Experimental Analysis of the Relevance of Fitness ...€¦ · Pilar Caamaño, Francisco Bellas,...

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

Overview

•  Introduction

•  Fitness Landscape Topographical Features

•  Evolutionary Algorithm Characterization

• Application and Results

• Conclusions

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.

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.

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

Overview

•  Introduction




• Conclusions

•  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

•  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

•  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)

•  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)

Overview

•  Introduction




• Conclusions

•  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

⎧

⎨⎪

⎩⎪

•  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

•  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

Overview

•  Introduction




• Conclusions

• 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

•  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

•  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

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%


These results confirm our assump9ons

•  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.


Overview

•  Introduction




• Conclusions

• 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

Experimental Analysis of the Relevance of Fitness ...€¦ · Pilar Caamaño, Francisco Bellas,...

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Transcript of Experimental Analysis of the Relevance of Fitness ...€¦ · Pilar Caamaño, Francisco Bellas,...