Dyna Dynamic mic Analysis Analysis of the of the Co-evo Co - - PowerPoint PPT Presentation

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Dyna Dynamic mic Analysis Analysis of the

• f the

Co Co-evo evoluti lution

• n of Ex
• f Exac

act t Cove Covers rs

Jeffrey Horn

Nort Northern hern Mi Michigan Univer chigan Universit sity

Depa epartm rtment of ent of Ma Mathematics thematics and C and Computer

• mputer Science

Science Marq arque uette, tte, MI I USA USA

jhorn@ jhorn@nmu.edu nmu.edu http:/ http://cs cs.nmu.edu/ .nmu.edu/~jeff ~jeffhorn horn

EVOLVE EVOLVE Bour Bourlinst linster er Cas Castle, tle, Luxembourg Luxembourg May May 25 25-27, 2011 27, 2011

NERL

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• 1. The RFS Algorithm
• 2. Static Analysis (previous results)
• 3. Dynamic Analysis (new results)
• 4. Discussion

OUTL OUTLINE INE

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• 1. The RFS Algorithm

– The Shape Nesting Problem

OUTL OUTLINE INE

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Nestin ing g of Pieces es on Substrat ate Descript iption ion of Piec ece e Shape pe Description iption of Subs bstrat ate e Shape ape GA with h Resourc

• urce-de

defin ined ed Fitne ness Shari ring ng (RFS) S)

The The Prac Practica tical l Pro Problem blem of Shap

• f Shape

e Nes Nesting ting

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Exa Example mple Run Run, , Ar Arbitr bitrar ary y Sha Shape pes

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Mor More e Exa Example mple Run Runs

species_count ≥ 19

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species_count ≥ 4

generation 1

(one generation beyond the initial population )

species_count ≥ 1 (shows all species)

Evo Evolution lution Can Can Be Se Be Select lection ion Only Only

generation 209

(8 cooperative species)

species_count ≥ 36 species_count ≥ 2

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generation 709

(9 cooperative species)

generation 609

(almost 9 cooperative species)

species_count ≥ 22 species_count ≥ 20

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RFS RFS Can Can Out Outpe perfor rform m Some Some Comme Commerc rcial ial Sha Shape pe Nes Nesting ting Soft Softwar ware

(Horn, IJCCI, 2010)

RFS, 12 disks

ProNest, 11 disks (optimized settings)

ProNest, 10 disks (default settings) OptiNest, 11 disks (default settings)

OptiNest, 12 disks (optimized settings)

ArtCam, 11 disks

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• 1. The RFS Algorithm

– The Shape Nesting Problem – GA with RFS shared fitness

OUTL OUTLINE INE

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Fitn Fitnes ess s Ter Terms ms

fa

fb fab ( = fba )

fa is area of shaped piece “a”. It is used as the

• bjective fitness of

“a” in RFS. Similarly for fb . fab is area of overlap (intersection) of “a” and “b”. It is used to calculate the shared fitness of “a” and “b” in RFS.

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Sha Share red d Fitn Fitnes ess s Calcu Calculation lation

• Under RFS, shared fitness of an individual x is a

function of all individuals in the population:

• This can be re-written in terms of “species” y:

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What is a “species”?

• DEFINITION: An individual is a member of

the population

• DEFINITION: A species is the set of all copies
• f an individual
• DEFINITION: the species_count, nX , is the

number of copies of a species X in the current population

• IMPLICATIONS:

– If two individuals differ in their chromosomes, they are members of two distinct species – Any two members of the same species have identical placements on the substrate, and therefore overlap (compete) 100% – We seek the largest possible set of feasible (on substrate), cooperative (non-overlapping) species.

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Res Resou

• urc

rce-de defi fine ned d Fitn Fitnes ess s Sha Sharing ring

.







species Y Sh,X

XY X X

f n f f

AC AB A A

f n f n f n f f

C B A Sh,A

  

Shared fitness Example for three

• verlapping niches

Example for two

• verlapping niches

AB A A

f n f n f f

B A Sh,A

 

A B A B C

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• 1. The RFS Algorithm

– The Shape Nesting Problem – GA with RFS shared fitness – The Exact Cover Problem

OUTL OUTLINE INE

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Beg Beginning innings: s: Ni Nich ches es on

• n a Flat

a Flat La Land ndsc scap ape

• Horn’s 1997 dissertation
• Fitness sharing on a “hat” function (1D flat fitness)

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Con Conve verg rges es to to Exac Exact t Cov Cover er

• Pop. Size is 2560. Width of “hat” is 160. Niche

radius is 20. So nine “niches” exactly cover the hat.

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• Sets of non-competing (i.e., non-overlapped) species.

Evo Evolved lved Coo Coope pera ration tion

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Shape Shape Nest Nesting ing in in One One Dmension Dmension

• RFS is equivalent to Fitness Sharing* in one dimension
• Convergence to exact cover for 1D shape nesting

(* Goldberg & Richardson, 1987)

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RFS RFS Can Can Find Find Exa Exact ct Cov Cover ers s in 2D in 2D

Parallel Problem Solving From Nature VII (2002)

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Distribution

• f

Entire Population

RFS RFS wi with th Mut Mutat ation ion (Horn (Horn, , 20 2002 02)

Much smaller pop size (N=500). Some globals must be discovered by mutation (some are NOT in Initial pop.) Pop has converged

• n the 16

globals, with mutation still producing some “misfits”

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• 1. The RFS Algorithm
• 2. Static Analysis (previous results)
• 3. Dynamic Analysis (new results)
• 4. Discussion

OUTL OUTLINE INE

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Niching Niching Equ Equil ilibrium ibrium

• Equilibrium under selection:
• Under proportionate selection:
• Solving,
• Meaning that all shared fitnesses must be the same:

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Fo For Exa r Example, mple, The The Thr Three ee-Niche Niche Cas Case

Finite population size N

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The I The Infi nfinite nite Population Population Model Model

Divide both sides of equations by N Substitute px for proportion nx/N

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A A Sys Syste tem m of Linea

• f Linear

r Equ Equat ations ions

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Intr Introd

• duc

ucing ing Pro Prope pert rty y I

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Int Intro rodu ducin cing g Pr Prop

• per

erty ty II II

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App Applying lying Ass Assump umption tions s of

• f Pro

Prop.s p.s I I an and d II II

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In 2 In 200 008 8 Gen Gener eraliz alized ed to to Arb Arbitra itrary ry Nu Numb mber er of Sp

• f Spec

ecies ies

Horn, Parallel Problem Solving From Nature X (2008)

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• 1. The RFS Algorithm
• 2. Static Analysis (previous results)
• 3. Dynamic Analysis (new results)
• 4. Discussion

OUTL OUTLINE INE

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• For species u, we ask if ?
• Let substitute for
• In our specific case, we ask if

Con Conve verg rgen ence ce to to Equilibrium Equilibrium

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• Let be the current distance of species u

from equilibrium

• Then is the distance of u from

equilibrium in the next time step.

• We ask if this distance decreases with each time

step: ?

Dista Distanc nce e to to Equ Equil ilibrium ibrium

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• Species a and b form an exact cover:
• At equilibrium:
• Looking first at species a:
• We solve for feasible values of

(pa,pb,pc), and we find them.

• But we also find feasible values for

Fo For th r the Thr e Three ee-Niche Niche Cas Case

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Non Non-mon monot

• ton
• nic

ic Spe Species cies Con Conve verg rgen ence ce

• This means that species a sometimes moves closer to

equilibrium but sometimes moves further away.

• For example, with fac = fbc = ½ , and pa starting at

equilibrium, pa can move away from equilibrium.

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• Results for species a hold for species b, by symmetry
• For species c, must solve
• Solutions exist, e.g., with fac = fbc = ½ and the initial

values shown below.

No Non-mo mono noto tonic nic for for All All (Thr (Three ee) ) Sp Spec ecies ies

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• Note that in both examples of non-monotonic

convergence, the species moving away from equilibrium were the species closest to equilibrium.

• What about the species furthest from equilibrium?
• If the species furthest from equilibrium always

moves closer to equilibrium, and no other species moves as far away, then the population must converge to equilibrium.

• That is, all species in the population must converge

to equilibrium.

Pop Populat ulation ion Con Conve verg rgen ence ce

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• We define the maximum distance of any species to

its equilibrium proportion:

• This distance is strictly decreasing if and only if

More Precisely…

condition Φ

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• Let a be the species furthest from equilibrium. (At

least, no other species is further from equilibrium.)

• Then we ask if all species are closer than this, after

the next selection event.

Fo For th r the Thr e Three ee Niche Niche Exa Exact ct Cov Cover er Cas Case

a is no closer than c a is no closer than b Will a be closer ? Will b be closer ? Will c be closer ?

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• We find that this is always true, inside of the

simplex 0 < pa,pb,pc < 1, because

• there exists no solution to any of the three

equations:

• Thus condition Φ holds whenever a is the furthest

species.

• By symmetry, this analysis holds for b as well.
• A similar analysis finds that Φ holds for c as well.

Res Result ult

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• 1. The RFS Algorithm
• 2. Static Analysis (previous results)
• 3. Dynamic Analysis (new results)
• 4. Discussion

OUTL OUTLINE INE

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• Strictly Monotonic Convergence: Thus the

population under RFS is continuously converging to the unique equilibrium population that represents the exact cover.

• The strictly decreasing distance is measured as the

maximum difference of any species proportion and its equilibrium proportion.

• This population distance is equivalent to the

maximum Holder norm of the vector

Summa Summary ry

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Que Quest stions ions

• Maximum Holder Norm:
• What about other Holder norms ( h ≠ ∞ ) ?

– Manhattan distance ( h = 1 ) – Euclidean distance ( h = 2 ) – etc. – Does the result hold for other norms?

• What about other metrics? (e.g., Π pu )

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• Generalize to all three-niche situations

– i.e., not just exact cover, two-against-one – So far, can find no solutions to either direction of inequality (i.e., can prove neither condition Φ or its negation)

• Generalize to many-versus-many exact covers
• Other dynamic analyses

– e.g., rate of convergence

• Introduction of variation operators

– mutation – recombination

Oth Other er Ext Exten ension sions

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• Deterministic algorithm
• Origins in stochastic algorithms (e.g., cGA)
• But in process of modeling, discovered that the

model captured a powerful algorithm.

• So, how do we add stochasticity?

– selection – initialization – variation

Impli Implica cation tions s for Va for Variat riation ion

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th than ank yo k you vielen vielen Dan Dank me merc rcί vill villmols mols mer merci ci muc mucha has gr grac acias ias