

Fitness

- A measure of the goodness of the organism
- Expressed as the probability that the organism

will live another cycle (generation) - Basis for the natural selection simulation
- Organisms are selected to mate with probabilities

proportional to their fitness - Probabilistically better solutions have a better

chance of conferring their building blocks to the

next generation (cycle)

A Simple GA

- Generate initial population
- do
- Calculate the fitness of each member
- // simulate another generation
- do
- Select parents from current population
- Perform crossover add offspring to the new

population - while new population is not full
- Merge new population into the current

population - Mutate current population
- while not converged

How do GAs work

- The structure of a GA is relatively simple to

comprehend, but the dynamic behavior is complex - Holland has done significant work on the

theoretical foundations of Gas - GAs work by discovering, emphasizing, and

recombining good building blocks of solutions

in a highly parallel fashion. - Melanie Mitchell, paraphrasing John Holland
- Using formalism
- Notion of a building block is formalized as a

schema - Schemas are propagated or destroyed according to

the laws of probability

Schema

- A template, much like a regular expression,

describing a set of strings - The set of strings represented by a given schema

characterizes a set of candidate solutions

sharing a property - This property is the encoded equivalent of a

building block

Example

- 0 or 1 represents a fixed bit
- Asterisk represents a dont care
- 1100 is the set of all solutions encoded in 8

bits, beginning with two ones and ending with two

zeros - Solutions in this set all share the same variants

of the properties encoded at these loci

Schema qualifiers

- Length
- The inclusive distance between the two bits in a

schema which are furthest apart (the defining

length of the previous example is 8) - Order
- The number of fixed bits in a schema (the order

of the previous example is 4)

Not just sum of the parts

- GAs explicitly evaluate and operate on whole

solutions - GAs implicitly evaluate and operate on building

blocks - Existing schemas may be destroyed or weakened by

crossover - New schemas may be spliced together from existing

schema - Crossover includes no notion of a schema only

of the chromosomes

Why do they work

- Schemas can be destroyed or conserved
- So how are good schemas propagated through

generations? - Conserved good schemas confer higher fitness

on the offspring inheriting them - Fitter offspring are probabilistically more

likely to be chosen to reproduce

Approximating schema dynamics

- Let H be a schema with at least one instance

present in the population at time t - Let m(H, t) be the number of instances of H at

time t - Let x be an instance of H and f(x) be its fitness
- The expected number of offspring of x is

f(x)/f(pop) (by fitness proportionate selection) - To know E(m(H, t 1)) (the expected number of

instances of schema H at the next time unit), sum

f(x)/f(pop) for all x in H - GA never explicitly calculates the average

fitness of a schema, but schema proliferation

depends on its value

Approximating schema dynamics

- Approximation can be refined by taking into

account the operators - Schemas of long defining length are less likely

to survive crossover - Offspring are less likely to be instances of such

schemas - Schemas of higher order are less likely to

survive mutation - Effects can be used to bound the approximate

rates at which schemas proliferate

Implications

- Instances of short, low-order schemas whose

average fitness tends to stay above the mean will

increase exponentially - Changing the semantics of the operators can

change the selective pressures toward different

types of schemas

Theoretical Foundations

- Empirical observation
- GAs can work
- Goal
- Learn how to best use the tool
- Strategy
- Understand the dynamics of the model
- Develop performance metrics in order to quantify

success

Theoretical Foundations

- Issues surrounding the dynamics of the model
- What laws characterize the macroscopic behavior

of GAs? - How do microscopic events give rise to this

macroscopic behavior?

Theoretical Foundation

- Hollands motivation
- Construct a theoretical framework for adaptive

systems as seen in nature - Apply this framework to the design of artificial

adaptive systems - Issues in performance evaluation
- According to what criteria should GAs be

evaluated? - What does it mean for a GA to do well or poorly?
- Under what conditions is a GA an appropriate

solution strategy for a problem?

Theoretical Foundation

- Hollands observations
- An adaptive system must persistently identify,

test, and incorporate structural properties

hypothesized to give better performance in some

environment - Adaptation is impossible in a sufficiently random

environment

Theoretical Foundation

- Hollands intuition
- A GA is capable of modeling the necessary tasks

in an adaptive system - It does so through a combination of explicit

computation and implicit estimation of state

combined with incremental change of state in

directions motivated by these calculations

Theoretical Foundation

- Hollands assertion
- The identify and test requirement is satisfied

by the calculation of the fitnesses of various

schemas - The incorporate requirement is satisfied by

implication of the Schema Theorem

Theoretical Foundation

- How does a GA identify and test properties?
- A schema is the formalization of a property
- A GA explicitly calculates fitnesses of

individuals and thereby schemas in the population - It implicitly estimates fitnesses of hypothetical

individuals sharing known schemas - In this way it efficiently manages information

regarding the entire search space

Theoretical Foundation

- How does a GA incorporate observed good

properties into the population? - Implication of the Schema Theorem
- Short, low-order, higher than average fitness

schemas will receive exponentially increasing

numbers of samples over time

Theoretical Foundation

- Lemmas to the Schema Theorem
- Selection focuses the search
- Crossover combines good schemas
- Mutation is the insurance policy

Theoretical Foundation

- Hollands characterization
- Adaptation in natural systems is framed by a

tension between exploration and exploitation - Any move toward the testing of previously unseen

schemas or of those with instances of low fitness

takes away from the wholesale incorporation of

known high fitness schemas - But without exploration, schemas of even higher

fitness can not be discovered

Theoretical Foundation

- Goal of Hollands first offering
- The original GA was proposed as an adaptive

plan for accomplishing a proper balance between

exploration and exploitation

Theoretical Foundation

- GA does in fact model this
- Given certain assumptions, the balance is

achieved - A key assumption is that the observed and actual

fitnesses of schemas are correlated - This assumption creates a stumbling block to

which we will return

Traveling Salesperson Problem

Initial Population for TSP

(5,3,4,6,2)

(2,4,6,3,5)

(4,3,6,5,2)

(2,3,4,6,5)

(4,3,6,2,5)

(3,4,5,2,6)

(3,5,4,6,2)

(4,5,3,6,2)

(5,4,2,3,6)

(4,6,3,2,5)

(3,4,2,6,5)

(3,6,5,1,4)

Select Parents

(5,3,4,6,2)

(2,4,6,3,5)

(4,3,6,5,2)

(2,3,4,6,5)

(4,3,6,2,5)

(3,4,5,2,6)

(3,5,4,6,2)

(4,5,3,6,2)

(5,4,2,3,6)

(4,6,3,2,5)

(3,4,2,6,5)

(3,6,5,1,4)

Try to pick the better ones.

Create Off-Spring 1 point

(5,3,4,6,2)

(2,4,6,3,5)

(4,3,6,5,2)

(2,3,4,6,5)

(4,3,6,2,5)

(3,4,5,2,6)

(3,5,4,6,2)

(4,5,3,6,2)

(5,4,2,3,6)

(4,6,3,2,5)

(3,4,2,6,5)

(3,6,5,1,4)

(3,4,5,6,2)

Create More Offspring

(5,3,4,6,2)

(2,4,6,3,5)

(4,3,6,5,2)

(2,3,4,6,5)

(4,3,6,2,5)

(3,4,5,2,6)

(3,5,4,6,2)

(4,5,3,6,2)

(5,4,2,3,6)

(4,6,3,2,5)

(3,4,2,6,5)

(3,6,5,1,4)

(3,4,5,6,2)

(5,4,2,6,3)

Mutate

(5,3,4,6,2)

(2,4,6,3,5)

(4,3,6,5,2)

(2,3,4,6,5)

(4,3,6,2,5)

(3,4,5,2,6)

(3,5,4,6,2)

(4,5,3,6,2)

(5,4,2,3,6)

(4,6,3,2,5)

(3,4,2,6,5)

(3,6,5,1,4)

(3,4,5,6,2)

(5,4,2,6,3)

Mutate

(5,3,4,6,2)

(2,4,6,3,5)

(4,3,6,5,2)

(2,3,4,6,5)

(2,3,6,4,5)

(3,4,5,2,6)

(3,5,4,6,2)

(4,5,3,6,2)

(5,4,2,3,6)

(4,6,3,2,5)

(3,4,2,6,5)

(3,6,5,1,4)

(3,4,5,6,2)

(5,4,2,6,3)

Eliminate

(5,3,4,6,2)

(2,4,6,3,5)

(4,3,6,5,2)

(2,3,4,6,5)

(2,3,6,4,5)

(3,4,5,2,6)

(3,5,4,6,2)

(4,5,3,6,2)

(5,4,2,3,6)

(4,6,3,2,5)

(3,4,2,6,5)

(3,6,5,1,4)

(3,4,5,6,2)

(5,4,2,6,3)

Tend to kill off the worst ones.

Integrate

(5,4,2,6,3)

(5,3,4,6,2)

(2,4,6,3,5)

(2,3,6,4,5)

(3,4,5,2,6)

(3,4,5,6,2)

(3,5,4,6,2)

(4,5,3,6,2)

(5,4,2,3,6)

(4,6,3,2,5)

(3,4,2,6,5)

(3,6,5,1,4)

Restart

(5,3,4,6,2)

(2,4,6,3,5)

(5,4,2,6,3)

(2,3,6,4,5)

(3,4,5,2,6)

(3,4,5,6,2)

(3,5,4,6,2)

(4,5,3,6,2)

(5,4,2,3,6)

(4,6,3,2,5)

(3,4,2,6,5)

(3,6,5,1,4)

Genetic Algorithms

- Facts
- Very robust but slow
- Can make simulated annealing seem fast
- In the limit, optimal

Other GA-TSP Possibilities

- Ordinal Representation
- Partially-Mapped Crossover
- Edge Recombination Crossover
- Problem
- Operators are not sufficiently exploiting the

proper building blocks used to create new

solutions.

Genetic Algorithms

- Some ideas
- Parallelism
- Punctuated equilibria
- Jump starting
- Problem-specific information
- Synthesize with simulated annealing
- Perturbation operator

Heuristic H

Length(MST) lt Length(T)

Let T be the optimal tour.

Heuristic H

Tour T

Tour T

Perturbation of points

Perturbation of a Point

Mutation Operator

Points are perturbed in a normal distribution

centered around the original location and a

standard deviation which is a function of the

original interpoint distances.

Crossover Operator

Characteristics of Operators

- Perturbed points tend to stay close to original

locations, hence distances remain reasonable. - Small shifts in point position can have an effect

on the MST, hence see many different solutions.

Improvement Over EC

Average Improvement 32.1

Improvement Over H

Average Improvement 15.1