Sandpile lion'11
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1
The Sandpile Mutation Operator for Genetic
Algorithms
Carlos M. Fernandes1,2
J.L.J. Laredo1
J.J. Merelo1
Agostinho C. Rosa2
1Department of Architecture and Computer Technology, University of Granada, Spain
2 LaSEEB-ISR-IST, Technical Univ. of Lisbon (IST), Portugal
LION’11 — Rome, January 2011
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Summary1. Motivation: Dynamic Optimization
2. Self-Organized Criticality and the Sandpile
3. Sandpile Mutation GA (GGASM)
4. Results
5. Mutation Rates
6. Conclusions
LION’11 — Rome, January 2011
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Dynamic Optimization Industrial applications (non-linearities,
multi-objective) Some applications have dynamic
components• Fitness function depends on time t• Non-stationary (or dynamic) optimization
LION’11 — Rome, January 2011
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Dynamic Optimization
LION’11 — Rome, January 2011
• Classification of dynamic problems• severity• period between changes• others...
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Dynamic Optimization and EC
Reaction to Changes
Memory Schemes
Multi-Population Schemes
Diversity Maintenance• Random Immigrants GA (RIGA)
one additional parameter
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Parameter Control
Deterministic: parameter values change according to deterministic rules
Adaptive: variation depends indirectly on the problem and the search stage
Self- adaptive: the values to evolve together with the solutions to the problem
LION’11 — Rome, January 2011
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Sand Pile Mutation Operator
“Sand” is dropped on top of 2D lattice
When the number of grains exceed the critical value, there is an avalanche of sand
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Self-Organized Criticality (SOC)
SOC is a state of criticality
formed by self-
organization in a long
transient period at the
border of order and chaos.
LION’11 — Rome, January 2011
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SOC in Evolutionary Computation
Krink and co-authors
• the power-law is computed offline
Self-Organized Random Immigrants GA
• uses a SOC model to introduce random
immigrants in the population
Sandpile Mutation
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Sand Pile Mutation Operator
l1
l2
l3
…
0
1
2
3
4
n1
n2
n3
…
Z
0
1
2
3
4Z Drop (g) grains (g is grain
rate) If h(x,y) > 3, topple Maximization: mutates if rand (0,1.0) > (normalized) fitness
LION’11 — Rome, January 2011
Parents’ fitness
The lattice is the population
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Dynamic Optimization Problems
[Yang & Yao] problem generator• period between changes (ε)• severity (ρ)
ε : 1200, 2400, 24000, 48000 ρ : 0.05, 0.3, 0.6, 0.95
• each stationary problem -> 16 different dynamic problems
fast and severe changes
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Dynamic Optimization Problems
m-k trap functions (30 and 40 bits)
Royal Road (64 bits)
Knapsack (100 bits)
non-deceptive, nearly-deceptive, deceptive,
combinatorial (constrained)
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Performance Metrics
Offline performance
• average of the best fitness throughout the run
Statistical tests
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Algorithms
Generational Standard GA (GGA)
Self-organized Criticality RIGA (SORIGA)
Elitism-based Immigrants GA (EIGA)
GGA with Sandpile Mutation (GGASM)
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Tests Several mutation probability and
population size values.
binary tournament 2-elitism uniform crossover (p=1.0)
• Balance disruptive effect and selective pressure
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Sand Pile Mutation
ρ →ε = 1200 ε = 2400 ε = 24000 ε = 48000.05 .3 .6 .95 .05 .3 .6 .95 .05 .3 .6 .95 .05 .3 .6 .95
order-4
GGA − − ≈ ≈ ≈ ≈ ≈ ≈ + + + + + + + +
SORIGA ≈ + + + ≈ + + + + + + + + + + +
EIGA − − ≈ − − ≈ ≈ − + + + + + + + +
R. Road
GGA + ≈ ≈ ≈ ≈ + ≈ ≈ ≈ + + + + + + +
SORIGA + + + ≈ + + + ≈ + + + + + + + +
EIGA ≈ ≈ + ≈ + + + ≈ + + + + + + + +
Knapsack
GGA − − + + − ≈ + + + + ≈ − + + + ≈
SORIGA − ≈ + + ≈ ≈ + + + + ≈ ≈ + + + ≈
EIGA − − + + − − + + + + − − + + + −
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Sand Pile Mutation Rate
3 246 489 732 975 1218146117041947219024332676291931623405364838910
0.1
0.2
0.3
0.4
0.5ρ = 0.05
online m
uta
tion r
ate
3 246 489 732 975 1218146117041947219024332676291931623405364838910
0.1
0.2
0.3
0.4
0.5ρ = 0.6
online m
uta
tion r
ate
ε = 24,000order-4 trap
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Conclusions GGASM is able to improve other GAs’
performance on dynamic optimization
Parameter set is not increased
Sandpile mutation rate self-regulates: mutation rates’ distribution depends on the problem and dynamics
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Future Research The Sandpile mutation may be hybridized
with any kind of Evolutionary Algorithm (and other bio-inspired algorithms)
Study the mutation rates and mutation distribution.
Test the Sandpile Mutation on stationary problems and constrained dynamic optimization
LION’11 — Rome, January 2011