cs 484 – artificial intelligence1 announcements lab 3 due tuesday, november 6 homework 6 due...

CS 484 – Artificial Intelligence 1

Announcements

• Lab 3 due Tuesday, November 6• Homework 6 due Tuesday, November 6• Lab 4 due Thursday, November 8• Current Event

• Chelsea - today• Beth - Tuesday, November 6

Genetic Algorithms

Lecture 13


Evolutionary Computation

• Computational procedures patterned after biological evolution

• Search procedure that probabilistically applies search operators to a set of points in the search space• Generate successor hypothesis by repeatedly

mutating and recombining parts of the best currently known hypothesis


General Procedure

• At each stage has a collection of hypotheses known as the current population

• Update population by replacing some fraction of it by offspring of the most fit current hypotheses

• Process forms a generate-and-test search


Reason for Popularity

• Evolution is known to be successful, robust method for adaptation within biological systems

• GAs can search spaces of hypotheses containing complex interacting parts

• Genetic algorithms are easily parallelized and can take advantage of the decreasing cost of powerful computer hardware


The ParametersGA(Fitness, Fitness_threshold, p, r, m)• Fitness - function evaluates how good a

hypothesis is• Fitness_threshold - minimum acceptable

hypothesis• p - size of the population• r - fraction of population to be replaced• m - mutation rate

2

pr⋅


The AlgorithmGA(Fitness, Fitness_threshold, p, r, m)• Initialize: P ← p random hypotheses• Evaluate: for each h in P, compute Fitness(h)• While [maxh Fitness(h)] < Fitness_threshold

• Select: Select (1 – r) members of P to add to PS based on fitness

• Crossover: Probabilistically select pairs of hypotheses from P. For each pair, <h1, h2>, produce two offspring by applying the Crossover operator. Add all offspring to PS

• Mutate: Invert a randomly selected bit in m · p random members of PS

• Update: P ← PS

• Evaluate: for each h in P, compute Fitness(h)• Return the hypothesis from P that has the highest fitness

( ) ( )( )∑

=

= P

jj

ii

hFitness

hFitnessh

1

Pr

€

r

2


Fitness• Fitness is an important concept in genetic

algorithms.• The fitness of a chromosome determines how

likely it is that it will reproduce.• Fitness is usually measured in terms of how well

the chromosome solves some goal problem.• E.g., if the genetic algorithm is to be used to sort

numbers, then the fitness of a chromosome will be determined by how close to a correct sorting it produces.

• Fitness can also be subjective (aesthetic)


Representing Hypotheses

• From the EnjoySport (simplified) example• Represent (Sky = Sunny v Cloudy) Λ

(AirTemp = Warm) by

• Represent IF AirTemp = Warm THEN EnjoySport = yes by

Sky AirTemp

011 10

Sky AirTemp EnjoySport

111 10 10


Single-Point Crossover

• Crossover is applied as follows:1) Select a random crossover point.

2) Break each chromosome into two parts, splitting at the crossover point.

3) Recombine the broken chromosomes by combining the front of one with the back of the other, and vice versa, to produce two new chromosomes.


Other types of Crossover• Usually, crossover is applied with one crossover

point, but can be applied with more, such as in the following case which has two crossover points:

• Uniform crossover involves using a probability to select which genes to use from chromosome 1, and which from chromosome 2.


Mutation

• A unary operator – applies to one chromosome.

• Randomly selects some bits (genes) to be “flipped” • 1 => 0 and 0 =>1

• Mutation is usually applied with a low probability, such as 1 in 1000.


Termination Criteria• A genetic algorithm is run over a number of

generations until the termination criteria are reached.

• Typical termination criteria are:• Stop after a fixed number of generations.• Stop when a chromosome reaches a specified fitness

level.• Stop when a chromosome succeeds in solving the

problem, within a specified tolerance.• Human judgment can also be used in some more

subjective cases.


Optimizing a mathematical function• A genetic algorithm can be

used to find the highest value for f(x) = sin (x).

• Each chromosome consists of 4 bits, to represent the values of x from 0 to 15.

• Fitness ranges from 0 (f(x) = -1) to 100 (f(x) = 1).• f'(x) = 50 * (f(x) + 1) - determines fitness

• By applying the genetic algorithm it takes just a few generations to find that the value x =8 gives the optimal solution for f(x).


What is a Schema?• How to characterize evolution of population in GA?• As with the rules used in classifier systems, a schema is a string

consisting of 1’s, 0’s and *’s.E.g.: 1011*001*0Matches the following four strings:1011000100101100011010111001001011100110

• A schema with n *’s will match a total of 2n chromosomes.• Each chromosome of r bits will match 2r different schemata.

• Example: 0010 is a representative of 24 distinct schemas• 00**, 0*10, ****, etc.


Schema Theorem

• Characterizes evolution in the population in terms of the number of instances representing each schema

• The fitness of a schema is defined as the average fitness of the chromosomes that match the schema.

• The fitness of a schema, S, in generation tth is written as follows:

f(S, t) • The number of occurrences of S in the population at time t

is :m(S, t)

• Looking for the expected value of m(S, t+1)• Defined in terms of m(S, t) and other properties


Selection Step• Evolution of GA depends on selection step,

recombination step, and mutation step• Selection step

• Cloned individuals become part of next generation• How many strings in schema S will be present in the

next generation?• A particular string h, is selected with a probability

• Let a(t) be the average fitness of all individuals∑=

= n

i ihf

hfh

1)(

)()Pr(

)(

)()Pr(

ian

hfh

⋅=


Fitness of the Next Generation• Interested in the probability that a single hypothesis

selected by the GA will be an instance of schema S• Probability that we select a representative of S

• Given that a(S,t) is the average fitness of instances of schema S

• The expected number of instances of S resulting from n independent selection steps

∑∈ ⋅

=Sh tan

hfS

)(

)()Pr(

),()(

),()Pr( tSm

tan

tSaS

⋅=

[ ] ),()(

),()1,( tsm

ta

tSatsmE =+


Schemata Properties• The defining length dL(S) of a schema, S, is the distance

between the first and last defined bits. For example, the defining length of each of the following schemata is 4:

**10111*1*0*1**

• The order O(S) is number of defined bits in S. The following schemata both have order 4:

**10*11*1*0*1**1

• A schema with a high order is more specific than one with a lower order.


The Effect of Single-Point Crossover

• For a schema S to survive crossover, the crossover point must be outside the defining length of S.

• Hence, the probability that S will survive crossover is:

• L(S) is the length of the bit strings and pc is the probability that single-point crossover will be applied to an individual

• This tells us that a short schema is more likely to survive crossover than a longer schema.

1)(

)(1)(Pr

−−=

SLSd

pS Lcs


The Effect of Mutation

• The probability that mutation will be applied to a bit in a string is pm

• Hence, the probability that a schema S will survive mutation is:

• We can combine this with the effects of selection, crossover, and reproduction to give:

)()1()(Pr SOms pS −=

[ ] ( ) )(11)(

)(1),(

)(

),()1,( sO

mL

c pSL

SdptSm

ta

tSatSmE −⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−≥+


The Schema Theorem

• Holland’s Schema Theorem, represented by the above formula, can be written as:More fit schemas will tend to grow in influence, especially schemas containing small number of defined bits, and defined bits that near one another within the bit string.

• This helps to explain why genetic algorithms work.

• It does not provide a complete answer.

[ ] ( ) )(11)(

)(1),(

)(

),()1,( sO

mL

c pSL

SdptSm

ta

tSatSmE −⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−≥+


The Building Block Hypothesis• Genetic algorithms manipulate short, low-order,

high fitness schemata in order to find optimal solutions to problems.”

• These short, low-order, high fitness schemata are known as building blocks.

• Hence genetic algorithms work well when small groups of genes represent useful features in the chromosomes.

• This tells us that it is important to choose a correct representation.


Messy Genetic Algorithms (1)• An alternative to standard genetic algorithms that

avoid deception.• Each bit in the chromosome is represented as a

(position, value) pair. For example:((1,0), (2,1), (4,0))

• In this case, the third bit is undefined, which is allowed with MGAs. A bit can also overspecified:((1,0), (2,1), (3,1), (3,0), (4,0))


Messy Genetic Algorithms (2)

• Underspecified bits are filled with bits taken from a template chromosome.

• The template chromosome is usually the best performing chromosome from the previous generation.

• Overspecified bits are usually dealt with by working from left to right and using the first value specified for each bit.


Messy Genetic Algorithms (3)

• MGAs use splice and cut instead of crossover.• Splicing involves simply joining two chromosomes together:

((1,0), (3,0), (4,1), (6,1))

((2,1), (3,1), (5,0), (7,0), (8,0))

((1,0), (3,0), (4,1), (6,1), (2,1), (3,1), (5,0), (7,0), (8,0))

• Cutting involves splitting a chromosome into two:

((1,0), (3,0), (4,1))((6,1), (2,1), (3,1), (5,0), (7,0), (8,0))


Co-Evolution• In the real world, the presence of predators is

responsible for many evolutionary developments.• Similarly, in many artificial life systems, introducing

“predators” produces better results.• This process is known as co-evolution.• For example, Ramps, which were evolved to sort

numbers: “parasites” were introduced which produced sets of numbers that were harder to sort, and the ramps produced better results.

cs 484 – artificial intelligence1 announcements lab 3 due tuesday, november 6 homework 6 due...

Documents

fitness crossover

p p random hypothesesevaluate

r members of p

crossover points

random crossover point

crossover operator

uniform crossover

p random members of