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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Swarm Intelligence:An Introduction
Nathan Bell
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
What is Swarm Intelligence?Why is Swarm Intelligence Interesting?
Introduction to Swarm Intelligence
This presentation will cover the following topics:What is Swarm Intelligence?Origins of Swarm IntelligenceCore ConceptsApplications and SI based algorithms
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
What is Swarm Intelligence?Why is Swarm Intelligence Interesting?
What is Swarm Intelligence?
What is “Swarm Intelligence” (SI)?http://www.youtube.com/watch?v=jEGV4ZSP22A
“The collective behaviour of a decentralized, self-organizedsystem”What is “Decentralization”?What is “Self-Organization”?
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
What is Swarm Intelligence?Why is Swarm Intelligence Interesting?
What is Swarm Intelligence?
What does this definition mean to us?System consisting of a population of “agents”Simple Interactions between agentsLeading to complex high-level behaviour
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
What is Swarm Intelligence?Why is Swarm Intelligence Interesting?
Why is Swarm Intelligence Interesting?
Why is SI interesting to us?Framework for decentralized and scalable problem solvingUnique perspective for addressing many problems
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
What is Swarm Intelligence?Why is Swarm Intelligence Interesting?
Why is Swarm Intelligence Interesting?
SI naturally lends itself to alternative computational models:Distributed modelsMassively parallel modelsFlexible, dynamic modelsRoboticsetc...
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
What is Swarm Intelligence?Why is Swarm Intelligence Interesting?
Why is Swarm Intelligence Interesting?
Useful in simulating many real-life systemsBehaviour of crowdsTraffic simulationSpread of diseaseetc...
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Origins, Observations of Nature
Where did the idea of SI originate?Inspired by the success of swarming creatures in natureIn particular: ants, termites and bees
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Ant Colonies
What’s so interesting about ants?One of the most successful species on the planetColonies can range from tens, to millions, of antsIndividual ants are extremely basic creaturesColony displays a complex structure and behaviourHow do they achieve this success?
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Colony Behaviour
A basic look at ant colony behaviour:Coordinated among specialized workersLabour tasks include:
DefenceFood CollectionBrood CareNest CleaningNest Construction
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Colony Behaviour
Ants achieve complex behaviours:Division of labour, adaptive task allocationPath finding and optimizationClustering and sortingStructure formationRecruitment for foraging and collective transport of food
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Colony Behaviour
How does the colony address these tasks?Ants, as individuals, are extremely basicNo single ant could plan these complex tasksThese tasks are easily completed by the ant colonyHow is this possible?
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Colony Behaviour
Colony behaviour:Labour within ant colonies is decentralizedNo central planning or controlLocal interactions between antsLaying of chemical pheromonesComplex behaviour arisesA swarm intelligence is displayed
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Ant Foraging
Ant Foraging:Ants effectively locate and exploit food sourcesFood sources are discovered by random explorationAnts discover efficient paths to known food sourcesAnts have no high-level knowledge of the situationHow do ants form these efficient paths?
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Ant Foraging
The key is pheromoneWhile searching for food, ant movement is largely random
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Ant Foraging
Ants will return to the colony with foodAnts with food will lay a trail of pheromone along their path
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Ant Foraging
Ants are sensitive to pheromoneThe pheromone deposited attracts other antsAnts encountering a pheromone trail will tend to follow itThese ants are more likely to reach the food source
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Ant Foraging
Over Time:More ants discover the food sourceMore pheromone is depositedExisting trails to the food source are strengthened
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Trail Optimization
At the high level:Ants will seem to choose more optimal pathsHow is this achieved?Combination of:
Randomness of movementNature of pheromone
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Trail Optimization
Properties of pheromone:Pheromones are chemicals subject to:
EvaporationDiffusionOther environmental effects
Pheromone weakens over timePheromone trails will dissipate and spread out
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Trail Optimization
Example:A food source has been discovered by two antsOne ant happens to take a more efficient path
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Trail Optimization
Each trail initially attracts a roughly equal number of antsAnts on the more efficient path arrive earlier
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Trail Optimization
Ants return, depositing additional pheromonePheromone trail is strengthened on return
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Trail Optimization
Ants following the more efficient trail have shorter tripsTrips are completed with a higher frequencyHigher frequency leads to stronger pheromone
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization
Trail Optimization
Ants will favour the stronger pheromone trailLess efficient trail slowly dissipatesThe colony has chosen the more efficient pathNo knowledge of the global situation!
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Core Concepts and PrinciplesDecentralizationSelf-OrganizationEmergent Behaviour
Core Concepts and Principles
What differentiates Swarm Intelligence from otherPopulation-based methods?
We must understand the core components:DecentralizationSelf-OrganizationEmergent Behaviour
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Core Concepts and PrinciplesDecentralizationSelf-OrganizationEmergent Behaviour
Decentralization and Self-Organization
Decentralization:Population of roughly homogeneous agentsControl is fully distributed among the populationNo central “brain” controlling the agentsEach agent has roughly the same level of influence
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Core Concepts and PrinciplesDecentralizationSelf-OrganizationEmergent Behaviour
Decentralization and Self-Organization
Self-Organization:Agents each act according to their own behaviourAgents communicate and interactCommunications can include:
Direct contactLocal exchange of informationLocal broadcastsStigmergyetc...
Global behaviour arises from the interactions of the agents
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Core Concepts and PrinciplesDecentralizationSelf-OrganizationEmergent Behaviour
Emergent Behaviour
“Emergent Behaviour” is the heart of SIA High-level behaviour of a population based systemMust arise from the local interactions within the populationSelf-organization of a decentralized populationTwo categories: Weak and Strong
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Core Concepts and PrinciplesDecentralizationSelf-OrganizationEmergent Behaviour
Emergent Behaviour
Weak emergent behaviour of a population:can be reduced to the behaviour of a single individual
Strong emergent behaviour of a population:can not be reduced to the behaviour of a single individual
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Core Concepts and PrinciplesDecentralizationSelf-OrganizationEmergent Behaviour
Emergent Behaviour
Weak Emergent Behaviour:Extremely commonCan be easily predicted by looking at a single individualOften simple to engineerFor example, if individuals simply “move forward”, thepopulation will, as a whole, move forward
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Core Concepts and PrinciplesDecentralizationSelf-OrganizationEmergent Behaviour
Emergent Behaviour
Strong Emergent Behaviour:Heart of SI and very interesting phenomena itselfHard to predict from the behaviour of an individualMay seem to “transcend” the capabilities of the individual“The whole is greater than the sum of its parts”It is often quite difficult to engineerFor example, path optimization of foraging ants
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
SI and Artificial LifeSI for Problem Solving
SI and Artificial Life
SI and Artificial Life:SI was first explored in the field of artificial lifeSimulations of swarming creatures“Boids” algorithmMotivation: learning more about emergent behaviours
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
SI and Artificial LifeSI for Problem Solving
SI based Algorithms
SI for problem solving:Relatively new fieldActive field of researchSI algorithms have been used to address many problemsMost commonly, optimization problems
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization
Ant Colony Optimization (ACO):Meta-heuristic method for combinatorial optimizationOne of the first SI algorithms for optimizationModelled after the foraging behaviour of antsACO finds good paths through graphsApplicable to a wide variety of problems
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
How ACO works
How does ACO work?ACO simulates the way ants communicate via pheromoneIteratively generates paths through graphsPheromone “deposited” on graph edgesPheromone levels effect edge choicesGood paths will emerge over time
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
How ACO works
Generic ACO Process:Generic ACO consists of a two phase loop
Edge selection phasePheromone update phase
Loop iterates until reaching some termination criteria
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Generic Ant Colony Optimization
Generic edge selection phase:Population of ants placed into the graphAnts move randomly through the graphMovement influenced by edge weights and pheromonePhase ends when all ants have created some kind of path
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Generic Ant Colony Optimization
Generic ant movement:For current node x with set of neighbours YChoose edge xy , y ∈ Y with probability pxy
pxy =(τxy )(ηxy )∑
yi∈Y (τxyi )(ηxyi )
ηxy - Contribution of edge weightτxy - Contribution of pheromone
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Generic Ant Colony Optimization
Generic pheromone update phase:Existing pheromone levels are reduced via "evaporation"
τxy = (1− ρ)τxyρ - Evaporation coefficient parameter
Ants return along the path taken through the graphAt each edge, ants deposit pheromonePheromone deposited according to: ∆τ k
Path of each ant is evaluated using heuristic∆τ k ∝ heuristic path value of ant k
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Travelling Salesman Problem (TSP):Given a set of citiesVisit each city exactly onceReturn to the originFind the tour with shortest total length
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Travelling Salesman Problem (TSP):NP-HardModelled as a complete graphEach node represents a cityEdges have weight equal to the distance between citiesACO is applied to find good solutions in polynomial time
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
ACO for TSP:Convert edge distance values into attractiveness value ηFor each pair of cities x , y :
ηxy = P/dxyP - Initial attractiveness parameterdxy - Distance between cities x and y
Each ant will form a valid tour during edge selectionPheromone deposited according to tour values
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Edge selection for TSP:Each ant randomly assigned some city as their originAnts travel through the graph visiting all citiesAnts blacklist cities they have previously visitedBlacklisted cities do not contribute to pxy valuesAfter visiting all cities, ants return to their origin
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Selecting the next city in the tour:Current city xSet of unvisited cities YMove to city y ∈ Y with probability pxy
pxy =(τxy )(nxy )∑
yi∈Y (τxyi )(nxyi )
If Y is empty, move to origin then stop
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
For example, consider the following situation:
A
B
C
D
E
F
Gd=8.00
d=16.12
d=14.14
d=14.56
T=3.21
T=1.10
T=0.81
T=2.98
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
X d η τ pA – - – – - – – - – – - –B – - – – - – – - – – - –C 14.56 2.98D – - – – - – – - – – - –E 14.14 0.81F 16.12 1.10G 8.00 3.21
Calculate η for each move:ηxy = P/dxy
P = 100 for example
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
X d η τ pA – - – – - – – - – – - –B – - – – - – – - – – - –C 14.56 6.87 2.98D – - – – - – – - – – - –E 14.14 7.07 0.81F 16.12 6.20 1.10G 8.00 12.5 3.21
Calculate p for each move:
pxy =(τxy )(ηxy )∑
yi∈Y (τxyi )(ηxyi )
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
X d η τ pA – - – – - – – - – 0B – - – – - – – - – 0C 14.56 6.87 2.98 0.28D – - – – - – – - – 0E 14.14 7.07 0.81 0.08F 16.12 6.20 1.10 0.09G 8.00 12.5 3.21 0.55
Choose the next moverandomly according to p:
C is chosen
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Move to C and repeat:
A
B
C
D
E
F
G
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
When all cities have been visited, a valid tour has been created
A
B
C
D
E
F
G
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Pheromone Update Phase:Each ant calculates the cost of their tourAnts travel along their tours depositing pheromoneFor each ant k :
∆τ k = Q/Lk
Lk - Value of ant k ’s tourQ - Pheromone attractiveness parameter
For each edge xy :τxy = (1− ρ)τxy +
∑k ∆τ k
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Consider this tour:
A
B
C
D
E
F
G
d=20
d=10.77
d=14.56
d=14.56
d=23.32d=10
d=18.44
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Pheromone deposit is calculated:
L = dAB + dBD + dDC + dCF + dFE + dEG + dGA
= 10.77 + 14.56 + 14.56 + 23.32 + 10 + 18.44 + 20= 111.65
Q = 100 for example
∆τ = Q/L= 100/111.65= 0.89
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Algorithm:For each edge xy :
Initialize ηxy = P/dxyInitialize τxy = 0
Run simulation loop until reaching termination criteriaEdge SelectionPheromone Update
Return best tour found as output
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Behaviour in early iterations:Little to no pheromoneAnts generate tours in a greedy wayClosest cities chosen with high probabilityHigh variety in TSP tours generated
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Behaviour in later iterations:Pheromone levels build up on popular edgesEdges used by “good” tours will have more pheromonePheromone levels begin to have greater influenceAnts generate similar tours using these “good” edges
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Early iterations:
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Later iterations:
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Ant Colony Optimization for Travelling Salesman
Convergence:
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
ACO for Classification
ACO finds applications in a wide variety of fieldsFor example: data miningRule-based classification
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Rule Based Classification
Given an object with a set of attributes:Assign a predefined class to the object
Based on a set of rulesFind a rule matching the attributesAssign a class using this rule
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Rule Based Classification
Classification Rule:A rule assigns a class with the form:
IF < conditions > THEN < class >IF A and B THEN 0IF (A and C) or D THEN 1etc...
A condition has the form:Attribute, Operator, Value
Colour = BlueWidth < 2etc...
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New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
ACO for Rule Based Classification
ACO can discover rules from a data set:These rules are of the form:
IF < condition AND condition AND ... > THEN < class >These conditions are restricted to “categorical” attributes
Attribute = Value
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Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Rules as Paths
Rules can be interpreted as paths through a graph:
Attribute A
Value 1
Value 2
Value 3
Attribute B
Value 1
Value 2
Value 3Attribute C
Value 1
Value 2
Value 3
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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Ant Colony OptimizationParticle Swarm Optimization
New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Rules as Paths
Vertices represent conditions:Attribute-value pair“Attribute = Value”
Edges represent “AND” relations between conditions
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New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Rules as Paths
IF A=1 AND B=2 AND C=3:
Attribute A
Value 1
Value 2
Value 3
Attribute B
Value 1
Value 2
Value 3Attribute C
Value 1
Value 2
Value 3
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Rules as Paths
IF B=1 AND C=1:
Attribute A
Value 1
Value 2
Value 3
Attribute B
Value 1
Value 2
Value 3Attribute C
Value 1
Value 2
Value 3
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New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Applying ACO
To apply ACO to this graph representation, we require:Initial edge weightsHeuristic for depositing pheromone
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New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Initial Edge Weights
Edge weights are assigned using the training set:Edges leading to an Attribute-Value pair are assignedweightings based on that Attribute-Value pairWeights are determined according to the normalizedShannon Entropy of that Attribute-Value pair within thetraining set
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Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Initial Edge Weights
Shannon Entropy:
H(W |Aa = Vav ) = −∑
w∈W
(P(w |Aa = Vav ) ∗ log2 P(w |Aa = Vav ))
W : Set of possible classesAa: Attribute a ∈ AVav : Value v ∈ Va
Represents the information gained by observing a givenAttribute-Value pair according to the training set
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Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Edge Weights
For edges leading to value v ∈ Va of attribute a ∈ A:
η =log2 |W | − H(W |Aa = Vav )∑
i∈A∑
j∈Vi
(log2 |W | − H(W |Ai = Vij)
)A normalized and inverted Shannon Entropy
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Pheromone Heuristic
New rules are evaluated using the “quality” measure:
Q =TP
TP + FN∗ TN
FR + TN
TP : True PositivesTN : True Negatives
FP : False PositivesFN : False Negatives
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Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
ACO Rule Generation Algorithm
Until termination:Initialize edge weights using the uncovered training setUntil convergence:
Generate a new rule as a pathDeposit and evaporate pheromone
Add the best-found rule to the final set of rulesDiscard training cases covered by the new rule
Output: A list of classification rules
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New Work
Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Convergence and Termination
Convergence occurs when either:The same rule is generated a specified number of timesThe maximum number of ants per iteration is reached
Termination occurs when:The number of uncovered cases reaches a threshold
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Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification
Result
ACO has successfully been applied to generate a set ofclassification rules from a training set.These rules can now be applied to perform classification.
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Particle Swarm Optimization
Particle Swarm Optimization (PSO):SI algorithm for real-valued, black-box optimizationDiscovered accidentallyOriginally a simulation of “social flocking” behaviourPopulation made of “particles” in the solution spaceCollision free “bird flocking” behaviour
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New Work
Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Real-valued Black-box Optimization
Real-valued optimization problems:Find the optimal input to some objective functionEach parameter is a real number
Solution space:Parameters define a “solution space”Each parameter corresponds to a dimensionA point in this space represents a function input
Black-box optimization problems:Objective function provided as a “black-box”Nothing is known about the functionMust learn about the function through evaluations
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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New Work
Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
How Particle Swarm Optimization Works
Particles:Particles exist as points in the solution spaceThese points represent input values to objective functionAssigned values by evaluating the objective function“Fly” through solution spaceCommunicate via broadcast
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
How Particle Swarm Optimization Works
Each particle is simply made up of:Position ~XVelocity ~VPersonal Best Point ~Pi
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
How Particle Swarm Optimization Works
Particle behaviour:Particles move according to their velocityVelocity is updated through accelerationsInertial weighting is applied to prevent explosive speedsAcceleration towards the particle’s best found pointAcceleration towards the swarm’s best found point
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
How Particle Swarm Optimization Works
0x1
-10 10
-10
0x2
10
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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Ant Colony OptimizationParticle Swarm Optimization
New Work
Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
How Particle Swarm Optimization Works
Acceleration towards personal best:Self influenceInfluence of the particle’s own knowledge
Acceleration towards global best:Social influenceInfluence of the swarm’s collective knowledgeGlobal best points are communicated via broadcast
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
How Particle Swarm Optimization Works
Accelerations:Randomly weighted according to parameters
φp - Personal influence parameterφg - Social influence parameter
Separate random weighting per dimensionMore “explorative”
Constant deceleration is appliedω - Inertial weighting parameterAllows stronger accelerations without explosive speedsPromotes convergence
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Effects of Parameters
Low personal influence, high social influence:Faster convergenceMore susceptible to local optimaExploitation > Exploration
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Effects of Parameters
High personal influence, low social influence:Slower convergenceLess susceptible to local optimaExploitation < Exploration
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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Ant Colony OptimizationParticle Swarm Optimization
New Work
Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Effects of Parameters
NO personal influence, high social influence:“Social only” swarmAll particles immediately converge to the best found pointBehaviour degrades to simple repeated local search
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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New Work
Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Effects of Parameters
High personal influence, NO social influence:“Self only” swarmEach particle acts completely independentlyBehaviour degrades to multiple local searchNo “swarm intelligence” present
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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New Work
Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Effects of Parameters
Parameter values:Matter of exploration vs. exploitationExploration required for more “difficult” problemsExploitation required for convergence and efficiency
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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New Work
Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Effects of Parameters
Parameters:Typically, both φp and φg are set to 2ω set to around 0.7Must be tweaked for each problem to get best resultsThis tweaking is often unintuitiveRequires knowledge of the solution space
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Particle Swarm Optimization Algorithm
Velocity and position update equations:For each dimension d :
Vd = ωVd + U[0, φp](Pid − Xd ) + U[0, φg](Pgd − Xd )
Xd = Xd + Vd
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Example Particle Update
Consider a particle with:X = [5,9]
V = [7,−5]
P = [0,2]
And global best found point:Pg = [10,11]
With PSO Parameters:ω = 0.7φp = 2, φg = 2
1
2
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Example Particle Update
Inertial weighting is applied:
V1 = ωV1
= 0.7 ∗ 7= 4.9
V2 = ωV2
= 0.7 ∗ −5= −3.5
1
2
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Example Particle UpdateRandom acceleration towards P:
V1 = V1 + U[0, φp](P1 − X1)
= 4.9 + U[0,2](0− 5)
= 4.9 + (1.1 ∗ −5)
= −0.6
V2 = V2 + U[0, φp](P2 − X2)
= −3.5 + U[0,2](2− 9)
= −3.5 + (0.2 ∗ −7)
= −4.9
1
2
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Example Particle UpdateRandom acceleration towards Pg :
V1 = V1 + U[0, φg](P1 − X1)
= −0.6 + U[0,2](10− 5)
= 4.9 + (0.4 ∗ 5)
= 6.9
V2 = V2 + U[0, φg](P2 − X2)
= −4.9 + U[0,2](11− 9)
= −4.9 + (0.9 ∗ 2)
= −3.1
1
2
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Example Particle Update
Update position:
X1 = X1 + V1
= 5 + 6.9= 11.9
X2 = X2 + V2
= 9− 3.1= 5.9
1
2
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Particle Swarm Optimization Algorithm
Algorithm:Initialize particles:
Random positionRandom velocityEvaluate initial positions
While termination criteria not met:Update Pg via communicationFor each particle i :
Update velocityUpdate positionEvaluate new positionUpdate Pi
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Swarm Behaviour
Behaviour in early stages:Particles will fly through the solution spaceOvershoot the global and personal best pointsArc and loop around these pointsCoarse-grained searchSwarm finds “good” areasParticles slow over time, swarm begins to converge
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Swarm Behaviour
Behaviour in later stages:Stagnation of global best point leads to convergenceParticles gather around this point, slowing over timeFine-grained search of the “good” areaSwarm finds the best point within the “good” area
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Benefits of Particle Swarm Optimization
Benefits of PSO:Very simple to implementNo costly computationsEasily extendable to problems of any dimensionEffective on difficult, noisy problemsCan be applied to any black-box real-valued functionCan even be used in meta-optimizationPSO finding optimal parameters for PSO on some problem
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Example Application
PSO can be applied to solve a problem if:A solution to the problem can be represented as a numberof real-valued parametersA solution’s quality can be represented by a single valueA function is provided which evaluates any given solution
As an example, consider the problem of maximizing thecoverage of a number of broadcast towers.
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Simple Broadcast Coverage Problem
Given a set number of broadcast towers:Assign each tower an x , y coordinateAssign each an amount of power
In order to:Maximize coverageMinimize power costs
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Simple Broadcast Coverage Problem
In order to apply PSO we must:Form a real-valued solution spaceRepresent the problem as a function which:
Assigns values to points in the solution spaceEvaluates the quality of a given solution point
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Tower Representation
Each tower has 3 parametersx coordinatey coordinatepower r
(x,y) r
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Solution Space
Multiple towers can be respresented:[Tower1,Tower2, ...,Towern]
[x1, y1, r1, x2, y2, r2, ..., xn, yn, rn]
A solution space with 3n dimensionsA particle’s position in this space describes:
The positions and power values of n towersA candidate solution to the tower coverage problem
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Example Point
For example, X = [3,3,5,8,11,3,13,4,2] corresponds to:
(3,3) 5
(8,11) 3
(13,4)
2
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Evaluating a Solution
PSO requires a function to evaluate these solutions:Total coverage of the towersPenalize for power costs
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Coverage
For simplicity:Houses withinbroadcast rangeEach house is worthsome set valueFor example:
10 per house
: Covered House
: Uncovered House
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Power Cost
For simplicity, the cost of a tower is:A flat initial cost, for example 100
If r <= 0, the tower is considered unusedNo initial cost in this case
Power cost scaling exponentially with rFor example: r2
Cost of n towers =∑n
i=0
{0, ri <= 0100 + r2
i , ri > 0
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Final Evaluation Function
The final evaluation function is then:
f (...) = HouseCoverage − PowerCosts
= 10 ∗ |Covered | −n∑
i=0
{0, ri <= 0100 + r2
i , ri > 0
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Example
f (7,7,7,21,10,6)
=10 ∗ |Covered |
−n∑
i=0
{0, ri <= 0100 + r2
i , ri > 0
=10 ∗ 34
−(
72 + 100)
−(
62 + 100)
=55
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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Example
f (13,10,13,21,11,0)
=10 ∗ |Covered |
−n∑
i=0
{0, ri <= 0100 + r2
i , ri > 0
=10 ∗ 49
−(
132 + 100)
− (0)
=121
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
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Ant Colony OptimizationParticle Swarm Optimization
New Work
Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage
Applying PSO
Applying PSO:Plug in the evaluation functionProvide appropriate parameter ranges
PSO will search for the optimal solution:According to the provided evaluation function
Accuracy of this function is essentialPSO will exploit errors in this function
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Thank You
Questions?
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IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
New Work
My research:Particle Field Optimization (PFO)Based on the PSO algorithmAbstraction of the “Bare-Bones” PSO model
Exploring:New high-level conceptNew behaviourNew avenues for development
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Bare-Bones PSO
Bare-Bones Particle Swarm (BBPSO):Abstraction of the core PSOSimplifies the particle update processRemoval of velocity and accelerationRetains roughly the same behaviour
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Bare-Bones PSO
From PSO to BBPSO:Observations of a single particle during swarm stagnationEach dimension shows a distinct bell-curve histogramCan a similar histogram be achieved in a simpler way?
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Bare-Bones PSO
New method of position update:Update particles according to Gaussian distributionDistribution constructed to mimic histogram bell-curveVelocity and accelerations discarded entirelyParticle movement abstracted to a random sampling
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Bare-Bones PSO
BBPSO particle update:For each particle i:
Pm =Pi+Pg
2For each dimension d :
Xid = N (Pmd , |Pid − Pgd |)
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Bare-Bones PSO
Result of BBPSO:Roughly the same high-level behaviourRoughly the same level of performanceParticles no longer “fly” through the spaceMuch simpler particle behaviourMore predictable, easier to analyze
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Abstracting the BBPSO Algorithm
Abstracting the BBPSO:It is possible to further abstract the BBPSORecall the components of the canonical PSO particle:
PositionVelocityPersonal best found pointCommunicated knowledge of global best
Each component is required to update the particle
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Abstracting the BBPSO Algorithm
Abstracting the BBPSO:The BBPSO algorithm removes the velocity componentPosition is updated by random samplingSampled distribution is independent of current positionCommunication of global bests depend on personal bestsCurrent position is used only to update the personal best
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Abstracting the BBPSO Algorithm
Particles without positions:Particles can be updated without storing a positionSample the random distributionEvaluate new pointUpdate the best found point if neededDiscard the new point
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Abstracting the BBPSO Algorithm
Effects of removing particle position:Identical behaviour to BBPSODifferent conceptsParticles no longer exist as explicit pointsEach particle defined by its best found pointParticle’s construct and sample a random distributionProbability of that particle existing at a given point
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
New Model
Moving forward with this concept:Individuals are no longer candidate solutionsEach represents a probability fieldSimple Gaussian distributionsNo longer “Particles”Instead, “Particle Fields”
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
New Model
At the population level:Population forms a complex probability fieldSum of simple individual fieldsProbability field of all possible particle locations
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
New Model
−4 −2 0 2 4
0.0
0.2
0.4
0.6
0.8
1.0
Population Probability Field
x
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sity
p1p2p3p4p5p6population
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
New Model
Population probability field:Probability field updated as individuals are updatedShows how PSO explores the solution spaceDemonstrates how PSO “learns”Similarities to a limited Bayesian learning process
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
New Model
Generating candidate solutions:Solutions have been separated from particlesMaintain a population of candidate solutionsSolutions generated by sampling complex populationdistributionFor each candidate solution:
Randomly select a particle field individualGenerate position from that individual’s distribution
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
New Model
Updating the particle field individuals:Each solution chooses an individual during generationIndividual’s are updated using associated solutionsSimilar to the standard PSO methodAdapted to handle any number of associated solutions
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
New Model
Moving forward again:Still roughly the same behaviour as the BBPSOSlightly more randomBiggest difference is the high-level conceptNew concept provides new avenues for development
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
New Model
Modifying the population distribution:Now possible to modify the population probability fieldConsider complex population distributionSum of simple individual distributionsApply simple weighting schemeDrastic change to population distributionFocus search to better areas?
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Weighting Schemes
Effects of weighting schemes:Introduce new behaviourChanges the population probability fieldChanges how the swarm “learns”Incorporate different information into the search
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Effects of Weighting
−4 −2 0 2 4
0.0
0.2
0.4
0.6
0.8
1.0
Population Probability Field
x
Den
sity
p1p2p3p4p5p6population
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Effects of Weighting
−4 −2 0 2 4
0.0
0.2
0.4
0.6
0.8
1.0
Weighted Population Probability Field
x
Den
sity
p1: 0.01p2: 0.4p3: 0.01p4: 0.1p5: 0.3p6: 0.08population
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Relative Population Sizes
Modifying relative population sizes:Model now consists of two separated populationsPossible to modify population sizes independentlyRelative sizes effects the high level behaviourMore particle field individuals:
May be more exploratoryLess particle field individuals:
Faster convergence
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Final PFO Algorithm
Final PFO Algorithm:Combining all these changes we have a distinct algorithmNew behaviour and perspectiveNew avenues for future development
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
PFO Simulation Step
For each candidate solution to be generated:Sample complex distribution defined by PF populationRandomly choose a particle field individualAccording to weighting valuesSample that individual’s simple distribution
For each PF individual:Choose best associated solutionUpdate personal best point if neededCalculate new weighting value
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Results
Preliminary experimental results:Weighting scheme for individuals:
Average value of candidate solutions generated by eachBetter information than just the best found?
Tests done with different ratios of population sizesCompared to BBPSOBetter performance on all test problems
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Introduction to Swarm IntelligenceOrigins, Observations of Nature
Core Concepts and PrinciplesSI based Algorithms
Ant Colony OptimizationParticle Swarm Optimization
New Work
IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm
Thank You
Questions?
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