COSC 3P71 Particle Swarm Optimization (PSO) Brock University - - PowerPoint PPT Presentation

COSC 3P71

Particle Swarm Optimization (PSO)

Brock University

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Swarm Intelligence

Recall Swarm Intelligence in general, and Ant Colony Optimization in specific. What do we remember? Biologically inspired (yawn) etc

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Particle Swarm Optimization

(PSO)

Inspired by flocking birds and schooling fish Develops multiple solutions in parallel Provides both elements of independent exploration and social cooperation/collaboration

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Particle Swarm Optimization

So... ?

More practically, it’s an optimization algorithm that seeks to find a vector of floating point values that best solves some N-dimensional target function

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The Swarm

swarm consists of multiple particles flying independently through search space each particle acts as a separate very simple agent we’ll better define a ‘particle’ in just a bit

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particles...

find new solutions that are ‘similar’ to their current solutions at the moment usually have a tendency to stay in motion search for solutions with awareness of past personal success search for solutions with consideration of progress made by the collective etc.

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so... what IS a particle?

each has a position: single solution; analogous to chromosome velocity: tendency to change position (and thus solution) at each step/increment neighbourhood: particles with which a particle collaborates

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searching

Updating the position (and thus solution) is trivial:

• x′ =

x + v intelligence normally comes from velocity update rule change the manner by which the velocity is updated, and you change the means of training the position (candidate solution)!

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initialization

All positions and velocities are randomized (per dimension) Positions may be chosen anywhere within the expected bounds of the search space

◮ We’ll talk a bit more about these bounds later

Choosing velocity bounds is trickier, and mostly isn’t standardized

◮ I’m partial to half the bounds of the search space (or less) ◮ Obviously the magnitude of the velocity bounds is strictly positive, but

the selectable initial velocity components should be permitted to be negative

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Conventional Velocity Update Rule

• v′ = ω

v + c1r1( xb − x) + c2r2( xgb − x)[+c3 r] The +c3 r is optional The rs are at least generated per particle, per iteration

◮ Optionally, they may be per dimension, per particle, per iteration ◮ i.e.

r

◮ Values typically range from 0 to 1 Brock University ((PSO)) Particle Swarm Optimization 10 / 23

Explanation of Terms

ω v: Inertia (or momentum) c1( xb − x): Cognitive component — tendency to drift back towards ‘past personal glory’ c2( xgb − x): Social component — tendency to drift towards global or neighbourhood best thus far c3 r: Explorative/random — to discourage stagnation

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Selection of Parameters

General decent initial guesses

ω — less than 1 (obviously) c1 and c1 — varies, but 2 and 2 isn’t unheard of Of course, all are normally determined empirically, and are normally static, but can be dynamic, or may be trained by another algorithm

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Neighbourhoods

Remember that the social component dictates the likelihood that a particle will rely on the work of its peers — those other particles within its neighbourhood. The simplest neighbourhood is the entire swarm

◮ That might inhibit exploration, and cause premature convergence

Alternatively, you can choose a neighbourhood size and a mechanism for choosing neighbours

◮ The initial question is whether you should choose the neighbourhood

• nce, at the beginning of the algorithm, or if you should decide each

iteration

⋆ If decided per-iteration, it will typically be based on Euclidean proximity ⋆ It’s up to you whether swarm associations are symmetric or not — e.g.

whether pA ∈ sB = ⇒ pB ∈ sA

⋆ When you choose your neighbourhoods probably won’t really matter,

because being assigned to the same neighbourhood a priori will tend to also encourage proximity

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Bounds and Restrictions

Solutions often have upper/lower bounds When this is true, it must be enforced on the positions Particles can simply be unable to exceed the bounds, may wrap to the

• ther side, or may bounce off

Note that, when such restrictions don’t exist, there may be some risk

• f the particles just flying away, never to be seen again

As mentioned earlier, velocities should also be “clamped” to some maximum magnitude.

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Limitations of Canonical PSO

First and foremost, PSO is not suitable for problems with “holes” in the search space

◮ e.g. x between 0 and 1, or between 2 and 3, but not betwen 1 and 2 ⋆ Of course, for a problem like this, one could simply adapt the

transcription

⋆ e.g. the aforementioned range could be mapped to a [0..2]; continuous

within the particle’s space, but discontinuous when evaluated for fitness

Not suitable for problems with highly constrained choices

◮ e.g. What’s legal in dimension 1 depends on what was chosen in

dimension 0

◮ This can be particularly problematic if trying to adapt to a

combinatorial problem

Not really appropriate when adjacent values in the search space aren’t “similar” in the solution space

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Limitations of Canonical PSO

One final thought...

For problems/functions that can scale into larger versions, dimensionality might eventually become a limiting factor. Remember that our fitness function will normally just give us a single value Declaring one particle’s 300-dimensional position better than another particle’s 300-dimensional position might not ascribe much significance to each individual dimension

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Benefits

PSOs are easy to code and (outside of possibly the fitness function) very fast to execute The lack of a need for differentiability

◮ Unlike, for example, gradient descent!

PSOs are easy to code and fast to execute There are relatively few parameters to choose

◮ Consequently, there may be less bias from the user/experimenter

PSOs are easy to code and fast to execute

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Applications

The most common application is function optimization Minimization/maximization is an example It can also include things like training weights for ANNs

◮ As such, it can be a viable alternative to BackProp Brock University ((PSO)) Particle Swarm Optimization 18 / 23

Applications

Hmmm...

Could we apply this to a combinatorial problem without changing the continuous nature of the pso itself? Could we apply it to something like TSP? Could we apply it to a highly-constrained problem, like, say, two-connected networks with bounded rings?

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Variations

Modifications to inertia

◮ The easiest is to start momentum high, and then gradually it reduce

• ver time

⋆ Compare to simulated annealing ⋆ It might help to avoid local minima, and then allow for refinement ◮ Personally, I’m partial to oscillation

Binary or combinatorial PSO

◮ If velocity is the propensity to change position, then one could relax the

definitions to include combinatorial versions

◮ Asterisk*

I even saw one paper that used special operators like crossover and mutation...

◮ ...wait a minute...

Anyhoo, in general, outside of screwy representations and such, much as the ‘intelligence’ comes from the velocity rule, that’s also where it’s easiest to introduce novel variations

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Example

Minimization

Let’s take a look at an example! Suppose we want to minimize the function: f (x, y) = cos(6.28×(3x+2y))+cos(6.28×(2x+3y))−2×sin(6.28×(x+y)) within the range of [−0.4..0.4] Just... because of reasons. Okay?

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Example

Additional samples

You can find some additional interesting test functions here:

http://www.sfu.ca/~ssurjano/optimization.html

Particularly neat ones are: Ackley Function Schaffer Function Eggholder Function Cross-in-Tray Function Langermann Function Rastrigin Function Also check out the Hartmann functions listed. They go up to 6D! That’s 4 more Ds!

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Questions?

Comments?

Catchy tunes?

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