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Tutorial on Particle Swarm Optimization

Jim Kennedy

Russ Eberhart

IEEE Swarm Intelligence Symposium 2005

Pasadena, California USA

June 8, 2005

Jim Kennedy

Bureau of Labor Statistics

U. S. Department of Labor

Washington, DC

kennedy_jim@bls.gov

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Particle SwarmsPart 1: Sociocognitive Optimization

Artificial IntelligenceAttempted to elicit intelligence from a computing machineby simulating human thought – good idea!

Early AI derived in the Dark Ages of psychology, when study of mind was taboo in science. Based on naïve introspectionism.

Computational IntelligencePerhaps we can apply more scientific concepts of mind.• Self-report gives an unsatisfactory account of “real” cognition• Sociocognition: thought as a social act• Self-organization of societies, cultures

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Sociocognition

SpanosLevine, Resnick, and HigginsLewin’s topological space

B=f(P,E)(P,E)=LS

Cognitive DissonanceHypothesis 1: There are two major sources of cognition, namely, own experience and communication from others.

Leon Festinger, 1954/1999, Social Communication and Cognition (draft)

-Types of cognitive relations-Minimization of dissonance-Fundamental Attribution Error-Exploration/exploitation

Note: vector=point

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Latané’s Social Impact Theory

i=f(SIN); i=sNt, t<1

Latané (1981)

Nowak, Szamrej, and Latané Psychological Review, 1990

Complex Adaptive Systems

Cellular automata• Fixed point attractors• Periodic attractors• Chaotic attractors• The “edge of chaos”

The Game of LifeAlife

Complexity (NK landscapes)

Self-organizationEmergence(Immergence – downward causation)

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Flocks, Herds, and Schools

•Heppner & Grenander•Craig Reynolds

•Steer toward the center•Match neighbors’ velocity•Avoid collisions•(Seek roost)

Evolutionary Computation

Variation operators•Mutation•Crossover

Selection

Population of proposed problem solutions

Emulates natural evolution to “breed” solutions to hard problems, write computer programs, build robots, etc.

D.T. Campbell: “Blind variation and selective retention”

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Particle Swarms

Sociocognitive space

•Certainly high-dimensional (i.e., Burgess & Lund)•Abstract – attitudes, behaviors, cognition•Heterogeneous with respect to evaluation (dissonance)•Multiple individuals

Individual has position=“mental state”: Individual changes:

x ir

vir

Particle Swarms

Individuals learn from their own experience

This formula, iterated over time, causes each individual i ’s trajectory to oscillate around its previous best point pi in the sociocognitive space.

It stochastically adjusts i ’s velocity depending on previous successes, and occasionally updates pi – the previous best.

⎪⎩

⎪⎨⎧

+←

−⊗+←

vxxxpvv

iii

iiii Urrr

rrrr )(),0( ϕ for i=1 to popsizefor j=1 to dimension

v[i][j]=v[i][j]+rand()*(p[i][j]-x[i][j])x[i][j]=x[i][j]+v[i][j]

next jnext i

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2 .9

-3

-2

-1

0

1

2

3 p h i=

3 .6

-4-3-2-101234 p h i=

3 .95

-1 0

-5

0

5

1 0 p h i=

3 .99

-3 0

-2 0

-1 0

0

1 0

2 0

3 0 p h i=

4

-300

-200

-100

0

100

200

300 ph i=

0.01

-30

-20

-10

0

10

20

30 phi=

0.5

-4-3-2-101234 phi=

1.3

-3

-2

-1

0

1

2

3 phi=

1.9

-3

-2

-1

0

1

2

3 phi=

Oscillating trajectories (nonstochastic)

⎪⎩

⎪⎨⎧

+←

−+←

vxxxpvv

iii

iiiirrrrrrr )(

Particle SwarmsSociocognitive space can contain many individualsThey influence one another

Evaluate your present positionCompare it to your previous best and neighborhood bestImitate self and others

(g is neighborhood best)

⎪⎩

⎪⎨

⎧

+←

−⊗+−⊗+←

vxxxpxpvv

iii

igiiii UUrrr

rrrrrr )())2

,0()()2

,0( 21 ϕϕ

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The “Drunkard’s Walk”The particle will explode out of control if it is not limited in some way.Three methods have been widely used:

Vmax

Inertia weight

Constriction coefficient

(note that these last two are equivalent)

))()2

,0()()2

,0( 21( xpxpvv idgdidididid UU −⊗+−⊗+=ϕϕχ

)(),0()(),0(21 xpCxpCvv idgdidididid

UU −⊗+−⊗+=α

maxmax

max;max

)()2

,0()()2

,0( 11

VthenVifelse

VthenVif

UU

vvvv

xpxpvv

idid

idid

idgdidididid

−=−<

=>

−⊗+−⊗+=ϕϕ

Binary Particle Swarms

)exp(11)(

vvs

−+=where logistic function

keeps it in (0..1)

Kennedy and Eberhart (1997)Kennedy and Spears (1998)

Being looked at closely, expanded

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

=<

−⊗+−⊗+←

0

1)()1,0(

)()2

,0()()2

,0( 11

xxv

xpxpvv

i

ii

igiiii

else

thensUif

UU

rrr

rrrrrr

r

ϕϕ

(Usu. Vmax, too)

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Topology

Traditional ones: gbest, lbest

Topology determines how solutions spread through the population.

Gbest: immediateLbest: slowed

Affects the rate of convergence, parallelism of the search

Innovative topologies

Lots of variables to work with:• Mean degree• Clustering• Heterogeneity

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FIPS“Fully Informed Particle Swarm” (Rui Mendes)

Should become the new standard

Distributes total φ across n terms

vxxN

xpUvv

iii

i

N i

n i

innbri

rrr

rrrr

+←

=← ⎟⎟

⎠

⎞⎜⎜⎝

⎛∑

−⊗+

1

)( )(),0( ϕχ

Best neighbor is not selectedIndividual not included in neighborhoodDependent on topology

FIPS ResultsTwo performance metrics

Red: Topologies with average degree in the interval (4, 4.25).

Green: Topologies with average degree in the interval (3, 3.25) and clustering coefficient in the interval (0.1, 0.6).

Blue: Topologies with average degree in the interval (3, 3.25) and clustering coefficient in the interval (0.7, 0.9).

Light Blue; Topologies with average degree in the interval (5, 6) and clustering coe�cient in the interval (0.025, 0.4).

Black: All other topologies.

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Evolutionary Computation and Particle Swarms

Change vs. selectionFitness and dissonanceCooperation vs. competition

Culture as evolution (anthropology)Adaptation / learningMemeticsEvolutionary epistemology

“Cultures”

1. 3.2.

4. 5. 6.

7. 8. 9.

+

+ +

+ --

-

+-

+

+ ++-

-

- +

+

---

++

+

1 1 1 1 0 0 0 1 11 1 1 1 0 0 0 1 11 1 1 1 0 0 0 1 11 0 0 1 1 1 1 0 01 0 0 1 1 1 1 0 01 0 0 1 1 1 1 0 01 0 0 1 1 1 1 0 01 1 1 1 0 0 0 1 11 1 1 1 0 0 0 1 11 1 1 1 0 0 0 1 1

Kosko’s “South African” FCM

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The Future of Particle Swarms

Trajectory Analysis2.9

-3

-2

-1

0

1

2

3 phi=

3.6

-4-3-2-101234 phi=

3.95

-10

-5

0

5

10 phi=

3.99

-30

-20

-10

0

10

20

30 phi=

4

-300

-200

-100

0

100

200

300 phi=

0.01

-30

-20

-10

0

10

20

30 phi=

0.5

-4-3-2-101234 phi=

1.3

-3

-2

-1

0

1

2

3 phi=

1.9

-3

-2

-1

0

1

2

3 phi=

Includes constriction, inertia, convergence, etc.

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Interaction AnalysisIndividual trajectories very weak.Optimization is a function of interparticle interactions.The swarm as a whole, and as an aggregation of

subpopulationsEffect on trajectory when new “bests” are found“Immergence” and the effect of culture.

Probability Distribution Analysis

Theory: PS’s place among the EAsPractice: New versions

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Initialization methodsPopulation sizePopulation diameterAbsolute vs. signed velocitiesPopulation topologyBirths, deaths, migrationLimiting domain (XMAX, VMAX)Multiobjective optimization“Subvector” techniquesComparison over problem spacesHybrids

Parameters, Conditions, & Tweaks

Particle Swarms As a GeneralProblem-Solving Methodology

Out of the computer...Replace eval() with humanReplace particles with humansNetworked computers (with or without users)Scientific research strategyManagement & innovation

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Sociocognition

Particle swarms as social-psychological theoryNot just “what,” but “why”Social behavior → improved knowledge → adaptationPS as a “simulation”