Sandpile evo star 2011
-
Upload
carlos-m-fernandes -
Category
Technology
-
view
890 -
download
0
Transcript of 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
3Evo*2011 – Torino, Italy, April 2011
4Evo*2011 – Torino, Italy, April 2011
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)
6Evo*2011 – Torino, Italy, April 2011
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
9Evo*2011 – Torino, Italy, April 2011
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
Evo*2011 – Torino, Italy, April 2011
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
[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.
11Evo*2011 – Torino, Italy, April 2011
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+ + + + + + + + + + + + + + + +
Evo*2011 – Torino, Italy, April 2011
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
Evo*2011 – Torino, Italy, April 2011
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
Evo*2011 – Torino, Italy, April 2011
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
Evo*2011 – Torino, Italy, April 2011
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
Evo*2011 – Torino, Italy, April 2011
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
Evo*2011 – Torino, Italy, April 2011
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.
20Evo*2011 – Torino, Italy, April 2011
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.
21Evo*2011 – Torino, Italy, April 2011
3-trap, 60 bits, n=60
22