a self-organized criticality mutation operator for dynamic optimization problems
DESCRIPTION
The Sandpile Genetic Algorithm, in Atlanta, 2008. Genetic and Evolutionary Computation Conference. GECCO'08. Mixing Genetic Algorithms and the Self-Organized Criticality Theory.TRANSCRIPT
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
A Self-Organized Criticality Mutation Operator for Dynamic Optimization
Problems
Carlos Fernandes1,2
J.J. Merelo2
Vitorino Ramos1
Agostinho C. Rosa1
1LaSEEB-ISR-IST, Technical Univ. of Lisbon (IST), Portugal2 Department of Architecture and Computer Technology, University of Granada, Spain
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Motivation and Objectives
•Develop an adaptive mutation operator to deal with Dynamic Optimization Problems (DOPs).
We aim at designing a mutation operator which may be able to give rise to small and large mutation rates in a self-regulated manner, non-deterministic.
Mutation operator should be able to react to changes, without detecting those changes. (When a change occurs, mutation should increase.)
Keep it simple! Avoid new parameters or complex parameter control.
DOPs require diversity. It is not mandatory that the algorithm finds the optimum, but, at least, to track it.
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Self-Organized Criticality•Self-Organized Criticality was identified by Bak, Tang and Wiesenfeld in 1987. “The Sandpile” model.
*Image taken from Kauffman’s Investigations
•Cellular automaton.
•“Sand” is randomly dropped on a lattice (2D). When the slope exceeds a specific threshold (zc = 4), the cell colapses and transfers the sand to the adjacent cells.
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Self-Organized Criticality•SOC: fractal, scale-invariant, 1/f noise, power-laws
*Image taken from Kauffman’s Investigations
Considerer a lattice (x,y) and a function z(x,y) which represents the number o grains in the cells.Starting with a flat surface z(x,y) = 0 for all x and y:Add a grain of sand: if z(x,y) > zc then an avalanche occurs
if z(x, y±1) = 4 or z(x±1, y) = 4Update z recursively
Sandpile: likehood of an avalanche is in power-law proportion to its size.
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Previous Related Work
•Tinós, R. and Yang, S. 2007. A self-organizing random immigrants genetic algorithm for dynamic optimization problems. Genetic Programming and Evolvable Machines 8, 255-286. SORIGA
• Krink, T., Rickers P., René T. 2000. Applying self-organised criticality to evolutionary algorithms. Proceedings of the 6th International Conference on Parallel Problem Solving from Nature, 375-384.
•Boettcher, S., Percus A.G. 2003. Optimization with extremal dynamics. Complexity 8(2), 57-62.
SORIGA introduces SOC in a GA by simulating a Bak-Sneppen model.
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
The Sandpile MutationThe 2D lattice is “connected” with the population (NxL size, where N is population size and L is chromosome size)
In each generation g grains are dropped into the lattice.
Individuals are ranked. This create a kind of slope for the sandpile, with sand collapsing with higher probability towards the worst chromosomes.
If the number of grains in a cell exceed 4, then an avalanche may occur depending on the fitness of the chromosome.
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
The Sandpile Mutationfor g grains do drop grain at random if zc = 4 compute normalized fitness:
Note: if bestFitness = worstFitness, fn is set to 0.5 (1.0?) if randomValue(0, 1.0) > fn and cell (n, l) not active mutate avalanche
and update lattice z recursively
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Test Set Severity of change: This criterion establishes how strongly the problem is changing
Speed of change: This criterion establishes how often the environment changes
•Yang and Yao’s dynamic problems generator*
•By using a binary mask, dynamic environments are created by applying the mask to each solution before its evaluation.
•Severity of change is controlled by setting the number of 1’s in the mask.
•Speed of change is controlled by defining the number of generations between the application of a different mask.
*Yang, S. and Yao, X. 2005. Experimental study on PBIL algorithms for dynamic optimization problems. Soft Computing 9(11), 815-834.
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Test Set
•Royal Road function
•Deceptive functions
•As in: Tinós, R. and Yang, S. 2007. A self-organizing random immigrants genetic algorithm for dynamic optimization problems. Genetic Programming and Evolvable Machines 8, 255-286
•Massively Multimodal Deceptive Problem (MMDP)
Speed was set to = 10, 100, 1000 (generations)𝜏Severity was set to ρ = 0.05, 0.6 and 0.95
9 different scenarios
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Test SetTwo-point crossover, pc = 0.7
N = 120
Tournament Selection (Kts = 0.9)
Sand pile Mutation Genetic Algorithm (SMGA)
SMGA: g = 10xL
SMGA (deceptive): g = 50xL
Performance is measured by the mean best-of-generation values, i.e., best fitness averaged over all generations, and then over all runs
30 runs for each configuration
Compared SMGA with Standard Generational GA (SGA: pm = 1/L) and Random Immigrants GA (RIGA: rr = 3, rr = 12)
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Results – Avalanches and Mutation
Avalanches Mutations
L = 24
L = 90
MMDP
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Results
Royal Road Deceptive 1 Deceptive 2
τ ρ SGA RIGA 1 RIGA 2 SMGA SGA RIGA 1 RIGA 2 SMGA* SGA RIGA1 RIGA2 SMGA*
10 0.05 (~) (~) (~) 31.41 (+) (+) (+) 0.855 (~) (+) (+) 0.752
10 0.60 (+) (+) (+) 13.40 (~) (~) (~) 0.793 (+) (+) (~) 0.594
10 0.95 (~) (~) (~) 17.12 (−) (−) (−) 0,922 (+) (~) (−) 0.558
200 0.05(−) (−) (−)
57.80 (+) (+) (+)
0.957(−) (+) (+)
0.7973
200 0.60 (+) (+) (+)
42.15 (+) (+) (+)
0.908 (+) (+) (+)
0.7832
200 0.95 (+) (+) (+)
46.43 (−) (−) (−)
0.939 (+) (+) (+)
0.7808
1000
0.05 (~) (~) (~)
62.36 (+) (+) (+)
0.994 (−) (+) (+)
0.79947
1000
0.60 (~) (~) (~)
54.85 (+) (+) (+)
0.984 (+) (+) (+)
0.79670
1000
0.95 (+) (+) (+)
56.62 (+) (+) (+)
0.988 (~) (+) (+)
0.79623
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Results
ρ = 0.05 ρ = 0.6 ρ = 0.95
Comparing SGA and SMGA’s dynamic behavior on Royal Road. 𝜏 = 200
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Results – Mutation rate
ρ = 0.05 ρ = 0.6 ρ = 0.95
𝜏 = 10
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Results – SMGA and SORIGA3-trap functions (between deception and non-deception): 10 traps, 30 bits
Uniform crossover
pc = 1.0
Binary tournament
Several pm and g values
SORIGA: rr = 3
Speed was set to 2400, 24000, 48000 evaluations
Population size N = 30 and N = 240
Severity was set to ρ = 0.05, 0.3, 0.6 and 0.95
12 different scenarios
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Results – SMGA and SORIGA
N = 240
N = 30
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Results – SMGA and SORIGAN = 30
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Results – SMGA and SORIGA
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Conclusions
SMGA is capable of outperforming SGA and RIGA on a wide range of problem settings, namely when severity is high.
It is at least competitive with the state-of-the-art SORIGA.
SMGA reacts to changes by increasing mutation rate (the occurrence of large avalanches is due to a sudden decrease in the average fitness).
Self-regulated (SOC?), non-deterministic.
Parameter g replaces pm
GECCO08 - AtlantaFernandes, Merelo, Ramos and Rosa – “Sandpile Mutation GA”
Future work
Change sandpile structure. Small-world, scale-free
Compare SMGA with SORIGA
Study SMGA response to different g.
Design different DOPs.