Genetic algorithms

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 GAGenerate initial populationdo

Calculate the fitness of each member// simulate another generationdo

Select parents from current populationPerform crossover add offspring to the

new populationwhile 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 “don’t care”

11****00 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 Holland’s 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 Holland’s 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 Holland’s 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 Holland’s 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 Holland’s 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 Holland’s 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)

(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)

(5,4,2,6,3)

(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)

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)

Tend to kill off the worst ones.

(3,4,5,6,2) (5,4,2,6,3)

Integrate

(5,3,4,6,2) (2,4,6,3,5)

(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)

Restart

(5,3,4,6,2) (2,4,6,3,5)

(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)

Genetic Algorithms Facts

Very robust but slowCan 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) < 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 centeredaround the original location and a standard deviation which is a function of the original interpoint distances.

Crossover Operator

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.

Characteristics of Operators

% Improvement Over EC

0

10

20

30

40

50

60

OSH Berlin KroA Pr TspAverage Improvement: 32.1%

% Improvement Over H

0

5

10

15

20

25

OSH Berlin KroA Pr TspAverage Improvement: 15.1%