L ECTURE 16: S WARM I NTELLIGENCE 2 / P ARTICLE S WARM O PTIMIZATION - - PowerPoint PPT Presentation

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May 24, 2023 166 likes •507 views

15-382 C OLLECTIVE I NTELLIGENCE - S18 L ECTURE 16: S WARM I NTELLIGENCE 2 / P ARTICLE S WARM O PTIMIZATION 2 I NSTRUCTOR : G IANNI A. D I C ARO BACKGROUND: REYNOLDS BOIDS Reynolds created a model of coordinated animal motion in which the

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15-382 COLLECTIVE INTELLIGENCE - S18

LECTURE 16: SWARM INTELLIGENCE 2 / PARTICLE SWARM OPTIMIZATION 2

INSTRUCTOR: GIANNI A. DI CARO

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BACKGROUND: REYNOLDS’ BOIDS

Reynolds, C.W.: Flocks, herds and schools: a distributed behavioral model. Computer Graphics, 21(4), p.25-34, 1987

Reynolds created a model of coordinated animal motion in which the agents (boids) obeyed three simple local rules:

Separation: steer to avoid crowding local flockmates Alignment: steer towards the average heading of local flockmates Cohesion: steer to move toward the average position

  • f local flockmates

https://www.youtube.com/watch?v=QbUPfMXXQIY

Field of view Play with Boids, PSO, CA, and … with NetLogo https://ccl.northwestern.edu/netlogo/

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BOIDS + ROOSTING BEHAVIOR

Kennedy and Eberhart included a roost, or, more generally, an attraction point (e.g, a prey) in a simplified Boids-like simulation, such that each agent: Eventually, (almost) all agents land on the roost What if:

  • roost = (unknown) extremum of a function
  • distance to the roost = quality of current

agent position on the optimization landscape

  • is attracted to the location of the roost,
  • remembers where it was closer to the roost,
  • shares information with its neighbors about its closest

location to the roost

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PARTICLE SWARM OPTIMIZATION (PSO)

  • PSO consists of a swarm of bird-like particles ➔ Multi-agent system
  • At each discrete time step, each particle is found at a position in the search space:

it encodes a solution point 𝒚 ∈ 𝛹𝑜 ⊆ ℝ𝑜 for an optimization problem with objective function f(𝒚): 𝛹𝑜 ⟼ ℝm (black-box or white-box problem)

  • The fitness of each particle represents the quality of its position on the optimization

landscape (note: this can be virtually anything, think about robots looking for energy)

  • Particles move over the search space with a certain velocity
  • Each particle has: Internal state + (Neighborhood ⟷ Network of social connections)

{~ x,~ v, ~ xpbest, N(p)}

  • At each time step, the velocity (both direction and speed) of

each particle is influenced + random, by:

  • pbest: its own best position found so far
  • lbest: the best solution that was found so far by the

teammates in its social neighbor, and/or

  • gbest: the global best solution so far
  • “Eventually” the swarm will converge to optimal positions
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NEIGHBORHOODS

Geographical Social Global

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VECTOR COMBINATION OF MULTIPLE BIASES

~ xpbest ~ xlbest − ~ xt !~ vt ~ xpbest − ~ xt ~ xt+1 ~ xt r2 · (~ xlbest − ~ xt) ~ vt r1 · (~ xpbest − ~ xt) ~ xlbest

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PARTICLE SWARM OPTIMIZATION (PSO)

~ r1 = U(0, 1) ~ r2 = U(0, 2)

ɸ are acceleration coefficients determining scale of forces in the direction of individual and social biases

element-wise multiplication operator

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VECTOR COMBINATION OF MULTIPLE BIASES

  • Makes the particle move in the same

direction and with the same velocity

  • Improves the individual
  • Makes the particle return to a previous

position, better than the current

  • Conservative
  • Makes the particle follow the best

neighbors direction

  • 1. Inertia
  • 2. Personal

Influence

  • 3. Social

Influence

Exploits what good so far Search for new solutions

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PSO AT WORK (MAX OPTIMIZATION PROBLEM)

Example slides from Pinto et al.

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PSO AT WORK

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PSO AT WORK

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PSO AT WORK

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PSO AT WORK

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PSO AT WORK

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PSO AT WORK

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PSO AT WORK

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PSO AT WORK

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PSO AT WORK

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GOOD AND BAD POINTS OF BASIC PSO

  • Advantages
  • Quite insensitive to scaling of design variables
  • Simple implementation
  • Easily parallelized for concurrent processing
  • Derivative free / Black-box optimization
  • Very few algorithm parameters
  • Very efficient global search algorithm
  • Disadvantages
  • Tendency to a fast and premature convergence in mid optimum points
  • Slow convergence in refined search stage (weak local search ability)
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GOOD NEIGHBORHOOD TOPOLOGY?

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GOOD NEIGHBORHOOD TOPOLOGY?

  • Also considered were:
  • Clustering topologies (islands)
  • Dynamic topologies
  • …
  • No clear way of saying which topology is the best
  • Exploration / exploitation dilemma
  • Some neighborhood topologies are better for local search others

for global search

  • lbest neighborhood topologies seems better for global search,
  • gbest topologies seem better for local search
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ACCELERATION COEFFICIENTS

  • The boxes show the distribution of the random vectors of the

attracting forces of the local best and global best

  • The acceleration coefficients determine the scale distribution
  • f the random individual / cognitive component vector 𝜚1 and

the social component vector 𝜚2

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ACCELERATION COEFFICIENTS

  • ɸ1 >0, ɸ2=0 particles are independent hill-climbers
  • ɸ1=0, ɸ2>0 swarm is one stochastic hill-climber
  • ɸ1=ɸ2>0 particles are attracted to the average of pi and gi
  • ɸ2>ɸ1 more beneficial for unimodal problems
  • ɸ1>ɸ2 more beneficial for multimodal problems
  • low ɸ1, ɸ2 smooth particle trajectories
  • high ɸ1, ɸ2 more acceleration, abrupt movements
  • Adaptive acceleration coefficients have also been proposed, for

example to have ɸ1, ɸ2 decreased over time (e.g., Simulated Annealing)

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ORIGINAL PSO: ISSUES

  • The acceleration coefficients should be set sufficiently high to

cover large search spaces

  • High acceleration coefficients result in less stable systems in

which the velocity has a tendency to explode!

  • To fix this, the velocity v is usually kept within the range [-vmax, vmax]
  • However, limiting the velocity does not necessarily prevent particles

from leaving the search space, nor does it help to guarantee convergence :(

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INERTIA COEFFICIENT

  • The inertia weight ω was introduced to control the velocity explosion
  • If ω, ɸ1, ɸ2 are set “correctly”, this update rule allows for

convergence without the use of vmax

  • The inertia weight can be used to control the balance between

exploration and exploitation:

  • ω ≥ 1: velocities increase over time, swarm diverges
  • 0 < ω < 1: particles decelerate, convergence depends on ɸ1, ɸ2

~ ~ + ~ r1 ~ ndd + ~ r2 ~ soc

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CONSTRICTION COEFFICIENT

  • Take away some ‘guesswork’ for setting ω, ɸ1, ɸ2
  • The constriction coefficient is an “elegant” method for preventing

explosion, ensuring convergence and eliminating the parameter vmax

  • The constriction coefficient was introduced as:
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FULLY INFORMED PSO (FIPS)

  • Best → Average: Each particle is affected by all of its K neighbors by

taking the average (personal best can or cannot be included)

  • The velocity update in FIPS is:
  • FIPS outperforms the canonical PSO’s on most test-problems
  • The performance of FIPS is generally more dependent on the

neighborhood topology (global best neighborhood topology is recommended)

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TYPICAL BENCHMARK FUNCTIONS

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PERFORMANCE VARIANCE

  • Minimization problems
  • Best solution found over

the iterations

  • Improvement in

performance differs according to the different strategic choices

  • No a single winner
  • Early stagnation of

performance can exist

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BINARY / DISCRETE PSO

  • A simple modification to the continuous one
  • Velocity remains continuous using the original update rule
  • Positions are updated using the velocity as a probability threshold

to determine whether the j-th component of the i-th particle is a zero or a one

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ANALYSIS, GUARANTEES

  • Hard because:

  • Stochastic search algorithm

  • Complex group dynamics

  • Performance depends on the search landscape
  • Theoretical analysis has been done with simplified PSOs on

simplified problems

  • Graphical examinations of the trajectories of individual particles

and their responses to variations in key parameters

  • Empirical performance distributions
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SUMMARY PSO

  • Inspired by social and roosting behaviors in bird flocking
  • Easy to implement, easy to get good results with “wise” parameter

tuning (but just a few parameters)

  • Computationally light
  • Exploitation-Exploration dilemma
  • A number of variants
  • A few theoretical properties (hard to derive for general cases)
  • Mostly applied to continuous function optimization, but also to

combinatorial optimization, and robotics / distributed systems

  • References:
  • Swarm Intelligence, J. Kennedy, R. Eberhart, Y. Shi, Morgan Kaufmann, 2001
  • Computational Intelligence, A. Engelbrecht, Wiley, 2007