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
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: ( x i , σ i , C i ) , 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
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 p extreme • Otherwise select parent i with probability δ Vol S ( f ( x i )) α p i = ∑ j δ Vol S ( f ( x j )) 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
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 • C i Covariance matrix of parent • i − 1 , i + 1 neighbours of the parent on the front in f -value • Covariance matrix of offspring � T � x i − 1 − x i �� x i − 1 − x i C = ( 1 − c r ) C i + c r σ i σ i 2 � T � x i + 1 − x i �� x i + 1 − x i + c r σ i σ i 2 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
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 ( C i , σ i , x i ) , Offspring ( C , σ , x ) • Adapt Covariance matrix of offspring by � T � x − x i �� x − x i C ← ( 1 − c cov ) C + c cov . σ i σ i • 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
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
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 10 4 D 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
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 Sphere/Sphere 1.0 5-D bbob-biobj - f1 Proportion of function+target pairs 5 instances 0.8 20-D 0.6 10-D 0.4 3-D 0.2 2-D 1.1 0.0 0 1 2 3 4 5 6 7 8 log10 of (# f-evals / dimension) 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
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 10 Sphere/Gallagher 101 1.0 20-D bbob-biobj - f10 Proportion of function+target pairs 5 instances 0.8 10-D 0.6 5-D 0.4 3-D 0.2 2-D 1.1 0.0 0 1 2 3 4 5 6 7 8 log10 of (# f-evals / dimension) 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
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 7 Sphere/Rastrigin 1.0 2-D bbob-biobj - f7 Proportion of function+target pairs 5 instances 0.8 3-D 0.6 5-D 0.4 20-D 0.2 10-D 1.1 0.0 0 1 2 3 4 5 6 7 8 log10 of (# f-evals / dimension) 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
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 1.0 2-D bbob-biobj - f1-f55 Proportion of function+target pairs 5 instances 0.8 3-D 0.6 5-D 0.4 20-D 0.2 10-D 1.1 0.0 0 1 2 3 4 5 6 7 8 log10 of (# f-evals / dimension) 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
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
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