Using Self-Organized Criticality for Adjusting the Parameters of a Particle Swarm

USING SELF-ORGANIZED CRITICALITY FOR ADJUSTING THE

PARAMETERS OF A PARTICLE SWARM

Carlos M. Fernandes1,2

J.J. Merelo1

Agostinho C. Rosa2

1Department of computers architecture and technology, University of Granada, Spain

2 LaSEEB-ISR-IST, Technical Univ. of Lisbon (IST), Portugal

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Why?

Particle Swarm and Self-Organized Criticality

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Exploration

Exploitation

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Parameter Controlin bio-inspired computation

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Deterministic: parameter values change according to

deterministic rulesAdaptive: variation depends indirectly on the problem and

search stage

Self- adaptive: values evolve together with the solutions to the problem Hand-tuning

SOC

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Particle Swarm Optimization (PSO)

Cultural and social interaction: cognitive, social and random factors.

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Bio-inspired: bird flock and fish school.

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PSO – equations and parameters

X i(t) =

X i – position of particle i (vector)V i – velocity of particle i (vector)

Vi(t) =

Xi(t-1)+Vi(t)

Vi(t-1)+c1 r1(pi-xi(t-1))+c2

r2(pg-xi(t-1))ω Vi(t-1)+c1 r1(pi-xi(t-1))+c2

r2(pg-xi(t-1))

ρ Xi(t-1)+Xi(t-1)+Vi(t)

c

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SOC: a state of criticality formed by self-organization in a long transient period at the border of order and chaos.

Self-Organized Criticality (SOC)

The Sandpile Model

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SOCPower-laws

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[1] BK-inspired Extremal Optimization, Boettcher and

Percus, 2003

[2] Sandpile in Evolutionary Algorithms, Krink et al., 2000-

2001

[3] SOC in PSO, Løvbjerg and Krink, 2004

[4] BK model in Evolutionary Algorithms – Self-Organized

Random Immigrants GA (SORIGA), Tinós and Yang, 2008

[5] Sand Pile Mutation for Genetic Algorithms, Fernandes et

al., 2008-2012

SOC in Bio-inspired Computation

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Bak-Sneppen Model

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f = 0.14

f = 0.41 f = 0.55

f = 0.79

f = 0.23

f = 0.90

f = 0.91

f = 0.32

f = 0.16

Per Bak (How Nature Works):Random numbers are arranged in a circle. At each time step, the lowest number, and the numbers at its two neighbours, are replaced by new random numbers.

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BK Model: punctuated equilibrium and power-laws

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The Bak-Sneppen PSO (BS-PSO)

X i(t) =Vi(t) =

ω Vi(t-1)+c1r1(pi-xi(t-1))+c2r2(pg-xi(t-1))

ρ Xi(t-1)+Xi(t-1)+Vi(t)

ω = 1-bs_fitness(i)

c1 =c2=1+bs_fitness(i)

ρ =random [0, 1-bs_fitness(i)]

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General Scheme

BS model PSO

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BS species

Particles

bs_fitness

ω

c

ρ

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o Sphere, Rastrigin, Rosenbrock, Griewank

o lbest and gbest topologies.

o TVIW-PSO, RANDIW- PSO, GLbestIW-PSO and IA-PSO

o Population size: n = 20

o 3000 generations

Test Set

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BS-PSO

bs, 1.49, 0 bs, 2.0, 0 bs, bs, 0 bs, bs, 0.25 bs. bs, bs

f1 3.35e+01(1.90e+02)

1.38e-15(3.21e-15)

8.30e-32(3.47e-31)

0.00e+00(0.00e+00)

0.00e+00(0.00e+00)

f2 1.67e+05(1.17e+06)

1.88e+02(2.53e+02)

8.56e+01(7.98e+01)

2.61e+01(2.66e-01)

2.60e+01(1.58e-01)

f3 2.82e+02(4.44e+01)

1.11e+02(2.75e+01)

2.02e+02(4.16e+01)

4.88e+00(7.73e+00)

3.32e+00(7.09e+00)

f4 1.63e+00(5.93e+00)

1.25e-02(1.26e-02)

1.65e-02(2.24e-02)

3.79e-03(2.29e-03)

4.51e-03(4.00e-03)

(ω, c, ρ)

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PSO 1 vs. PSO 2 f1 f2 f3 f4

BS-PSO vs TVIW-PSO + + + +BS-PSO vs RANDIW-PSO + + + +BS-PSOvs GLbestIW-PSO + + + +

Results – lbest

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PSO 1 vs. PSO 2 f1 f2 f3 f4

BS-PSO vs TVIW-PSO + + – –BS-PSO vs RANDIW-PSO + ~ – ~BS-PSO vs GLbestIW-PSO + + ~ +

full control

without perturbation of position

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Results – lbest

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PSO 1 vs. PSO 2 f1 f2 f3 f4

BS-PSO vs IA-PSO + + ~ +

BS-PSOvs IA-PSO (bs controled ) ~ ~ + ~

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Results – gbest

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PSO 1 vs. PSO 2 f1 f2 f3 f4

BS-PSO vs TVIW-PSO + + + +BS-PSO vs RANDIW-PSO + + + +BS-PSOvs GLbestIW-PSO + + + +

BS-PSO) vs IA-PSO + + ~ +

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Distribution of parameter values

0 27 54 81 108 135 162 189 216 243 270 297 324 351 378 405 432 459 486 513 540 567 594 621 648 675 702 729 756 783 810 837 864 891 918 945 972 999

00.10.20.30.40.50.60.70.80.9

1

iterations

0.01 0.1 1

1

10

100

1000

ωi =ρi

num

ber

of

sam

ple

s

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o With a simple set of equations we are able to control three (four) parameters of the PSO.

o The resulting algorithm is competitive with other variants of the PSO.

o Full control of the PSO by BS attains good performance.

o Hand-tuning is not required.

Conclusions

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o Information (state) from PSO into the model.

o Test BS-PSO on dynamic environments.

o Scalability.

o BS critical state: investigate the behaviour before and after the system reaches the critical state.

Future Research

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Questions?