tutorial on particle swarm optimization
TRANSCRIPT
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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
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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”