Grant Dick Department of Information Science, School of Business, University of Otago, Dunedin, NZ ...
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Australasian Computational Intelligence Summer School, 2009
Spatially-Structured Evolutionary Computation
Grant DickDepartment of Information Science,
School of Business,University of Otago, Dunedin, NZ
http://www.otago.ac.nz/informationscience/
21/11/2009
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Overview
• Basic evolutionary algorithm (EA) model• Spatial models of evolution• Theory of spatially-structured EAs (SSEAs)• SSEA examples and applications• Conclusion
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Background
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Basic Terms
• Individual – candidate solution to problem• Population – collection of individuals• Deme – subset of population
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Evolutionary Algorithms (EAs)
• Stochastic population-based search methods• “Generate and test” metaphor• Inspired by principles of neo-Darwinian
evolution:– Variation (e.g. mutation)– Fitness– Selection (“Survival of the fittest”)
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Basic EA Model
pop = generate and evaluate random populationwhile not done do
parents = selection(pop)gen = recombine(parents)evaluate(gen)pop = replacement(pop gen)
end while
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Concepts of Evolution
• Mutation• Selection:– Takeover time– Probability of fixation to mutant
• Genetic Drift:– “Drunkard’s Walk”– Fixation through chance in a finite population
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Space
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Populations in Space
• Populations evolve through time and space:– Spatial structure of populations often ignored
• Space divides a population and provides:– Clines (Gradients)– Local competition for resources– Reproductive isolation– Restricted gene flow
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Spatial Models of Speciation
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Allopatric Parapatric Sympatric
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Spatially-Structured Evolutionary
Algorithms (SSEAs)
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Spatial Structure vs. Parallel Exec.
• Spatial structure does NOT imply parallel execution:– Parallel execution (e.g. Fitness evaluation)
needs no structure (e.g. master-slave model)• However, spatial structure provides logical
framework for parallelisation
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SSEAs
• Inspired by models of biological evolution• Limit interactions between individuals
through topological constraints• Two main approaches:– Island models– Diffusion models
• Basic EA can be thought of as a special case SSEA (called a panmictic EA)
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Island Models
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Island Model EAs
• Multiple population extension to simple EA• Basic operators unchanged:– e.g. selection, recombination, mutation
• Introduction of migration operators
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Island Model EAs
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Islands
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Island Model EApop = generate and evaluate random populationwhile not done do
for each islandimmigrants = select_migrants(pop, topology)insert_immigrants(island, immigrants)
nextfor each island
/* perform simple EA iteration within island */next
end while
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Migration
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Migration
• Exchange of individuals between islands– Promote restoration of intra-island diversity
• Parameters:– Topology– Rate (how often, how many)– Selection strategy (e.g. best or random)– Replacement strategy– Copy or true exchange?
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Migration Topologies
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Fully-Connected Stepping Stone Directed
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Punctuated Equilibrium
• Islands converge quickly:– Long periods of stasis
• Migration disrupts equilibrium:– Introduces diversity– Temporary drop in
fitness
• Disruption allows evolution to continue
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Island Model EAs
Benefits• Easily parallelised on
cluster computing platforms
• Can help prevent premature convergence
• Can implement local policies within islands:– e.g. Injection island EAs
Challenges• Many parameters:
– Topology– Migration rate– Migration size– Migration selection– Migration replacement
• Parameter interaction:– e.g. Little and frequent– e.g. Large and infrequent
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Diffusion Models
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Diffusion Model EAs
• “Isolation by distance” extension to EAs• Basic operators unchanged:– e.g. selection, recombination, mutation
• Introduction of overlapping subpopulations:– a.k.a. demes or neighbourhoods
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Diffusion Model EApop = generate and evaluate random populationwhile not done do
for each location in spacedeme = construct_deme(pop, location)parents = selection(deme)offspring = recombine(parents)evaluate(offspring)insertion(offspring, location)
nextend while
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Diffusion Model Topologies
• Regular (common):– e.g. Ring, ladder, torus
• Irregular (uncommon):– e.g. Scale-free, small-
world, random
• Single individual at each location:– Contrast with island
models
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Topology Representation
• Graph representation:– Locations as vertices– Edges relate “close” locations
• Matrix representation:– NxN matrix– Entry in matrix represent shared connection
between locations:• 1 = relationship present (connected)• 0 = no relationship (not connected)
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Topology Representation
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 01 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 02 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 03 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 14 1 0 0 0 1 1 0 1 1 0 0 0 0 0 0 05 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 06 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0 07 0 0 0 1 1 0 1 1 0 0 0 1 0 0 0 08 0 0 0 0 1 0 0 0 1 1 0 1 1 0 0 09 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 0
10 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 011 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 112 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 113 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 014 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 115 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1
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0 1 2
4 5 6
8 9 10
12 13 14
3
7
11
15
0
4
8
12
12 13 14 15
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Demes (Neighbourhoods)
• Define boundaries of local interactions:– Selection/competition– Reproduction
• Each individual “belongs” to multiple demes
• Ratio between deme size and population size defines spatial characteristic
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0 1 2
4 5 6
8 9 10
12 13 14
3
7
11
15
Deme of location 10 = {6, 9, 10, 1, 14)(von Neumann neighbourhood)
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Deme Construction
/* construct deme at location x *//* connectivity matrix M */deme = For each location y
If Mxy == 1 then deme = deme individualxy
Next
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Example Deme Structures (Ring)
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D=1 D=2 D=3 D=6
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Example Deme Structures (Torus)
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“Compact” “Diamond” “Linear”
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Offspring Placement
• Offspring are placed within the population within the same deme as their parents
• Example strategies:– “Replace always” – offspring always replace
the individual that resides at the current location
– Elitism – offspring replace the current individual when they are strictly better, otherwise the current individual remains
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Properties ofDiffusion Models
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More Basic Terms
• Gene – single coded piece of information• Locus – position of gene in individual• Allele – unique value for a gene• Ploidy – number of complete sets of genes• Homozygous – same alleles (by value) for a
gene at a given locus• Heterozygous – different alleles (by value)
for a gene at a given locus
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Behavioural Changes of SSEAs
• Mutation – none, same as for simple EA• Genetic drift• Selection
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Drift in SSEAs
• Local divergence:– Produces homogeneous subpopulations that
eventually grow/die out• Measurement of interest:– Time to loss of variation in population (fixation
time)
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Wright-Fisher Model of Drift
• Use Markov chain and first-step analysis– Compute the fundamental matrix of transition
probabilities• Assumption: populations identical by
proportion of alleles• Assumption: transition between states is
binomial:–
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pij= ( )Nj ( )N - i
NN-j( )i
Nj
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Wright-Fisher Model and SSEAs
• Consider these two distinct populations:
• Wright-Fisher models assumes they are identical in terms of transition probability:– Need to model each unique population state!
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Extended Wright-Fisher Model
• Model transition from one unique (by placement) population to another:– Requires 2N 2N matrix (rather than N N)
• Gives exact model for behaviour:– Computationally prohibitive once N > 11
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Random Walk in Ring Structures
Raw fixation timeAfter subtracting “panmictic” fixation time
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Random Walk in Toroidal Structures
Raw fixation timeAfter subtracting “panmictic” fixation time
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Random-Walk Model(Ring Population Structure)
• Model drift behaviour on rings• Model drift as two components:– Contribution from “panmictic” fixation time– Contribution from “spatial effect”
• Time to fixation then becomes:– ring = panmixa + X
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Random Walk Model of Gene Flow
• Alleles enter and leave neighbouring demes
• Eventually a specific allele will enter the deme furthest away from its origin:– Distance b will be N/2– Call this an “absorbing
state”
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Source: Dick & Whigham (2005), 2005 IEEE Congress on Evolutionary Computation , pp. 1855-1860
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Drift with Multiple Loci
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Source: Dick & Whigham (2005), 2005 IEEE Congress on Evolutionary Computation , pp. 1855-1860
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Random Walk Model
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Source: Dick & Whigham (2005), 2005 IEEE Congress on Evolutionary Computation , pp. 1855-1860
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Selection in SSEAs – Takeover Time
• “The [...] time it takes for the single best individual to take over the entire population”
• SSEAs reduce selection intensity:– Quality of selection (i.e.
prob. of fixation) not measured.
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Source: Giaocobini et al. (2005), IEEE Transactions on Evolutionary Computation 9(5) , pp. 489-505
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Probability of Mutant Fixation
• Assume that a mutant is introduced into population with fitness f=1+r (r [-1:1])– What is the chance that this mutant will fix in
the population?• Can be modelled through a Moran process:– Continuous birth/death model
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Moran Process
• Pick individual a at random (pa = 1/N)
• Pick individual b with probability proportional to fitness (pb = fb/f)
• Remove a from population
• Create duplicate of b
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ab
Source: Moran (1962), The Statistical Processes of Evolutionary Theory . Oxford, Clarendon Press
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Moran Process in SSEAs
• Pick individual a at random (pa = 1/N)
• Construct deme around a• Pick individual b from
within deme with probability proportional to fitness (pb = fb/f)
• Replace a with b
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b
a
Source: Whigham and Dick (2008), Genetic Programming and Evolvable Machines 9(2) , pp. 157-170
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Moran Process in SSEAs
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Source: Komarova (2006), Bulletin of Mathematical Biology 68, pp. 1573-1599
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Behavioural aspects of SSEAs
• Selection quantifiably less but qualitatively similar to panmictic EA– Qualitative change minor (negligible?)
• Drift neither quantitatively nor qualitatively similar to panmictic EA– Drift promotes local divergence– May be useful for discovering multiple
solutions (more later ...)
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Applications of SSEAs
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Classical Function OptimisationSource: Mühlenbein et al. (1991), Proc. 4th Intl. Conf. Genetic Algorithms, pp. 271-278
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55Australasian Computational Intelligence Summer School, 2009
Multimodal Optimisation
• Problems may present several potential solutions:– Of same or varying fitness
• EAs typically return a single solution
• Ideally, EAs should return multiple useful solutions:– Within a single run!
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56Australasian Computational Intelligence Summer School, 2009
Global Multimodal Search Approach – Fitness Sharing
• Frequency-dependent selection:– e.g. Resource
contention
• Reduce fitness of common solutions
• Stabilise against selection and drift
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57Australasian Computational Intelligence Summer School, 2009
Multimodal Search using SSEAs
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Unequal Peaks
Equal Peaks
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58Australasian Computational Intelligence Summer School, 2009
Parapatric Speciation Revisited
• Parapatric speciation results from:– Reproductive isolation through distance– Divergence through adaptation to local
environment
• Most SSEAs promote isolation by distance• Local environments largely ignored!
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59Australasian Computational Intelligence Summer School, 2009
Gradient-Based SSEAs
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Source: Dick & Whigham (2006), 6th Intl. Conf. Simulated Evolution and Learning , pp. 505-512
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60Australasian Computational Intelligence Summer School, 2009
Gradient-Based SSEAs
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Source: Dick & Whigham (2006), 6th Intl. Conf. Simulated Evolution and Learning , pp. 505-512
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61Australasian Computational Intelligence Summer School, 2009
Gradient-Based SSEAs
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Source: Dick & Whigham (2006), 6th Intl. Conf. Simulated Evolution and Learning , pp. 505-512
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62Australasian Computational Intelligence Summer School, 2009
Local Sharing Methods
• Incorporate fitness sharing within the demes of an SSEA:– Alternatively weight sh(dij) according to
topological distances between i and j• Control elitism through shared fitness• Introduce localised stabilising selection
pressure:– Help preserve lower-valued desired optima
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63Australasian Computational Intelligence Summer School, 2009
Local Sharing Methods
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Maximum Peak Ratio performance (higher=better)
Source: Dick & Whigham (2008), 7th Intl. Conf. Simulated Evolution and Learning , pp.452-461
Equal-Valued Peaks Function Unequal-Valued Peaks Function
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64Australasian Computational Intelligence Summer School, 2009
Local Sharing Methods
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Chi-Square-Like performance (lower=better)
Source: Dick & Whigham (2008), 7th Intl. Conf. Simulated Evolution and Learning , pp.452-461
Equal-Valued Peaks Function Unequal-Valued Peaks Function
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65Australasian Computational Intelligence Summer School, 2009
Multiobjective Optimisation
• Selection based on multiple (conflicting) objectives
• Aim: evolve a covering set of non-dominated solutions
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66Australasian Computational Intelligence Summer School, 2009
Gradient-Based EMO
• Apply gradient of objective weightings
• Fitness then becomes sum of individually weighted objectives:– f(i,x,y) = w1(x,y)o1(i)
+ w2(x,y)o2(i)
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10 w1
01 w2
Source: Kirley (2001), 2001 IEEE Congress on Evolutionary Computation , pp. 949-956
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67Australasian Computational Intelligence Summer School, 2009
Algorithm 1 Algorithm 2 Comparison
T1MEA SPEA [100, 0]
MEA PAES [52.1, 0]
T4MEA SPEA [55.5, 0]
MEA PAES [96.3, 0]
MEA – Gradient-Based EMO
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Source: Kirley (2001), 2001 IEEE Congress on Evolutionary Computation , pp. 949-956
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68Australasian Computational Intelligence Summer School, 2009
Bloat Control in Genetic Programming
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69Australasian Computational Intelligence Summer School, 2009
Bloat Control via SSEAs
• Bloat is increase in program size:– Without corresponding fitness increase
• Elitist replacement known to reduce bloat:– But elitism stagnates search in panmictic EAs
• Inbreeding thought to reduce bloat:– Small demes in SSEAs promote inbreeding
• Solution: Spatial Structure + Elitism (SS+E)
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70Australasian Computational Intelligence Summer School, 2009
Bloat Control via SSEAs
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Source: Whigham and Dick. (TBA), IEEE Transactions on Evolutionary Computation (in press)
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71Australasian Computational Intelligence Summer School, 2009
Bloat Control via SSEAs
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Source: Whigham and Dick. (TBA), IEEE Transactions on Evolutionary Computation (in press)
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72Australasian Computational Intelligence Summer School, 2009
Parameter Control via SSEAs
• Terrain-based genetic algorithm (TBGA):– Gradient of parameter
space
• Each region specifies unique combination of crossover/mutation rate
• Hopefully, the ideal combination ofparameters is present in terrain
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2
1
3
6
4
5
7
8
0.050.00 0.10 0.250.15 0.20 0.30 0.35
Mutation Prob.#
Cros
sove
r Po
ints
Mutation rate = 0.15# crossover points = 6
Source: Gordon et al. (1999), Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 1999) , pp. 229-235
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73Australasian Computational Intelligence Summer School, 2009
Predator-Prey SSEAs
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• Sparsely populated space:– Individuals “roam” the
landscape
• Two “species”:– Predators, for selection– Prey, candidate solutions
for given problem
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74Australasian Computational Intelligence Summer School, 2009
Predator-Prey SSEAs
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Real-parameter function optimisation
Source: Li & Sutherland (2002), 4th Intl. Conf. Simulated Evolution and Learning , pp.76-80
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75Australasian Computational Intelligence Summer School, 2009
Predator-Prey SSEAs
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Multiobjective optimisation (predators targetspecific objectives, thus promoting coverage)
Source: Li (2003), Evolutionary Multi-Criterion Optimization, Second International Conference (EMO 2003) , pp. 207-221
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Conclusion
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Recommendations for SSEAs
• Torus topology good in general• Use strictly better elitism in all cases• Local selection method not so important• Use crossover probability of 1.0• Increase mutation probability slightly
compared to panmictic EA
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78Australasian Computational Intelligence Summer School, 2009
Possible Research Areas in SSEAs
• Drift/Selection in Non-Linear Structures• Scalability of Gradient-Based SSEAs to
higher-dimension problems• Applicability of alternative population
structures:e.g. scale-free
• Implementation of SSEAs on GPUs
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79Australasian Computational Intelligence Summer School, 2009
Thank you
Questions?
[email protected]://www.otago.ac.nz/informationscience/
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