chapter written, new 1d plots, new imgs
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arbeit/bibma.bib
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arbeit/bibma.bib
@ -1,128 +1,175 @@
|
||||
@article{anrichterEvol,
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author = "Richter, Andreas and Achenbach, Jascha and Menzel, Stefan and Botsch, Mario",
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booktitle = "IEEE Congress on Evolutionary Computation",
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location = "Vancouver, Canada",
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publisher = "IEEE",
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title = "Evolvability as a Quality Criterion for Linear Deformation Representations in Evolutionary Optimization",
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year = "2016",
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note={\url{http://graphics.uni-bielefeld.de/publications/cec16.pdf}, \url{https://pub.uni-bielefeld.de/publication/2902698}},
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% This file was created with JabRef 2.10.
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% Encoding: UTF-8
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@Article{back1993overview,
|
||||
Title = {An overview of evolutionary algorithms for parameter optimization},
|
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Author = {B{\"a}ck, Thomas and Schwefel, Hans-Paul},
|
||||
Journal = {Evolutionary computation},
|
||||
Year = {1993},
|
||||
Number = {1},
|
||||
Pages = {1--23},
|
||||
Volume = {1},
|
||||
|
||||
Publisher = {MIT Press},
|
||||
Url = {https://www.researchgate.net/profile/Hans-Paul_Schwefel/publication/220375001_An_Overview_of_Evolutionary_Algorithms_for_Parameter_Optimization/links/543663d00cf2dc341db30452.pdf}
|
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}
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||||
@article{spitzmuller1996bezier,
|
||||
title="Partial derivatives of Bèzier surfaces",
|
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author="Spitzmüller, Klaus",
|
||||
journal="Computer-Aided Design",
|
||||
volume="28",
|
||||
number="1",
|
||||
pages="67--72",
|
||||
year="1996",
|
||||
publisher="Elsevier",
|
||||
url={https://doi.org/10.1016/0010-4485(95)00044-5},
|
||||
|
||||
@Book{golub2012matrix,
|
||||
Title = {Matrix computations},
|
||||
Author = {Golub, Gene H and Van Loan, Charles F},
|
||||
Publisher = {JHU Press},
|
||||
Year = {2012},
|
||||
Volume = {3}
|
||||
}
|
||||
@article{hsu1991dmffd,
|
||||
title={A direct manipulation interface to free-form deformations},
|
||||
author={Hsu, William M},
|
||||
journal={Master's thesis, Brown University},
|
||||
year={1991},
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url={https://cs.brown.edu/research/pubs/theses/masters/1991/hsu.pdf},
|
||||
|
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@Article{hsu1991dmffd,
|
||||
Title = {A direct manipulation interface to free-form deformations},
|
||||
Author = {Hsu, William M},
|
||||
Journal = {Master's thesis, Brown University},
|
||||
Year = {1991},
|
||||
|
||||
Url = {https://cs.brown.edu/research/pubs/theses/masters/1991/hsu.pdf}
|
||||
}
|
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@article{hsu1992direct,
|
||||
title={Direct Manipulation of Free-Form Deformations},
|
||||
author={Hsu, William M and Hughes, John F and Kaufman, Henry},
|
||||
journal={Computer Graphics},
|
||||
volume={26},
|
||||
pages={2},
|
||||
year={1992},
|
||||
url={http://graphics.cs.brown.edu/~jfh/papers/Hsu-DMO-1992/paper.pdf},
|
||||
|
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@Article{hsu1992direct,
|
||||
Title = {Direct Manipulation of Free-Form Deformations},
|
||||
Author = {Hsu, William M and Hughes, John F and Kaufman, Henry},
|
||||
Journal = {Computer Graphics},
|
||||
Year = {1992},
|
||||
Pages = {2},
|
||||
Volume = {26},
|
||||
|
||||
Url = {http://graphics.cs.brown.edu/~jfh/papers/Hsu-DMO-1992/paper.pdf}
|
||||
}
|
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@inproceedings{Menzel2006,
|
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author = {Menzel, Stefan and Olhofer, Markus and Sendhoff, Bernhard},
|
||||
title = {Direct Manipulation of Free Form Deformation in Evolutionary Design Optimisation},
|
||||
booktitle = {Proceedings of the 9th International Conference on Parallel Problem Solving from Nature},
|
||||
series = {PPSN'06},
|
||||
year = {2006},
|
||||
isbn = {3-540-38990-3, 978-3-540-38990-3},
|
||||
location = {Reykjavik, Iceland},
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||||
pages = {352--361},
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||||
numpages = {10},
|
||||
url = {http://dx.doi.org/10.1007/11844297_36},
|
||||
doi = {10.1007/11844297_36},
|
||||
acmid = {2079770},
|
||||
publisher = {Springer-Verlag},
|
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address = {Berlin, Heidelberg},
|
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|
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@Article{gaussNewton,
|
||||
Title = {An Algorithm for Least-Squares Estimation of Nonlinear Parameters},
|
||||
Author = {Donald W. Marquardt},
|
||||
Journal = {Journal of the Society for Industrial and Applied Mathematics},
|
||||
Year = {1963},
|
||||
Number = {2},
|
||||
Pages = {431-441},
|
||||
Volume = {11},
|
||||
|
||||
Doi = {10.1137/0111030},
|
||||
Eprint = {https://doi.org/10.1137/0111030},
|
||||
Url = {https://doi.org/10.1137/0111030}
|
||||
}
|
||||
@book{golub2012matrix,
|
||||
title={Matrix computations},
|
||||
author={Golub, Gene H and Van Loan, Charles F},
|
||||
volume={3},
|
||||
year={2012},
|
||||
publisher={JHU Press}
|
||||
|
||||
@InProceedings{Menzel2006,
|
||||
Title = {Direct Manipulation of Free Form Deformation in Evolutionary Design Optimisation},
|
||||
Author = {Menzel, Stefan and Olhofer, Markus and Sendhoff, Bernhard},
|
||||
Booktitle = {Proceedings of the 9th International Conference on Parallel Problem Solving from Nature},
|
||||
Year = {2006},
|
||||
|
||||
Address = {Berlin, Heidelberg},
|
||||
Pages = {352--361},
|
||||
Publisher = {Springer-Verlag},
|
||||
Series = {PPSN'06},
|
||||
|
||||
Acmid = {2079770},
|
||||
Doi = {10.1007/11844297_36},
|
||||
ISBN = {3-540-38990-3, 978-3-540-38990-3},
|
||||
Location = {Reykjavik, Iceland},
|
||||
Numpages = {10},
|
||||
Url = {http://dx.doi.org/10.1007/11844297_36}
|
||||
}
|
||||
@article{weise2012evolutionary,
|
||||
title={Evolutionary Optimization: Pitfalls and Booby Traps},
|
||||
author={Weise, Thomas and Chiong, Raymond and Tang, Ke},
|
||||
journal={J. Comput. Sci. \& Technol},
|
||||
volume={27},
|
||||
number={5},
|
||||
year={2012},
|
||||
url={http://jcst.ict.ac.cn:8080/jcst/EN/article/downloadArticleFile.do?attachType=PDF\&id=9543}
|
||||
|
||||
@Article{minai2006complex,
|
||||
Title = {Complex engineered systems: A new paradigm},
|
||||
Author = {Minai, Ali A and Braha, Dan and Bar-Yam, Yaneer},
|
||||
Journal = {Complex engineered systems: Science meets technology},
|
||||
Year = {2006},
|
||||
Pages = {1--21},
|
||||
|
||||
Publisher = {Springer},
|
||||
Url = {https://www.researchgate.net/profile/Yaneer_Bar-Yam/publication/225104044_Complex_Engineered_Systems_A_New_Paradigm/links/59107f20a6fdccbfd57eb84d/Complex-Engineered-Systems-A-New-Paradigm.pdf}
|
||||
}
|
||||
@inproceedings{thorhauer2014locality,
|
||||
title={On the locality of standard search operators in grammatical evolution},
|
||||
author={Thorhauer, Ann and Rothlauf, Franz},
|
||||
booktitle={International Conference on Parallel Problem Solving from Nature},
|
||||
pages={465--475},
|
||||
year={2014},
|
||||
organization={Springer},
|
||||
url={https://www.lri.fr/~hansen/proceedings/2014/PPSN/papers/8672/86720465.pdf}
|
||||
|
||||
@Article{anrichterEvol,
|
||||
Title = {Evolvability as a Quality Criterion for Linear Deformation Representations in Evolutionary Optimization},
|
||||
Author = {Richter, Andreas and Achenbach, Jascha and Menzel, Stefan and Botsch, Mario},
|
||||
Year = {2016},
|
||||
Note = {\url{http://graphics.uni-bielefeld.de/publications/cec16.pdf}, \url{https://pub.uni-bielefeld.de/publication/2902698}},
|
||||
|
||||
Booktitle = {IEEE Congress on Evolutionary Computation},
|
||||
Location = {Vancouver, Canada},
|
||||
Publisher = {IEEE}
|
||||
}
|
||||
@article{gaussNewton,
|
||||
author = {Donald W. Marquardt},
|
||||
title = {An Algorithm for Least-Squares Estimation of Nonlinear Parameters},
|
||||
journal = {Journal of the Society for Industrial and Applied Mathematics},
|
||||
volume = {11},
|
||||
number = {2},
|
||||
pages = {431-441},
|
||||
year = {1963},
|
||||
doi = {10.1137/0111030},
|
||||
URL = {https://doi.org/10.1137/0111030},
|
||||
eprint = {https://doi.org/10.1137/0111030}
|
||||
|
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@InProceedings{richter2015evolvability,
|
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Title = {Evolvability of representations in complex system engineering: a survey},
|
||||
Author = {Richter, Andreas and Botsch, Mario and Menzel, Stefan},
|
||||
Booktitle = {Evolutionary Computation (CEC), 2015 IEEE Congress on},
|
||||
Year = {2015},
|
||||
Organization = {IEEE},
|
||||
Pages = {1327--1335},
|
||||
|
||||
Url = {http://www.graphics.uni-bielefeld.de/publications/cec15.pdf}
|
||||
}
|
||||
@article{minai2006complex,
|
||||
title={Complex engineered systems: A new paradigm},
|
||||
author={Minai, Ali A and Braha, Dan and Bar-Yam, Yaneer},
|
||||
journal={Complex engineered systems: Science meets technology},
|
||||
pages={1--21},
|
||||
year={2006},
|
||||
publisher={Springer},
|
||||
url={https://www.researchgate.net/profile/Yaneer_Bar-Yam/publication/225104044_Complex_Engineered_Systems_A_New_Paradigm/links/59107f20a6fdccbfd57eb84d/Complex-Engineered-Systems-A-New-Paradigm.pdf}
|
||||
|
||||
@InBook{Rothlauf2006,
|
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Title = {Representations for Genetic and Evolutionary Algorithms},
|
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Author = {Rothlauf, Franz},
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Chapter = {2},
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Pages = {9--32},
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Publisher = {Springer Berlin Heidelberg},
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Year = {2006},
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|
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Address = {Berlin, Heidelberg},
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|
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Abstract = {In this second chapter, we present an introduction into the field of representations for genetic and evolutionary algorithms. The chapter provides the basis and definitions which are essential for understanding the content of this work.},
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Booktitle = {Representations for Genetic and Evolutionary Algorithms},
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||||
Doi = {10.1007/3-540-32444-5_2},
|
||||
ISBN = {978-3-540-32444-7},
|
||||
Url = {https://doi.org/10.1007/3-540-32444-5_2}
|
||||
}
|
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@article{wagner1996complex,
|
||||
title={Complex adaptations and the evolution of evolvability},
|
||||
author={Wagner, Gunter P and Altenberg, Lee},
|
||||
journal={Evolution},
|
||||
volume={50},
|
||||
number={3},
|
||||
pages={967--976},
|
||||
year={1996},
|
||||
url={http://arep.med.harvard.edu/pdf/Wagner96.pdf},
|
||||
|
||||
@Article{spitzmuller1996bezier,
|
||||
Title = {Partial derivatives of Bèzier surfaces},
|
||||
Author = {Spitzmüller, Klaus},
|
||||
Journal = {Computer-Aided Design},
|
||||
Year = {1996},
|
||||
Number = {1},
|
||||
Pages = {67--72},
|
||||
Volume = {28},
|
||||
|
||||
Publisher = {Elsevier},
|
||||
Url = {https://doi.org/10.1016/0010-4485(95)00044-5}
|
||||
}
|
||||
@inproceedings{richter2015evolvability,
|
||||
title={Evolvability of representations in complex system engineering: a survey},
|
||||
author={Richter, Andreas and Botsch, Mario and Menzel, Stefan},
|
||||
booktitle={Evolutionary Computation (CEC), 2015 IEEE Congress on},
|
||||
pages={1327--1335},
|
||||
year={2015},
|
||||
organization={IEEE},
|
||||
url={http://www.graphics.uni-bielefeld.de/publications/cec15.pdf}
|
||||
|
||||
@InProceedings{thorhauer2014locality,
|
||||
Title = {On the locality of standard search operators in grammatical evolution},
|
||||
Author = {Thorhauer, Ann and Rothlauf, Franz},
|
||||
Booktitle = {International Conference on Parallel Problem Solving from Nature},
|
||||
Year = {2014},
|
||||
Organization = {Springer},
|
||||
Pages = {465--475},
|
||||
|
||||
Url = {https://www.lri.fr/~hansen/proceedings/2014/PPSN/papers/8672/86720465.pdf}
|
||||
}
|
||||
@article{back1993overview,
|
||||
title={An overview of evolutionary algorithms for parameter optimization},
|
||||
author={B{\"a}ck, Thomas and Schwefel, Hans-Paul},
|
||||
journal={Evolutionary computation},
|
||||
volume={1},
|
||||
number={1},
|
||||
pages={1--23},
|
||||
year={1993},
|
||||
publisher={MIT Press},
|
||||
url={https://www.researchgate.net/profile/Hans-Paul_Schwefel/publication/220375001_An_Overview_of_Evolutionary_Algorithms_for_Parameter_Optimization/links/543663d00cf2dc341db30452.pdf}
|
||||
|
||||
@Article{wagner1996complex,
|
||||
Title = {Complex adaptations and the evolution of evolvability},
|
||||
Author = {Wagner, Gunter P and Altenberg, Lee},
|
||||
Journal = {Evolution},
|
||||
Year = {1996},
|
||||
Number = {3},
|
||||
Pages = {967--976},
|
||||
Volume = {50},
|
||||
|
||||
Url = {http://arep.med.harvard.edu/pdf/Wagner96.pdf}
|
||||
}
|
||||
|
||||
@Article{weise2012evolutionary,
|
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Title = {Evolutionary Optimization: Pitfalls and Booby Traps},
|
||||
Author = {Weise, Thomas and Chiong, Raymond and Tang, Ke},
|
||||
Journal = {J. Comput. Sci. \& Technol},
|
||||
Year = {2012},
|
||||
Number = {5},
|
||||
Volume = {27},
|
||||
|
||||
Url = {http://jcst.ict.ac.cn:8080/jcst/EN/article/downloadArticleFile.do?attachType=PDF\&id=9543}
|
||||
}
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BIN
arbeit/img/weise_fig3.png
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arbeit/img/weise_fig3.png
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arbeit/ma.md
68
arbeit/ma.md
@ -44,7 +44,9 @@ etc.), the translation of the problem--domain into a simple parametric
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representation can be challenging.
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The quality of such a representation in biological evolution is called
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*evolvability*\cite{wagner1996complex} and is at the core of this thesis.
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*evolvability*\cite{wagner1996complex} and is at the core of this thesis, as the
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parametrization of the problem has serious implications on the convergence speed
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and the quality of the solution\cite{Rothlauf2006}.
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However, there is no consensus on how *evolvability* is defined and the meaning
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varies from context to context\cite{richter2015evolvability}.
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@ -196,59 +198,72 @@ however, is very generic and we introduce it here in a broader sense.
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\end{algorithm}
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The general shape of an evolutional algorithm (adapted from
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\cite{back1993overview}} is outlined in Algorithm \ref{alg:evo}. Here, $P(t)$
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\cite{back1993overview}) is outlined in Algorithm \ref{alg:evo}. Here, $P(t)$
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denotes the population of parameters in step $t$ of the algorithm. The
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population contains $\mu$ individuals $a_i$ that fit the shape of the parameters
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we are looking for. Typically these are initialized by a random guess or just
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zero. Further on we need a so-called *fitness-function* $\Phi : I \mapsto M$ that can take
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each parameter to a measurable space along a convergence-function $c : I \mapsto
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zero. Further on we need a so--called *fitness--function* $\Phi : I \mapsto M$ that can take
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each parameter to a measurable space along a convergence--function $c : I \mapsto
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\mathbb{B}$ that terminates the optimization.
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The main algorithm just repeats the following steps:
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- **Recombine** with a recombination-function $r : I^{\mu} \mapsto I^{\lambda}$ to
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- **Recombine** with a recombination--function $r : I^{\mu} \mapsto I^{\lambda}$ to
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generate new individuals based on the parents characteristics.
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This makes sure that the next guess is close to the old guess.
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- **Mutate** with a mutation-function $m : I^{\lambda} \mapsto I^{\lambda}$ to
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- **Mutate** with a mutation--function $m : I^{\lambda} \mapsto I^{\lambda}$ to
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introduce new effects that cannot be produced by mere recombination of the
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parents.
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Typically this just adds minor defects to individual members of the population
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like adding a random gaussian noise or amplifying/dampening random parts.
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- **Selection** takes a selection-function $s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu$ that
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- **Selection** takes a selection--function $s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu$ that
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selects from the previously generated $I^\lambda$ children and optionally also
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the parents (denoted by the set $Q$ in the algorithm) using the
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fitness-function $\Phi$. The result of this operation is the next Population
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fitness--function $\Phi$. The result of this operation is the next Population
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of $\mu$ individuals.
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All these functions can (and mostly do) have a lot of hidden parameters that
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can be changed over time. One can for example start off with a high
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mutation--rate that cools off over time (i.e. by lowering the variance of a
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gaussian noise). As the recombination and selection-steps are usually pure
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gaussian noise).
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## Advantages of evolutional algorithms
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\label{sec:back:evogood}
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\change[inline]{Needs citations}
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The main advantage of evolutional algorithms is the ability to find optima of
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general functions just with the help of a given error--function (or
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fitness--function in this domain). This avoids the general pitfalls of
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gradient--based procedures, which often target the same error--function as an
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evolutional algorithm, but can get stuck in local optima.
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general functions just with the help of a given fitness--function. With this
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most problems of simple gradient--based procedures, which often target the same
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error--function which measures the fitness, as an evolutional algorithm, but can
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easily get stuck in local optima.
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This is mostly due to the fact that a gradient--based procedure has only one
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point of observation from where it evaluates the next steps, whereas an
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Components and techniques for evolutional algorithms are specifically known to
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help with different problems arising in the domain of
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optimization\cite{weise2012evolutionary}. An overview of the typical problems
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are shown in figure \ref{fig:probhard}.
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\begin{figure}[!ht]
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\includegraphics[width=\textwidth]{img/weise_fig3.png}
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\caption{Fig.~3. taken from \cite{weise2012evolutionary}}
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\label{fig:probhard}
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\end{figure}
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Most of the advantages stem from the fact that a gradient--based procedure has
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only one point of observation from where it evaluates the next steps, whereas an
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evolutional strategy starts with a population of guessed solutions. Because an
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evolutional strategy modifies the solution randomly, keeps the best solutions
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and purges the worst, it can also target multiple different hypothesis at the
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same time where the local optima die out in the face of other, better
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candidates.
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If an analytic best solution exists (i.e. because the error--function is convex)
|
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an evolutional algorithm is not the right choice. Although both converge to the
|
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same solution, the analytic one is usually faster. But in reality many problems
|
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have no analytic solution, because the problem is not convex. Here evolutional
|
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optimization has one more advantage as you get bad solutions fast, which refine
|
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over time.
|
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If an analytic best solution exists and is easily computable (i.e. because the
|
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error--function is convex) an evolutional algorithm is not the right choice.
|
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Although both converge to the same solution, the analytic one is usually faster.
|
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|
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But in reality many problems have no analytic solution, because the problem is
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either not convex or there are so many parameters that an analytic solution
|
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(mostly meaning the equivalence to an exhaustive search) is computationally not
|
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feasible. Here evolutional optimization has one more advantage as you can at
|
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least get a suboptimal solutions fast, which then refine over time.
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|
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## Criteria for the evolvability of linear deformations
|
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\label{sec:intro:rvi}
|
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@ -427,8 +442,13 @@ linear equations.
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## Parametrisierung sinnvoll?
|
||||
|
||||
- Nachteile von Parametrisierung
|
||||
- wie in kap. \ref{sec:back:evo} zu sehen, ist Parametrisierung
|
||||
wichtig\cite{Rothlauf2006}.
|
||||
- Parametrisierung zwar lokal, aber nicht 1:1
|
||||
- Deformation ist um einen Kontrollpunkt viel direkter zu steuern.
|
||||
- => DM--FFD?
|
||||
- => DM--FFD kann abhelfen, further study.
|
||||
- Schlechte Parametrisierung sorgt dafür, dass CP u.U. nicht zur
|
||||
Parametrisierung verwendet werden.
|
||||
|
||||
|
||||
# Scenarios for testing evolvability criteria using \acf{FFD}
|
||||
@ -480,7 +500,7 @@ linear equations.
|
||||
## Results of 1D Function Approximation
|
||||
|
||||
\begin{figure}[!ht]
|
||||
\includegraphics[width=\textwidth]{img/evolution1d/20170830-evolution1D_5x5_100Times-all_appended.png}
|
||||
\includegraphics[width=\textwidth]{img/evolution1d/20171005-all_appended.png}
|
||||
\caption{Results 1D}
|
||||
|
||||
\end{figure}
|
||||
|
||||
BIN
arbeit/ma.pdf
BIN
arbeit/ma.pdf
Binary file not shown.
@ -189,8 +189,10 @@ into a simple parametric representation can be challenging.
|
||||
|
||||
The quality of such a representation in biological evolution is called
|
||||
\emph{evolvability}\cite{wagner1996complex} and is at the core of this
|
||||
thesis. However, there is no consensus on how \emph{evolvability} is
|
||||
defined and the meaning varies from context to
|
||||
thesis, as the parametrization of the problem has serious implications
|
||||
on the convergence speed and the quality of the
|
||||
solution\cite{Rothlauf2006}. However, there is no consensus on how
|
||||
\emph{evolvability} is defined and the meaning varies from context to
|
||||
context\cite{richter2015evolvability}.
|
||||
|
||||
As we transfer the results of Richter et al.\cite{anrichterEvol} from
|
||||
@ -331,10 +333,10 @@ optimization?}\label{what-is-evolutional-optimization}
|
||||
|
||||
\label{sec:back:evo}
|
||||
|
||||
\change[inline]{Write this section} In this thesis we are using an
|
||||
evolutional optimization strategy to solve the problem of finding the
|
||||
best parameters for our deformation. This approach, however, is very
|
||||
generic and we introduce it here in a broader sense.
|
||||
In this thesis we are using an evolutional optimization strategy to
|
||||
solve the problem of finding the best parameters for our deformation.
|
||||
This approach, however, is very generic and we introduce it here in a
|
||||
broader sense.
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{An outline of evolutional algorithms}
|
||||
@ -354,13 +356,13 @@ generic and we introduce it here in a broader sense.
|
||||
\end{algorithm}
|
||||
|
||||
The general shape of an evolutional algorithm (adapted from
|
||||
\cite{back1993overview}\} is outlined in Algorithm \ref{alg:evo}. Here,
|
||||
\cite{back1993overview}) is outlined in Algorithm \ref{alg:evo}. Here,
|
||||
\(P(t)\) denotes the population of parameters in step \(t\) of the
|
||||
algorithm. The population contains \(\mu\) individuals \(a_i\) that fit
|
||||
the shape of the parameters we are looking for. Typically these are
|
||||
initialized by a random guess or just zero. Further on we need a
|
||||
so-called \emph{fitness-function} \(\Phi : I \mapsto M\) that can take
|
||||
each parameter to a measurable space along a convergence-function
|
||||
so--called \emph{fitness--function} \(\Phi : I \mapsto M\) that can take
|
||||
each parameter to a measurable space along a convergence--function
|
||||
\(c : I \mapsto \mathbb{B}\) that terminates the optimization.
|
||||
|
||||
The main algorithm just repeats the following steps:
|
||||
@ -368,58 +370,73 @@ The main algorithm just repeats the following steps:
|
||||
\begin{itemize}
|
||||
\tightlist
|
||||
\item
|
||||
\textbf{Recombine} with a recombination-function
|
||||
\textbf{Recombine} with a recombination--function
|
||||
\(r : I^{\mu} \mapsto I^{\lambda}\) to generate new individuals based
|
||||
on the parents characteristics.\\
|
||||
This makes sure that the next guess is close to the old guess.
|
||||
\item
|
||||
\textbf{Mutate} with a mutation-function
|
||||
\textbf{Mutate} with a mutation--function
|
||||
\(m : I^{\lambda} \mapsto I^{\lambda}\) to introduce new effects that
|
||||
cannot be produced by mere recombination of the parents.\\
|
||||
Typically this just adds minor defects to individual members of the
|
||||
population like adding a random gaussian noise or amplifying/dampening
|
||||
random parts.
|
||||
\item
|
||||
\textbf{Selection} takes a selection-function
|
||||
\textbf{Selection} takes a selection--function
|
||||
\(s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu\) that
|
||||
selects from the previously generated \(I^\lambda\) children and
|
||||
optionally also the parents (denoted by the set \(Q\) in the
|
||||
algorithm) using the fitness-function \(\Phi\). The result of this
|
||||
algorithm) using the fitness--function \(\Phi\). The result of this
|
||||
operation is the next Population of \(\mu\) individuals.
|
||||
\end{itemize}
|
||||
|
||||
All these functions can (and mostly do) have a lot of hidden parameters
|
||||
that can be changed over time. One can for example start off with a high
|
||||
mutation--rate that cools off over time (i.e.~by lowering the variance
|
||||
of a gaussian noise). As the recombination and selection-steps are
|
||||
usually pure
|
||||
of a gaussian noise).
|
||||
|
||||
\section{Advantages of evolutional
|
||||
algorithms}\label{advantages-of-evolutional-algorithms}
|
||||
|
||||
\label{sec:back:evogood}
|
||||
|
||||
\change[inline]{Needs citations} The main advantage of evolutional
|
||||
algorithms is the ability to find optima of general functions just with
|
||||
the help of a given error--function (or fitness--function in this
|
||||
domain). This avoids the general pitfalls of gradient--based procedures,
|
||||
which often target the same error--function as an evolutional algorithm,
|
||||
but can get stuck in local optima.
|
||||
The main advantage of evolutional algorithms is the ability to find
|
||||
optima of general functions just with the help of a given
|
||||
fitness--function. With this most problems of simple gradient--based
|
||||
procedures, which often target the same error--function which measures
|
||||
the fitness, as an evolutional algorithm, but can easily get stuck in
|
||||
local optima.
|
||||
|
||||
This is mostly due to the fact that a gradient--based procedure has only
|
||||
one point of observation from where it evaluates the next steps, whereas
|
||||
an evolutional strategy starts with a population of guessed solutions.
|
||||
Because an evolutional strategy modifies the solution randomly, keeps
|
||||
the best solutions and purges the worst, it can also target multiple
|
||||
different hypothesis at the same time where the local optima die out in
|
||||
the face of other, better candidates.
|
||||
Components and techniques for evolutional algorithms are specifically
|
||||
known to help with different problems arising in the domain of
|
||||
optimization\cite{weise2012evolutionary}. An overview of the typical
|
||||
problems are shown in figure \ref{fig:probhard}.
|
||||
|
||||
If an analytic best solution exists (i.e.~because the error--function is
|
||||
convex) an evolutional algorithm is not the right choice. Although both
|
||||
converge to the same solution, the analytic one is usually faster. But
|
||||
in reality many problems have no analytic solution, because the problem
|
||||
is not convex. Here evolutional optimization has one more advantage as
|
||||
you get bad solutions fast, which refine over time.
|
||||
\begin{figure}[!ht]
|
||||
\includegraphics[width=\textwidth]{img/weise_fig3.png}
|
||||
\caption{Fig.~3. taken from \cite{weise2012evolutionary}}
|
||||
\label{fig:probhard}
|
||||
\end{figure}
|
||||
|
||||
Most of the advantages stem from the fact that a gradient--based
|
||||
procedure has only one point of observation from where it evaluates the
|
||||
next steps, whereas an evolutional strategy starts with a population of
|
||||
guessed solutions. Because an evolutional strategy modifies the solution
|
||||
randomly, keeps the best solutions and purges the worst, it can also
|
||||
target multiple different hypothesis at the same time where the local
|
||||
optima die out in the face of other, better candidates.
|
||||
|
||||
If an analytic best solution exists and is easily computable
|
||||
(i.e.~because the error--function is convex) an evolutional algorithm is
|
||||
not the right choice. Although both converge to the same solution, the
|
||||
analytic one is usually faster.
|
||||
|
||||
But in reality many problems have no analytic solution, because the
|
||||
problem is either not convex or there are so many parameters that an
|
||||
analytic solution (mostly meaning the equivalence to an exhaustive
|
||||
search) is computationally not feasible. Here evolutional optimization
|
||||
has one more advantage as you can at least get a suboptimal solutions
|
||||
fast, which then refine over time.
|
||||
|
||||
\section{Criteria for the evolvability of linear
|
||||
deformations}\label{criteria-for-the-evolvability-of-linear-deformations}
|
||||
@ -606,10 +623,18 @@ system of linear equations.
|
||||
\tightlist
|
||||
\item
|
||||
Nachteile von Parametrisierung
|
||||
\item
|
||||
wie in kap. \ref{sec:back:evo} zu sehen, ist Parametrisierung
|
||||
wichtig\cite{Rothlauf2006}.
|
||||
\item
|
||||
Parametrisierung zwar lokal, aber nicht 1:1
|
||||
\item
|
||||
Deformation ist um einen Kontrollpunkt viel direkter zu steuern.
|
||||
\item
|
||||
=\textgreater{} DM--FFD?
|
||||
=\textgreater{} DM--FFD kann abhelfen, further study.
|
||||
\item
|
||||
Schlechte Parametrisierung sorgt dafür, dass CP u.U. nicht zur
|
||||
Parametrisierung verwendet werden.
|
||||
\end{itemize}
|
||||
|
||||
\chapter{\texorpdfstring{Scenarios for testing evolvability criteria
|
||||
@ -710,7 +735,7 @@ Optimierung}\label{besonderheiten-der-optimierung}
|
||||
Approximation}\label{results-of-1d-function-approximation}
|
||||
|
||||
\begin{figure}[!ht]
|
||||
\includegraphics[width=\textwidth]{img/evolution1d/20170830-evolution1D_5x5_100Times-all_appended.png}
|
||||
\includegraphics[width=\textwidth]{img/evolution1d/20171005-all_appended.png}
|
||||
\caption{Results 1D}
|
||||
|
||||
\end{figure}
|
||||
|
||||
BIN
dokumentation/evolution1d/20171005-all_appended.png
Normal file
BIN
dokumentation/evolution1d/20171005-all_appended.png
Normal file
Binary file not shown.
|
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Reference in New Issue
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