The dynamics of the best individuals in co-evolution

Speaker: Ta-Chun Lien

Reference

Outline

• Background knowledge and motivation• Test functions and experimental result• Best-of-generation trajectories• Best-response functions• Three Hypothesis and experimental result• Conclusions

Coevolution

• An individual’s fitness is evaluated by other individuals in population

Coevolution(cont’d)

• Competitive co-evolutionary framework– To evolve behaviors in competitive environments.

• Cooperative co-evolutionary framework– To solve difficult function optimization problems.

Motivation

• Understand dynamics of CoEA• Give users insights to improve CoEA

performance

Test Functions

These landscapes are similar with respect to properties such as continuity, modality, ruggedness

Mechanisms

• Two populations, one evolving values for the x and the other evolving values for the y1. Real-valued representation2. Binary tournament selection3. Gaussian mutation operator with fixed sigma4. Single best collaboration strategy5. The two populations take turns in evolving

• Keeping a fixed budget in terms of number of evaluations

Experimental Results of offAxisQuadratic

Increasing population size and elitism improve the performance

Experimental Results of Rosenbrock

Increasing population size and elitism degrade the performance

Best-of-generation trajectories

• Plot the best individual of x and y populations of each generation.

• Two reasons:– The main concern of cooperative co-evolution

for optimization is the best individuals the algorithm produces.

– For easier understanding.

Best-of-generation trajectories(cont’d)

Vertical line: connecting an X generation with the following Y generationHorizontal line: connecting a Y generation with the following X generation

Experimental Results

Size 10 without elitism Size 10 with elitism Size 200 with elitism

Best-response curves

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Best-response functions

• The active population is trying to give the best response possible to the best individual in the frozen population

Suppose the problem function is

the active population is X and the y0 is the best individual in Y

X population is searching for an individual x*y0

such that

Define a function

RDDRDDf YXYX ,;:

),(min),( *oDxoy yxfyxf Xo

yXY xysponseXbestDDsponseXbest *)(Re,:Re

Best-response functions(cont’d) solve the equation for bestResponseX(y) solve the equation for bestResponseY(x) • For offAxisQuadratic bestResponseX(y)=-y/2 bestResponseY(x)=-x• For rosenbrock bestResponseX(y): solving the equation graphically by interpolation

0),(

y

yxf

0),(

x

yxf

Best-response curvesSize 200 with elitism

梗ㄍㄥ̌

Observations

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Size 100 Size 200 Size 10 with elitism

Observations(cont’d)

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Size 100 Size 200 Size 10 with elitism

Discuss

• Increasing the population size increased the precision

• For offAxisQuadratic, due to relative positions of the best-response curves, a deterministic system would advance like on a ladder

• A tradeoff between the accuracy of following the best-response curves and the number of steps taken along them

Discuss(cont’d)

• For rosenbrock, after the first or second step enter the region of almost-overlap and then be forced to take extremely small steps.

• With small population size, the algorithm’s best-of-generation trajectory take large steps.

• High accuracy doesn’t balance off a small number of steps

Test function

Test function

α=0 α=0.25 α=0.5

α=0.75 α=0.9 α=1

n=8

Best-response curves

Three hypotheses

• At low α, increasing population size and introduction elitism will have beneficial effects on performance

• As α increases, we will begin to see “curving” effects

• As α reached 1, increasing population size and introducing elitism will decrease performance

Experimental Resultspopulation sizing

α=0 α=0.25 α=0.5

α=0.75 α=0.9 α=1

Experimental Resultselitism

α=0 α=0.25

α=0.75

α=0.5

α=0.9 α=1

Conclusions

• A new methodology of analyzing the dynamics of CoEAs.

• Best response curves have a strong influence on co-evolutionary performance.

• Although formulas for the best-response curves will not be available for most practical applications, analyzing the trajectories of best-of-generation individuals will help infer their shapes.

THANKS FOR YOUR ATTENTION

謝 謝Please e-mail to epopvic@gmu.edu , if you have questions