sandpile evo star 2011

Carlos M. Fernandes1,2

Juan L.J. Laredo1

Antonio .M. Mora1

Juan Julián Merelo1

Agostinho C. Rosa2

1Department of Computer Architecture, University of Granada, Spain2 LaSEEB-ISR-IST, Technical Univ. of Lisbon (IST), Portugal

Evo*2011 – Torino, Italy, April 2011 1

1. Motivation: Dynamic Optimization Problems.

2. Self-Organized Criticality (SOC) and the

Sandpile Model.

3. Sandpile Mutation GA (GGASM).

4. Mutation Rates.

5. Conclusions and Future Work.

2Evo*2011 – Torino, Italy, April 2011

Non-stationary (or dynamic) fitness functions:• Fitness function depends on time t



Solve the stationary problem.

Characteristics of the changes:

• Severity

• Frequency

• Cyclic

• Predictability

Evolutionary Algorithms: full convergence must be avoided

5

0

16

32

48

64

0 1600 3200 4800 6400 8000

best

of

gen

era

tio

n

generations

ρ = 0.05 (low severity)

GGASM SORIGA

0

16

32

48

64

0 1600 3200 4800 6400 8000

best

of

gen

era

tio

n

generations

ρ = 0.95 (high severity)

Reaction to Changes• Increase Mutation Rate (Hypermutation)

Diversity Maintenance

• Maintain genetic diversity at a higher level

Random Immigrants Genetic Algorithm (RIGA)


Genetic Algorithm with a Self-Organized Criticality Mutation Operator (Sandpile Mutation)

7

SOC is state of criticality formed by

self-organization in a long transient

period at the border of order and

chaos.

8

o Cellular Automata

o “Sand” is dropped on top

of 2D lattice, increasing the

number of grains in the cell.

o When the slope exceeds

a critical value, the grains

topple to the neighbouring

cells - Avalanche

Evo*2011 – Torino, Italy, April 2011

Power-law relationship between the

size of the avalanches and their

frequency.

Krink et al. compute the sandpile offline, and

then use the avalanche size as the mutation

probabilities.

Self-Organized Random Immigrants GA

• uses a SOC model to introduce random immigrants in

the population

Sandpile Mutation: works on-line at the bit

level


10

n1

n2

n3

…

0

1

2

3

4

l1

l2

l3

…

Z

0

1

2

3

4Z Drop (g) grains (g is grain

rate)

If h(x,y) = 4, topple

Maximization: mutates if

rand (0,1.0) > (normalized)

fitness


Parents’ fitness

The lattice is the population

Wilson’s [14] algorithmic descriptionWilson’s [14] algorithmic description

Nathan Winslow (1997), Introduction to Self-Organized Criticality and Earthquakes, discussion

paper, Department of Geological Sciences, University of Michigan, 1997

http://www2.econ.iastate.edu/classes/econ308/tesfatsion/SandpileCA.Winslow97.htm

http://www2.econ.iastate.edu/classes/econ308/tesfatsion/SandpileCA.Winslow97.htm

[Yang & Yao] problem generator• Period (generations or function evaluations)

between changes. Frequency = 1/ε• severity : ρ Є [0, 1]

• ρ×lenght → number of variables that are affected by changes

• Trap Function, Onemax, Royal Road, Knapsack…

• Compute the offline performance: best fitness averaged over the entire run.


12

ρ

ε = 1200 ε = 4800 ε = 12000 ε = 24000

0.05 0.3 0.6 0.95 0.05 0.3 0.6 0.95 0.05 0.3 0.6 0.95 0.05 0.3 0.6 0.95

3-trap + + + + + + − + + + + + + + + +

4-trap − + + + + − + + + ≈ + + + ≈ + +

Royal

Road+ − − − + − − − + − − − + − − −

Knaps

ack+ + + + + + + + + + + + + + + +


GGASM vs SORIGA

(statistical tests)

Order-3 Trap Functions and Onemax

Population Size n = 30

Chromosome lenght l = 30• 30x30 sandpile

pc=1.0, 2-elitism; uniform crossover.

Several g valuesVarying frequency (1/ε) and severity (ρ)Compare the population before and after g

grains are dropped.

13

14

1E+00

1E+01

1E+02

1E+03

1 10 100

Qu

an

tity

ρ = 0.05; ε = 1200

1 10 100

ρ = 0.5; ε = 1200

1 10 100

ρ = 0.95; ε = 1200

1E+00

1E+01

1E+02

1E+03

1 10 100

Qu

an

tity

% of the alleles

ρ = 0.05; ε = 12000

1 10 100

% of the alleles

ρ = 0.5; ε = 12000

1 10 100

% of the alleles

ρ = 0.95; ε = 12000


Order-3 trap function

15

0%

30%

60%

0 200 400 600 800 1000

% o

f th

e a

lle

les

generations

ρ = 0.05 (low severity)ε = 1200

0%

30%

60%

0 200 400 600 800 1000

generations

ρ = 0.5 (medium severity)ε = 1200


16

1E+00

1E+01

1E+02

1E+03

1E+04

1 10 100

Qu

an

tity

% of the alelles

ε = 6000

1 10 100

% of the alelles

ε = 24000

1 10 100

% of the alelles

ε = 120000


600000 evalutationsorder-3 trap functions

17

1E+00

1E+01

1E+02

1E+03

1E+04

1 10 100

Qu

an

tity

order-3 trap; ε = 1200

1E+00

1E+01

1E+02

1E+03

1E+04

1E+05

1 10 100

onemax; ε = 1200

1E+00

1E+01

1E+02

1E+03

1E+04

1 10 100

Qu

an

tity

% of the alleles

order-3 trap; ε = 12000

1E+00

1E+01

1E+02

1E+03

1E+04

1 10 100

% of the alleles

onemax; ε = 12000


ρ = random

18

1E+00

1E+01

1E+02

1E+03

1E+04

1 10 100

Qu

an

tity

% of the allelles

g = (n×l)/32

1 10 100

% of the alleles

g = (n×l)/16

1 10 100

% of the alleles

g = (n×l)/8


ε = 12000; ρ = random n= 30; l = 30

order-3 traps

19

21

22

23

24

25

26

1/(8×l) 1/(4×l) 1/(2×l) 1/l 2/l 4/l

me

an

bes

t-o

f-g

en

era

tio

n

mutation probability, pm

GGA21

22

23

24

25

26

(n×l)/32 (n×l)/16 (n×l)/8 (n×l)/4 (n×l)/2 n×l

grain rate, g

GGASM

ε = 1200

error = 0.71%error = 2.59%

The distribution of mutation rate varies with severity and frequency.

Different base-function may lead to different distributions.

The grain rate affects the distribution. The algorithmic description and the

topology impose a limit to the mutation rate.


The working mechanisms are not fully understood.

Study the distribution rate and the optimal grain rate values when the sandpile grows.

Variables’ linkage.

Sandpile Topology.


3-trap, 60 bits, n=60

sandpile evo star 2011

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