AI AI Department of Computer Science University of Calgary CPSC - - PowerPoint PPT Presentation

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 1

Particle Swarm Optimization Particle Swarm Optimization (PSO) (PSO)

Adaptive Swarms for Optimization Adaptive Swarms for Optimization

AI AI

Christian Jacob

Department of Computer Science University of Calgary

CPSC 565 — Winter 2003

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 2

PSO: An Overview PSO: An Overview

• Developed by

– Russ Eberhart, Purdue School of Engineering and Technology, Indianapolis – Jim Kennedy, Bureau of Labor Statistics, Washington, DC

• A concept for optimizing non-linear functions using

particle swarm methodology

• Has roots in Artificial Life and Evolutionary Computation
• Simple concept
• Easy to implement
• Computationally efficient
• Effective on a wide variety of problems

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 3

Evolution of Concept and Paradigms Evolution of Concept and Paradigms

• Discovered through simplified social model simulation
• Related to bird flocking, fish schooling and swarming

theory

• Related to evolutionary computation:

– Genetic algorithms – Evolution strategies

• Kennedy developed the “cornfield vector” for birds

seeking food

• Bird flock became swarm
• Expanded to multi-dimensional search
• Incorporated acceleration by distance
• Paradigm simplified

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 4

Flocks, Herds, and Schools Flocks, Herds, and Schools

• Avoid collisions
• Match neighbours’ velocity and orientation
• Steer toward the center

Separation Alignment Cohesion

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 5

PSO Algorithm PSO Algorithm

• 1. Initialize population in hyperspace.
• stochastically assign locations and velocities
• 2. Evaluate fitness of individual particles.
• 3. Keep track of location where individual had its highest

fitness.

• 4. Modify velocities based on previous best and global (or

neighbourhood) best positions.

• neighbourhoods don’t change
• 5. Terminate if some condition is met.
• 6. Go to step 2.

Fly solutions through problem space …

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 6

Particle Swarm Optimization Particle Swarm Optimization

DEMO

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 7

Particle Swarms Particle Swarms

• Sociocognitive Space

– High-dimensional – Abstract: attitudes, behaviours, cognition – Heterogeneous with respect to evaluation (dissonance) – Multiple individuals

• Individual is characterized by

– Position = “mental state”: xi – Changes = velocity: vi

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 8

Particle Swarms: Particle Swarms: “ “Code Code” ”

• Individuals (particles) learn from their own experience:

– vi := vi + j() · (pi - xi) – xi := xi + vi

• xi: current position of individual i
• vi: current velocity of individual i
• pi: so-far best position for individual I
• (pi - xi): acceleration towards previous best
• j(): generates random positive number
• This formula, iterated over time, causes the individual’s trajectory to
• scillate around their previous best point pi in sociocognitive space.
• The velocity of individual i is stochastically adjusted depending on

previous successes.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 9

Particle Swarms with Neighbourhood Best Particle Swarms with Neighbourhood Best

• Sociocognitive space can contain many individuals that influence one

another:

– vi := vi + j1() · (pi - xi) + j2() · (pg - xi) – xi := xi + vi

• pg: previous best position in the population
• (pi - xi): acceleration towards previous best
• (pg - xi): acceleration towards global best
• j1(), j2(): generate random positive numbers
• Evaluate your present position.
• Compare it to your previous best and neighbourhood best.
• Imitate self and others.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 10

General PSO Update Algorithm General PSO Update Algorithm

• Global version:

– vid := wi vid + c1 j1() · (pid - xid) + c2 j2() · (pgd - xid) – xid := xid + vid

• d: dimension
• c1, c2 : positive constants

set exploration vs. exploitation

• w: inertia
• j1(), j2(): generate random positive numbers
• For neighbourhood version:

– Change pgd to pld .

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 11

The The “ “Drunkard Drunkard’ ’s Walk s Walk” ”

• The particle will explode out of control if it is not limited

in some way. Three methods are widely used:

• Vmax:

vi := vi + j1() · (pi - xi) + j2() · (pg - xi) if vi > Vmax then vi := Vmax else if vi < - Vmax then vi := Vmax

• Inertia weight a:

vi := a vi + j1() · (pi - xi) + j2() · (pg - xi)

• Constriction coefficient c:

vi := c (vi + j1() · (pi - xi) + j2() · (pg - xi))

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 12

Important Parameters: V Important Parameters: Vmax

max

• An important parameter in PSO; typically the only one

adjusted

• Clamps particles’ velocities on each dimension
• Determines “finteness” with which regions are searched:

– If too high, can fly past optimal solutions – If too low, can get stuck in local minima

• Set Vmax to dynamic range of the variables.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 13

Important Parameters: Inertia Weight Important Parameters: Inertia Weight a a

• Inertia weight a:

vi := a vi + j1() · (pi - xi) + j2() · (pg - xi)

• It seems possible that one can get rid of Vmax by setting a

equal to the dynamic range of each variable.

• Then a must be selected carefully and/or decreased over

the run.

• Hence, the inertia weight a seems to have attributes of the

temperature in simulated annealing.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 14

Evolutionary Computation & Particle Swarms Evolutionary Computation & Particle Swarms

• Culture as evolution (anthropology)
• Adaptation / learning
• Memetics
• Evolutionary epistemology
• Change vs. selection
• Fitness and dissonance
• Cooperation vs. competition
• Evolution = competitive struggle
• PS = cooperation inherent

PS = 5th EC paradigm?

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 15

Basic Principles of Swarm Intelligence Basic Principles of Swarm Intelligence

• Proximity principle:

– The population should be able to carry out simple space and time computations.

• Quality principle:

– The population should be able to respond to quality factors in the environment.

• Diverse response principle:

– The population should not commit its activities along excessively narrow channels.

• Stability principle:

– The population should not change its mode of behaviour every time the environment changes.

• Adaptability principle:

– The population must be able to change its behaviour mode when it’s worth the computational price.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 16

Adherence to Swarm Intelligence Principles Adherence to Swarm Intelligence Principles

• Proximity:

– N-dimensional space calculations carried out over a series of time steps

• Quality:

– Population responds to quality factors pbest and gbest (or lbest)

• Diverse response:

– Responses allocated between pbest and gbest (or lbest)

• Stability:

– Population changes state only when gbest (or lbest) changes

• Adaptability:

– Population does change state when gbest (or lbest) changes

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 17

Enhancements of PSO Enhancements of PSO

• Elitist concept from GA might be helpful in PSO.

– Carry global best particle into next generation?

• Could incorporate Gaussian distribution into stochastic

velocity changes.

– Variance might then be like inertia weight. – Put noise on decrease of inertia weights (better convergence)

• Could assign Vmax on a parameter-by-parameter basis.

– Analogous to controlling severity of mutation in GA & EP

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 18

GAs vs. PSO: Crossover GAs vs. PSO: Crossover

• Does not have crossover.
• Acceleration toward personal and global best is similar

concept.

• Particles midway between swarms also exhibit crossover

features.

• Recombination operator in evolution strategies may be

more analogous.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 19

GAs vs. PSO: Mutation GAs vs. PSO: Mutation

• GAs are not actually ergodic:

– A number of mutations probably required – Low fitness individuals will not survive selection – Probability of survival decreases geometrically with generations

• EP (for parameter optimization) is ergodic: can reach any

point in one jump

• PSO seems to fall between GA and EP

– any particle can eventually go anywhere

• PSO mutation-like behaviour is directional

– GA and EP are omni-directional

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 20

GAs vs. PSO: Selection GAs vs. PSO: Selection

• GA selection supports survival or the fittest

(when using elitist strategy)

• There is no selection in PSO

– All particles survive for the length of the run. – Number of particles does not change.

• PSO is the only “evolutionary algorithm” that does not

remove candidate population members.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 21

PSO as an Evolutionary Algorithm PSO as an Evolutionary Algorithm

• Distinctions among EC paradigms continue to blur.
• New hybrid PSO approaches will be emphasized.
• Practical PSO applications will be emphasized, in addition

to benchmarking.

• Focus on how the PSO paradigms work.
• Many more (PSO) hybrids to come …

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 22

PSO Example PSO Example

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 23

References References

• Kennedy, J. and R. C. Eberhart (2001). Swarm Intelligence.

San Francisco, Morgan Kaufmann Publishers.

• Kennedy, J. and R. C. Eberhart (2002). Tutorial on

Particle Swarm Optimization. 2002 World Congress on Computational Intelligence, Hawaii, USA.