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Pilar Caamaño, Francisco Bellas, José A. Becerra, Vicente Díaz, Richard J. Duro Integrated Group for Engineering Research, University of Coruña, Spain
Experimental Analysis of the Relevance of Fitness Landscape Topographical Characterization
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Overview
• Introduction
• Fitness Landscape Topographical Features
• Evolutionary Algorithm Characterization
• Application and Results
• Conclusions
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Overview
• Introduction
• Fitness Landscape Topographical Features
• Evolutionary Algorithm Characterization
• Application and Results
• Conclusions
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Problem
• Users have a hard time selecting and tuning evolutionary algorithms for their particular needs because: – Most evolutionary algorithms are characterized by their
developers in ad hoc and usually optimistic settings. • Results are provided on their strong points. • Comparisons are given over benchmarks in which they
usually excel. • No or very little data is given on where they
underperform. – Even in algorithm competitions the function sets used are
usually biased and very little information is provided on the real performance of successful algorithms, except for the fact that they win the competition.
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Fitness landscape analysis (FLA)
• Two different point of views in FLA: – Statistical point of view: statistical “hardness
measures”. • Fitness distance correlation, fitness variance,
epistasis variance, etc. – Informational point of view: description of the
topographical features of the landscape. • Information landscapes, Exploratory
landscapes, etc.
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Previous work
• Exhaustive analysis of different benchmark fitness landscapes based on topographical features. – Experimental evidences of the relevance of
those features over the response of several EAs depending on the landscape features.
Two main features
Separability
Modality
Four Evolu9onary Algorithms
Differen9al Evolu9on
Covariance Matrix Adapta9on -‐ ES
Real Coded -‐ GA Macro-‐evolu9onary Algorithms
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Overview
• Introduction
• Fitness Landscape Topographical Features
• Evolutionary Algorithm Characterization
• Application and Results
• Conclusions
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• Graphical representation of the hyper-surface over with the individuals of the EA performs the search.
• Two main topographical features: – Separability
– Modality
Fitness Landscape Topographical Features
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• Can each gene of the chromosome be optimized separately from the rest?
• Typical binary classification: – Linearly separable – Non separable – Insufficient to explain the performance of EAs in all
cases. • We have proposed a third class:
– Non linearly separable
• We have developed a heuristic method to estimate the specific type of separability.
Separability
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• It is related with the number and the distribution of the optima throughout the landscape.
• Typical classification: – Unimodal
– Multimodal
• It is very rough and does not take into account the distribution of the optima.
• A heuristic method based on the attraction basins theory has been proposed to analyze the modality of a landscape.
Modality (1)
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• Topographical information obtained after the application of the heuristic for modality analysis: – Unimodal functions:
• Length of the longest path to the optimum – Multimodal functions:
• Size of the optimum attraction basin • Size of the largest attraction basin • Distance between the optimum a.b. and
other a.b. • Maximum distance between a.b.
Modality (2)
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Overview
• Introduction
• Fitness Landscape Topographical Features
• Evolutionary Algorithm Characterization
• Application and Results
• Conclusions
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• Four Evolutionary Algorithms. • 72 analytical functions and 22 real-world
application problems. • 50 independent runs. • 10000·n function evaluations ( being n the
dimension of the problem). • CPEM measure to evaluate the results
– It rewards the algorithms that performs less FEs.
Evolutionary Algorithms Characterization Experimental Setup
CPEM =10−6 ⋅ #FEs
#Max _FEsif abs_ error ≤10−6
abs_ error otherwise
⎧
⎨⎪
⎩⎪
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• The characterization of an EA depends on the combination of fitness landscape features.
• The success depending on separability is related with the search direction of the algorithm: – Orthogonal directions for L-Sep. and NL-Sep. – Diagonal directions for Non-Sep.
Evolutionary Algorithms Characterization Experimental Results
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• To obtain a satisfactory performance over unimodal functions: – A strategy to adapt the mutation step size automatically.
– An exploitative strategy to speed up the convergence speed.
• To reach a better performance in the multimodal functions: – An explorative behavior is highly recommended.
– To cover the maximum area when the a.b. are spread out over the landscape.
– To find the a.b. when they are narrow.
Evolutionary Algorithms Characterization Experimental Results
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Overview
• Introduction
• Fitness Landscape Topographical Features
• Evolutionary Algorithm Characterization
• Application and Results
• Conclusions
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• Objective: To obtain the blade geometry production optimal aerodynamic results.
• The geometry is obtained by providing, at a finite number of sections, the distribution of: – The twist angles. – The chords. – The thickness.
• Fitness: amount of energy produces by the blade.
HAWT Design Problem
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• The chromosome length is 11, it is considered a low-dimensional problem.
• There exists inter-relationships between the genes, the problem is Non-Separable.
• Modality analysis: – 94 attraction basins. – The largest a.b. is not the optima. – The optima are concentrated in a narrow area. – There are no plateaus. – The a.b. are narrow making them difficult to locate.
Fitness Landscape Analysis Results
10-5
10-4
10-3
10-2
10-1
100
0.00 0.25 0.50 0.75 1.00Att
ract
ion
bas
in s
ize
(a.u
., l
og
. sc
ale)
Distance to the optimum attraction basin (a.u.)
Attraction basins distribution
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• For being Non-Separable and Low-Dimensional: – DE or MA are recommended.
• For being multimodal with narrow a.b.: – An explorative strategy is highly recommended. – The DE is the more explorative of the four
analyzed EAs.
Evolutionary Algorithms Analysis Results
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10-6
10-4
10-2
100
10-3
10-2
10-1
100
101
102
CP
EM
val
ue
(a.u
., l
og. sc
ale)
Function Evaluations performed (a.u., log. scale)
Algorithms Comparison
RCGACMA - ES
DEMA
Algorithm CPEM Value Best Value SR
DE 1.34·∙10-‐4 ± 2.47·∙10-‐4 4.20·∙10-‐8 73%
CMA-‐ES 9.04·∙10-‐4 ± 3.31·∙10-‐4 3.02·∙10-‐4 0%
MA 3.69·∙10-‐4 ± 2.72·∙10-‐4 5.64·∙10-‐7 27%
RCGA 6.64·∙10-‐4 ± 3.84·∙10-‐4 1.90·∙10-‐7 13%
Evolutionary Algorithms Analysis Results
These results confirm our assump9ons
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• The highly explorative strategy of the DE outperforms the other algorithms.
• The mainly exploitative behavior of the CMA-ES causes a higher convergence rate at the beginning, but drives to stagnation in local optima.
• The low SR of the RCGA and MA are due to the non-separability. – These algorithms do not deal with genes
relationships.
Evolutionary Algorithms Analysis Results
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Overview
• Introduction
• Fitness Landscape Topographical Features
• Evolutionary Algorithm Characterization
• Application and Results
• Conclusions
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• We have shown the usefulness and relevance of a topographical features analysis before the application of an EA.
• We have focused our work in the information analysis: to provide an in-depth description of the fitness landscape. – Mainly, separability and modality.
• We have extended our previous work to the analysis of real-world applications.
Conclusions
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