Combining Competent Crossover and
Mutation Operators:  A Probabilistic
Model Building Approach

Claudio Limal,  Kumara Sas...
Overview

Motivation
Background

° Crossover VS Mutation
' Extended Compact GA
° BB-wise Mutation Algorithm

Probabilistic...
Motivation

0 EDAs outperform simple GA

0 EDAS solve decomposable problems within a low-
order polynomial number of funct...
Motivation

0 Efficiency-enhancement techniques
' Hybridization
‘ Time Continuation
° Parallclization
° Evaluation-Relaxat...
Background

0 Crossover versus Mutation
0 Extended Compact Genetic Algorithm

0 Building-Block-wise Mutation Algorithm
Crossover vs Mutation

o Assumption of knowledge of the problem
decomposition (Sastry and Goldberg,  2004)

o For determin...
Extended Compact GA

0 Estimation of distribution 5 Linkage learning

0 Estimate the distribution of the population using
...
Which Model is Better? 

0 Metric:  Minimum Description Length (MDL)

‘ Preference for accurate and simpler models

0 Sear...
Model Building

0 In each generation,  after selection,  perform a
greedy search for the best MPM

0 Start with the simple...
eCGA steps

Extended Compact Genetic Algorithm (eCGA)

(1) Create A mudoui population of n lll(ll’l(llIil. l:~’. 
(2) Eval...
BB-wise Mutation Algorithm

0 Induces good neighborhoods as linkage groups
(Sastry & Goldberg,  2004)

0 Use linkage learn...
BB Neighborhood

Consider a order-3 decomposable problem

with4 subfunctions (BB partitions):  331 332 333 334
/  101 011 ...
Extended Compact Mutation
Algorithm

mien e Cmnpacl.  'utati01i gorit mi e " A

(1) Create it rzunloiii population of n iu...
A Probabilistic Model Building
Hybrid GA

0 The probabilistic model of eCGA is used for two
distinct purposes: 

l .  Effe...
Joining Both. ..

0 Learn the model

o Perform BB-wise local search on the best individual of
the population

0 Update the...
Updating the Model

Model found:  [x1x2x3][x4x5x6][x7xS][x9]

Best individual of the population 000 011 01 0
Mutated versi...
Hybfid
eCGA

Hybrid Eattended Compact.  Genetic Algorithm
(heCGA)

(1) Create a random population of 11 individuals. 
(2) E...
Experiments

0 m concatenated deceptive trap functions of order-k

1 if u =  I. ‘
ftrup(U) : { l_d_ u * l—d

3 othc-rwisc
...
Experiments

0 Problem 1: Deception

III I
. fd(»’) =  Z frn-p(-1‘a. -s--1‘I. -.'+| - - - ---7'1.-«'+5:—l 1
i=0

0 Problem...
Uniformly Scaled BBs

   

 

k-4
5000 v V ' v '
L’
C
. ..‘
§
C man 3
v’ _
2 2
§ 300 S
5 ' 5
3 .5
8 *5
I00
8
’ E10 .  5;: ...
Number at iunclnon evalualrons,  nk

 

Behaviour of Hybrid eCGA

Uniformly Scaled BB5 .  More Flexible:  can

solve the p...
Uniformly Scaled BBs

o eCGA needs smaller populations,  but takes more
FES than eCMA and heCGA

l.  BBs discovery in a pr...
Population size.  n

 
   

Exponentially Scaled BBs

Number 01 function evaluations.  rye

    

I.  . .   _ 1 .1 . . .  ...
Exponentially Scaled BBs

o eCMA needs exponentially pop sizes and NFES
0 Hybrid eCGA performs similar to regular eCGA

o ...
Number of function evaluations,  nm

Behaviour of Hybrid eCGA

P’
in

1-! 
r

N
U! 

N

in
v

1

Exponentially Scaled BBs
...
BBs with additive Gaussian Noise

. ..
39-
': 

~ eCMA ~
-'°'. ".°‘?9é

   

Population size.  n
all

   

Number ol lunct...
BBs with additive Gaussian Noise

0 Trasition phase on Hybrid eCGA behaviour

0 For small values of noise,  similar case t...
Summary and Conclusions

0 Integration of BB-wise mutation on eCGA

o Probabilistic Model Building Hybrid GA
allows a more...
Extensions

0 Other hybridization configurations

0 Combination with other enhancement techniques for
EDAS

0 Problems wit...
Acknowledgements

o GECCO reviewers for helpful comments and
suggestions

0 FCT/  MCES under grants POSI /  SR1 /  42065 /...
Upcoming SlideShare
Loading in …5
×

Combining Competent Crossover and Mutation Operators: A Probabilistic Model Building Approach

1,348 views
1,213 views

Published on

This paper presents an approach to combine competent crossover and mutation operators via probabilistic model building. Both operators are based on the probabilistic model building procedure of the extended compact genetic algorithm (eCGA). The model sampling procedure of eCGA, which mimics the behavior of an idealized recombination—where the building blocks (BBs) are exchanged without disruption—is used as the competent crossover operator. On the other hand, a recently proposed BB-wise mutation operator—which uses the BB partition information to perform local search in the BB space—is used as the competent mutation operator. The resulting algorithm, called hybrid extended compact genetic algorithm (heCGA), makes use of the problem decomposition information for (1) effective recombination of BBs and (2) effective local search in the BB neighborhood. The proposed approach is tested on different problems that combine the core of three well known problem difficulty dimensions: deception, scaling, and noise. The results show that, in the absence of domain knowledge, the hybrid approach is more robust than either single-operator-based approach.

Published in: Economy & Finance, Technology
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
1,348
On SlideShare
0
From Embeds
0
Number of Embeds
50
Actions
Shares
0
Downloads
0
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

Combining Competent Crossover and Mutation Operators: A Probabilistic Model Building Approach

  1. 1. Combining Competent Crossover and Mutation Operators: A Probabilistic Model Building Approach Claudio Limal, Kumara Sastryz, David Goldbergz, and Fernando Lobe‘ ‘Department of Electronics and ‘Illinois Genetic Algorithm Laboratory C°mP“l°" Sdence E“3in¢¢"i“8 Department of General Engineering U“iV°’5"Y °f A183?“ University of Illinois at Urbana- Champaign
  2. 2. Overview Motivation Background ° Crossover VS Mutation ' Extended Compact GA ° BB-wise Mutation Algorithm Probabilistic Model-Building Hybrid GA Experiments Conclusions Extensions
  3. 3. Motivation 0 EDAs outperform simple GA 0 EDAS solve decomposable problems within a low- order polynomial number of function evaluations 0 However, for large-scale problems, even a low-order polynomial number of FE can be very demanding. ..
  4. 4. Motivation 0 Efficiency-enhancement techniques ' Hybridization ‘ Time Continuation ° Parallclization ° Evaluation-Relaxation 0 Use Model information to perform local search = > Building-Block-wise Local Search 0 A more general Hybridization
  5. 5. Background 0 Crossover versus Mutation 0 Extended Compact Genetic Algorithm 0 Building-Block-wise Mutation Algorithm
  6. 6. Crossover vs Mutation o Assumption of knowledge of the problem decomposition (Sastry and Goldberg, 2004) o For deterministic decomposable problems: ' Mutation outperforms Crossover ' Speedup of O( sqrt(k)*log(m) ) o For decomposable problems with additive Gaussian noise: ' Crossover outperforms Mutation ' Speedup of O( sqrt(k)*m / log(m) )
  7. 7. Extended Compact GA 0 Estimation of distribution 5 Linkage learning 0 Estimate the distribution of the population using Marginal Product Models (MPMS) ° For a linear problem: lxillxzllxsl [X4] [Xsl[X. «,l lX7llX3l lX9llXwllX11llX12l ‘ For an order-3 additively decomposable problem: lx1x2x3l lx4X5xel lX7x3X9l lx10x1lx12l ° For an order-4 additively decomposable problem: lX1x2"3x-ll lX5xex7"3l lX9x10X11"12l
  8. 8. Which Model is Better? 0 Metric: Minimum Description Length (MDL) ‘ Preference for accurate and simpler models 0 Search for the model that minimizes the storage needed to represent the population ° Overall Complexity = Model Complexity + Population Compression Complexity 1 Ill 2*’ C', ,, : log.2(n + ’l ) E(‘. .k’ — I) C', . : n —l}i_j l(_)g‘.2(])i_, ‘) : =l _)= l
  9. 9. Model Building 0 In each generation, after selection, perform a greedy search for the best MPM 0 Start with the simplest model: [x1][x2][x3]. ..[x, _1][xl] 0 Consider all possible merges of two subsets and choose the one that leads to a lower overall complexity 0 Stop when no further improvement is possible
  10. 10. eCGA steps Extended Compact Genetic Algorithm (eCGA) (1) Create A mudoui population of n lll(ll’l(llIil. l:~’. (2) Evaluate all lll(ll'l(llldlb' in the population. (3) Apply s-wise tournauieiit selc-ction (4) llo(, leI the S(! lL‘('[~L‘(, l ll. l(ll'l(, lllzll: i using it gl'Ut'(, l)' . lPll . s:_. -nrcll proce«lure. (5) Generate a new population according to the . IPIl found in step 4. ((5) If stopping criteria is not. satisfied. return to step 2.
  11. 11. BB-wise Mutation Algorithm 0 Induces good neighborhoods as linkage groups (Sastry & Goldberg, 2004) 0 Use linkage learning procedures developed for selectorecombinative GAS o Mutation: Bit-wise = > Building-Block-wise 0 Search: Hillclimbing = > Deterministic or Random
  12. 12. BB Neighborhood Consider a order-3 decomposable problem with4 subfunctions (BB partitions): 331 332 333 334 / 101 011 001 110 Solution A 101 011 010 110 Neighborhood of Solution A within BB partition #3 101 011 011 110 101 011 100 110 101 011 101 110 How can we get the best BB in partition #3? 101 011 110 110 Evaluate all the neighborhood and choose 101 011 111 110 the best individual
  13. 13. Extended Compact Mutation Algorithm mien e Cmnpacl. 'utati01i gorit mi e " A (1) Create it rzunloiii population of n iu<livi<lIml. ~ and eval- uate their fitness. ('2) Apply . -c-vvisu IU| ll‘Il? |Il| (.‘l| l. §| .'l(‘(‘l. lt)ll (3) . Io«lel the select. e«l inclivitlmils using :1 greedy llPll h0tll‘(‘ll proculure. (-I} C'vlIoo. ~‘e the heat. iI| (li'l<lII: l of the poplllution for BB- wise mutation. (5) For earl: (lvt('«‘t(-(l llli pnrtiiiioll: (-5.1) Create '2" — 1 unique lll(llVl(lllill. ' with all pU. ': ~’ll)lC selieuiata in the current BB partition. Note that the rest of the individual remains the 3211110 and equal to the lust solution found so far. (5.2) l‘: ’allllM. l(. ' nll 2* - I ll| (ll’l(lll2|l. 's' and I1-t. Mi11 the lu-st for inutatiou in the other BB pa1‘titions.
  14. 14. A Probabilistic Model Building Hybrid GA 0 The probabilistic model of eCGA is used for two distinct purposes: l . Effective recombination of BB3 that provide rapid global-search capabilities 2. Effective search in the BB neighborhod that locally provides high-quality solutions 0 The key idea is to obtain the benefits from both approaches
  15. 15. Joining Both. .. 0 Learn the model o Perform BB-wise local search on the best individual of the population 0 Update the model parameters (frequencies) according with the BB-wise mutated individual ° Increase the BB instances frequencies of the mutated individual by s ° Decrease the BB instances frequencies of the previous best solution by s
  16. 16. Updating the Model Model found: [x1x2x3][x4x5x6][x7xS][x9] Best individual of the population 000 011 01 0 Mutated version of the best individual 111 111 00 0
  17. 17. Hybfid eCGA Hybrid Eattended Compact. Genetic Algorithm (heCGA) (1) Create a random population of 11 individuals. (2) Evaluate all individuals in the population. (3) Apply s-wise tournament selection (4) . VIodel the selected individuals using a greedy MP. 'I Search procedure. (5) Apply BB-wise mutation to the best individual. (6) Update the frequencies of the MPM found on step 4 according to the BB5 instances present. on the mutated individual: (6.1) Increase the BB instmices frequencies of the mu- tated individual by s. (6.2) Decrease the BB Instances frequencies of the pre- vious best individual by 3. (7) Generate a new population according to the updated . 'IPl»[. (8) If stopping criteria is not satisfied, return to step 2.
  18. 18. Experiments 0 m concatenated deceptive trap functions of order-k 1 if u = I. ‘ ftrup(U) : { l_d_ u * l—d 3 othc-rwisc u - # of 15 in the string k — size of the trap function d — fitness signal For our experiments, k=4 and d= ]/k=0.25
  19. 19. Experiments 0 Problem 1: Deception III I . fd(»’) = Z frn-p(-1‘a. -s--1‘I. -.'+| - - - ---7'1.-«'+5:—l 1 i=0 0 Problem 2: Deception + Scaling m—1 fa. .(<': | = Z 1?ifr. .., .(J‘i. ..l'kun. -.. .J‘uu. - ll (:0 0 Problem 3: Deception + Noise fdr1(. .Y) = fd(. ') + G((). aj-1.-)
  20. 20. Uniformly Scaled BBs k-4 5000 v V ' v ' L’ C . ..‘ § C man 3 v’ _ 2 2 § 300 S 5 ' 5 3 .5 8 *5 I00 8 ’ E10 . 5;: 3;; : . : : '. . ': . 5° . J O ocGA. ou'm"’) : ' 1:‘ 1: I O ¢CGAO1m“} I: . V ,5 2 . . , ,5 ; IIoCMAO(m'I7 8 I : fleCIM. Cxm) _ I _ _ hoCGA"O{mx) O M; ‘ H ‘ 0 hoOGA.0[m”) 2 3 4 6 10 15 20 i 3 4 6 to 15 20 Number of building Mocks. m Number of building blocks. m k=4
  21. 21. Number at iunclnon evalualrons, nk Behaviour of Hybrid eCGA Uniformly Scaled BB5 . More Flexible: can solve the problem within a bigger range of pop size than eCMA k=4.m=10 o 4 lines, 4 different hitting times (# gen. ) 2500 Population 5128. n who was 2500 aocc
  22. 22. Uniformly Scaled BBs o eCGA needs smaller populations, but takes more FES than eCMA and heCGA l. BBs discovery in a progressive way 2. Mixing time o eCMA scales better than eCGA o heCGA behaves similarly to eCMA
  23. 23. Population size. n Exponentially Scaled BBs Number 01 function evaluations. rye I. . . _ 1 .1 . . . o . co». ;oqm“; 0 -csA: <:<m“'> . .. .. .. . . . . . . D QCMA; cum”; 9 ocMA'OIm'; |1 1°, ‘ O nece: Loum") ‘O. 9 MCGNOW I 2 3 4 5 1a 15 2o 2 3 4 6 10 15 20 Number of building blocks. m Number or building blocks. in
  24. 24. Exponentially Scaled BBs o eCMA needs exponentially pop sizes and NFES 0 Hybrid eCGA performs similar to regular eCGA o I-Iybrid eCGA changed his behaviour
  25. 25. Number of function evaluations, nm Behaviour of Hybrid eCGA P’ in 1-! r N U! N in v 1 Exponentially Scaled BBs I10‘ k=4.m=1O I . . . _l 1000 1500- 2000 2500 3000 3503 4000 450:) 5000 Population size. n 0 Number of PE grows almost linearly with pop size 0 Increasing pop size will not reveal much more BB information
  26. 26. BBs with additive Gaussian Noise . .. 39- ': ~ eCMA ~ -'°'. ".°‘?9é Population size. n all Number ol lunction evaluations. nu Noise-to-signal ratio aflxiogdi Noise-to-signal ratio. crznllcr/ cl)2 10'
  27. 27. BBs with additive Gaussian Noise 0 Trasition phase on Hybrid eCGA behaviour 0 For small values of noise, similar case to the deterministic function: ' Hybrid eCGA do similar to eCMA, which is the best e For larger values of noise: ’ Hybrid eCGA do similar to eCGA, which is the best
  28. 28. Summary and Conclusions 0 Integration of BB-wise mutation on eCGA o Probabilistic Model Building Hybrid GA allows a more general hybridization o In the absence of domain knowledge, the hybrid eCGA is more robust than either single- operator-based approach
  29. 29. Extensions 0 Other hybridization configurations 0 Combination with other enhancement techniques for EDAS 0 Problems with overlapping BBS 0 Application to real-world problems
  30. 30. Acknowledgements o GECCO reviewers for helpful comments and suggestions 0 FCT/ MCES under grants POSI / SR1 / 42065 / 2001 and SFRH / BD / 16980 / 2004, AF OSR/ USAF under grants F49620—O0-0163 and F49620-03-1-0129, NSF under grant DMI-9908252, and ITR grants DMR-99- 76550 and DMR-0121695.

×