Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite - - PowerPoint PPT Presentation

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite

Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel

Department of Computer Science

Slide 1/12 — July 20, 2016

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

UP-MO-CMA-ES in a nutshell

Population S of Individuals: (xi,σi,Ci) , i = 1,...

while Stopping criterion is not met do Select parent from S based on Hypervolume Contribution; Sample Offspring with Crossover; if Offspring is non-dominated in S then Adapt (σ,C) of parent and offspring; Add offspring to S; end else Adapt σ of parent; end end

Slide 2/12 — Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel — Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite — July 20, 2016

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

Parent Selection

Select parent based on Hypervolume Contribution
Select extremum points with probability pextreme
Otherwise select parent i with probability

pi = δVolS(f(xi))α ∑j δVolS(f(xj))α .

Slide 3/12 — Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel — Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite — July 20, 2016

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

Crossover

Ci Covariance matrix of parent
i−1, i+1 neighbours of the parent on the front in f -value
Covariance matrix of offspring

C = (1−cr)Ci + cr 2 xi−1 −xi σi xi−1 −xi σi T + cr 2 xi+1 −xi σi xi+1 −xi σi T

Slide 4/12 — Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel — Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite — July 20, 2016

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

Covariance-Matrix-Adaptation

Parent (Ci,σi,xi), Offspring (C,σ,x)
Adapt Covariance matrix of offspring by

C ← (1−ccov)C +ccov x−xi σi x−xi σi T .

Same for parent

Slide 5/12 — Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel — Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite — July 20, 2016

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

Step Size adaptation

Success based as in MO-CMA-ES
Running estimate of success rate
Adjust σ until success rate 1/2

Slide 6/12 — Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel — Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite — July 20, 2016

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

Multi-Objective Exploration

Dominance-based selection gets stuck in local optima
Run k = 100 instances in round robin fashion
D initial points per instance
Merge fronts after 104D iterations
Run single front until budget exhausted

Slide 7/12 — Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel — Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite — July 20, 2016

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

Results on Sphere/Sphere

1 2 3 4 5 6 7 8

log10 of (# f-evals / dimension)

0.0 0.2 0.4 0.6 0.8 1.0

Proportion of function+target pairs

2-D 3-D 10-D 20-D 5-D

bbob-biobj - f1 5 instances

1.1

1 Sphere/Sphere

Slide 8/12 — Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel — Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite — July 20, 2016

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

Results on Sphere/Rastrigin

1 2 3 4 5 6 7 8

log10 of (# f-evals / dimension)

0.0 0.2 0.4 0.6 0.8 1.0

Proportion of function+target pairs

2-D 3-D 5-D 10-D 20-D

bbob-biobj - f10 5 instances

1.1

10 Sphere/Gallagher 101

Slide 9/12 — Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel — Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite — July 20, 2016

SLIDE 10

U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

Results on Sphere/Rastrigin

1 2 3 4 5 6 7 8

log10 of (# f-evals / dimension)

0.0 0.2 0.4 0.6 0.8 1.0

Proportion of function+target pairs

10-D 20-D 5-D 3-D 2-D

bbob-biobj - f7 5 instances

1.1

7 Sphere/Rastrigin

Slide 10/12 — Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel — Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite — July 20, 2016

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

Overall Results

1 2 3 4 5 6 7 8

log10 of (# f-evals / dimension)

0.0 0.2 0.4 0.6 0.8 1.0

Proportion of function+target pairs

10-D 20-D 5-D 3-D 2-D

bbob-biobj - f1-f55 5 instances

1.1

Slide 11/12 — Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel — Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite — July 20, 2016

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U N I V E R S I T Y O F C O P E N H A G E N D E P A R T M E N T O F C O M P U T E R S C I E N C E

Thanks See you at BBComp Session!

Slide 12/12 — Oswin Krause, Tobias Glasmachers, Nikolaus Hansen, and Christian Igel — Unbounded Population MO-CMA-ES for the Bi-Objective BBOB Test Suite — July 20, 2016