Probabilistic Model Building and
Competent Genetic Programming

Kumara Sastry,  and David E.  Goldberg

Illinois Genetic A...
Motivation

oz.  Genetic Algorithms (GAS)
+ Design decomposition theory (Goldberg,  2002)

+ Design of competent GAS
no:  ...
Outline

«: ~ Background

oz» Objective

o: « Extended compact GA (eCGA)

«: - Probabilistic incremental program evolution...
Background

~: » Competent GAS
+ Messy genetic algorithm (Goldberg,  Korb,  & Deb,  1989)
+ Operators that identify and ex...
Objective

o: « Design a competent GP
o Identify and exchange substructures effectively
o Polynomial scale-up on boundedly...
Extended Compact Genetic Algorithm (eCGA)

~: » A Probabilistic model building GA (Harik,  1999)
+ Builds models of good s...
Marginal Product Model (MPM)

~: » Partition variables into clusters
-:  Product of marginal distributions on a partition ...
Minimum Description Length Metric

ex Hypothesis:  For an optimal model

+ Model size and error is minimum

: ~ Model comp...
.4>. °°! ’. -‘

Algorithmic Description of eCGA

Initialize the population
Evaluate the fitness of individuals
Perform sel...
9’

. '. °’. °‘: "‘

Building an Optimal MPM

.  Assume independent genes ([1], [2], ... ,[. .f: ])

Compute MDL metric,  ...
Probabilistic Incremental Program Evolution

~: » Salustowicz & Schmidhuber (1997)
+ Based on PBIL (Baluja,  1994)
4- Tree...
PIPE:  Model Representation & Sampling

  
   
       
   
         
    

     

 


.   ’

l‘:  2u 01)] ‘  ‘

HR) U U] ‘...
Extended Compact Genetic Programming

oz.  Extension of PIPE
+ Handle multivariate interactions
+ Identify and exchange im...
eCGP:  Model Representation

,  _ _ _ _ _ _ _ _ _ _ _ _ __

I’ '’f.  I '
l ‘I I
 / I I I 6) pl
"1 l _ _/ 
,- I 0,0 x ' x '...
Test Problem Design

~: » Design adversarial problems
+ Thwart the mechanism of the GP
oz» Building-block (BB) identificat...
GP-Easy Problem:  ORDER

~: » Primitive set:  {JOIN-.  X1-.  XL X2: X2.-.  ' - ' +X£~ Xe}
-:  Parse program tree inorder
o...
GP-Hard Problem:  Deceptive Trap

°: ’    X1): (1$ ‘X2: X2: ' ' ' 3 X59 
~: ~ Expression mechanism:  Same as ORDER
»:  BB ...
GP-Easy Problem:  Competent GP vs.  Simple GP

:  '0" simple GP:  O(l2'9‘)
.  -0- eCGP:  O(l2‘86)

4

10;

Average number ...
GP-Hard Problem:  eCGP vs.  Simple GP

- I I T 6 I I l ‘
7 -0- Simple GP:  o(i°-9‘) 

I -6- eCGP:  O(l3‘18) 

10

Average ...
0:0

0
0.0

Future Work

Handle more complex variable interactions
o Overlapping,  and hierarchical building blocks

Apply...
Summary & Conclusions

~: » Developed a competent genetic programming

+ Probabilistic model building GP
4: Combine of eCG...
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Probabilistic Model Building and Competent Genetic Programming

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This paper describes a probabilistic model building genetic programming (PMBGP) developed based on the extended compact genetic algorithm (eCGA). Unlike traditional genetic programming, which use fixed recombination operators, the proposed PMBGA adapts linkages. The proposed algorithms, called the extended compact genetic programming (eCGP) adaptively identifies and exchanges non-overlapping building blocks by constructing and sampling probabilistic models of promising solutions. The results show that eCGP scales-up polynomially with the problem size (the number of functionals and terminals) on both GP-easy problem and boundedly difficult GP-hard problem.

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Probabilistic Model Building and Competent Genetic Programming

  1. 1. Probabilistic Model Building and Competent Genetic Programming Kumara Sastry, and David E. Goldberg Illinois Genetic Algorithms Laboratory Department of General Engineering University of Illinois at Urbana-Champaign http: //www-i| |iga| .ge. uiuc. edu {ksastry, deg}@uiuc. edu Supported by: AFOSR F49620-00-0163. F49620-03-1-0129, NSF DMI-9908252, and CSE fellowship
  2. 2. Motivation oz. Genetic Algorithms (GAS) + Design decomposition theory (Goldberg, 2002) + Design of competent GAS no: Solve hard problems quickly, reliably, and accurately »o= Polynomial scale-up on boundedly-difficult problems ~: - Genetic Programming (GP) 4. Significant advances in application 4: Problem-specific operators + Limited attention to competent operator design >9: Generic operators that adapt linkage = « Solve hard problems quickly, reliably, and accurately
  3. 3. Outline «: ~ Background oz» Objective o: « Extended compact GA (eCGA) «: - Probabilistic incremental program evolution (PIPE) «: ~ Extended compact GP (eCGP) «: « Initial Results: GP-easy and GP-hard problems ~: ~ Future Work & Conclusions
  4. 4. Background ~: » Competent GAS + Messy genetic algorithm (Goldberg, Korb, & Deb, 1989) + Operators that identify and exchange building blocks -: « Perturbation techniques: 4» Messy GA, fast messy GA, GEMGA, LINC, LIMD, -: o Linkage adaptation techniques: LLGA «: o Probabilistic model building techniques: 4» PBIL, CGA, BMDA, BOA, hBOA, IDEA, eCGA, PIPE, -: - Goldberg (2002), Pelikan; Goldberg, & Lobo (2002); Pelikan (2003); Larranaga & Lozano (2002); Chen et a/ (In preparation).
  5. 5. Objective o: « Design a competent GP o Identify and exchange substructures effectively o Polynomial scale-up on boundedly difficult problem o: o Combine ideas from GAS & GP o Extended compact GA (eCGA) (Harik, 1999) o Probabilistic incremental program evolution (PIPE) (Sa| ustowicz 8. Schmidhuber, 1997) o: ~ Study Scale-up of competent GP 0 GP-easy and GP-hard problems
  6. 6. Extended Compact Genetic Algorithm (eCGA) ~: » A Probabilistic model building GA (Harik, 1999) + Builds models of good solutions as linkage groups -: » Key idea: 4. Good probability distribution 5 Linkage / earning . :~ Key components: 4. Representation: Marginal product model (MPM) 4: Marginal distribution of a gene partition 4. Quality: Minimum description length (MDL) as Occam's razor principle =0: All things being equal, simpler models are better 4» Search Method: Greedy heuristic search
  7. 7. Marginal Product Model (MPM) ~: » Partition variables into clusters -: Product of marginal distributions on a partition of genes oz» Gene partition maps to linkage groups MPM: [1,2,3], [4, 5,6], [r%—2,r-1,r%] x1X_2X3 _ X4 X5 X6 _ XL2 79-1 Xx: , Y {P0oos P001» P010: P100» Polls P101» P11oa P111}
  8. 8. Minimum Description Length Metric ex Hypothesis: For an optimal model + Model size and error is minimum : ~ Model complexity, Cm + # of bits required to store all marginal probabilities TH. Cm = |og2(-n) Z [xkt — 1] 1:1 «z. Compressed population complexity, Cp + Entropy of the marginal distribution over all partitions m. ,)(k‘l CD = ‘'1 Z Z [-Pry‘ ‘O92 (Pz‘. j)] -i=1j=1 -: « MDL metric, Cc = Cm + Cp
  9. 9. .4>. °°! ’. -‘ Algorithmic Description of eCGA Initialize the population Evaluate the fitness of individuals Perform selection Build Probabilistic models of selected solutions + Search for an optimal model »« Optimize both model Structure and parameters Generate new solutions using the model Repeat steps 2-5 till termination criteria are met
  10. 10. 9’ . '. °’. °‘: "‘ Building an Optimal MPM . Assume independent genes ([1], [2], ... ,[. .f: ]) Compute MDL metric, Cc All combinations of two subset merges + E9-. {(l1.2l. l3]. ---. [/fl). ([1.3l. [2]. ---. l/ill). ([1]. l2]. ---. l/:1‘-1.0)} Compute MDL metric for all model candidates Select the set with minimum MDL, Cg If Cc > C; , accept the model and go to step 2. Else, the current model is optimal
  11. 11. Probabilistic Incremental Program Evolution ~: » Salustowicz & Schmidhuber (1997) + Based on PBIL (Baluja, 1994) 4- Tree repsentation: Functions and Terminals -: « Univariate model: Each node is independent 4» Complete n—ary tree 1. Probability of selecting a function or terminal o: « Model metric and Search method are not needed ~: ~ Good for simple problems + Cannot handle complex interactions
  12. 12. PIPE: Model Representation & Sampling . ’ l‘: 2u 01)] ‘ ‘ HR) U U] ‘ ‘ , _ I’; U, : [V, |_| ‘|-1 ‘ R V 1'1.» - rm: ~ . »~. ; liq. » L, ___ l'1‘: u - 0.115 ‘. —' ‘ ’~ 3."_” 1'1'. ,:~ -0111 l P I ’ l'1<1n'1 : U U] l ' “ . . 1 %— exp 1 l‘1rcv<I -1111: _ +_ i, I I l'lexpl= 0.3 ‘ i 11 N R ' - ‘ I)l[h—)_. ) 7 U U; _ ’ A, ” : _,. --'‘—. f_: I » V‘ , — ’ ' i 1 (11.71 a ' P P 1 l’ 110 ] . _ , _ R ~ I ‘V2.11 R N. ‘ I R ~: o Start with root node «: o Depth first, left-to-right traversal o: » Choose either a function or terminal based on model
  13. 13. Extended Compact Genetic Programming oz. Extension of PIPE + Handle multivariate interactions + Identify and exchange important substructures -: « Extension of eCGA + Evolve programs instead of bitstrings + Handle variable-size problems «: o Key components: Complete n-ary tree for model building Model representation: Marginal product model Metric: Minimum description length Model Searcher: Greedy heuristic method *2» + ‘P +
  14. 14. eCGP: Model Representation , _ _ _ _ _ _ _ _ _ _ _ _ __ I’ '’f. I ' l ‘I I / I I I 6) pl "1 l _ _/ ,- I 0,0 x ' x ' I _. ‘ I / /-x I _/ ~ / / I X l I‘, I ’, l / l1,| I x: . I I0 , - I _/ ’ ’ I ’ / / I I , / l , ’ 2 "’ 2.1 ’/ 7’) 13' ' I / / //' . ’, / “" ’ . I ' ' l'< 1 [Ir ) I’ l I , _ X, , ‘ I _/ l, / _/ ' , I X1 ~ I ‘_____________ ~: o Complete n-ary tree (similar to PIPE) -: « Marginal product model (similar to eCGA) 4» Partition tree-nodes into clusters + Marginal probability distribution of each cluster
  15. 15. Test Problem Design ~: » Design adversarial problems + Thwart the mechanism of the GP oz» Building-block (BB) identification is critical «I BB structure not known to GP I: No a priori knowledge should be used ~: o Tunable problem difficulty 4- Without changing the functional form ~: o Should bound GP performance
  16. 16. GP-Easy Problem: ORDER ~: » Primitive set: {JOIN-. X1-. XL X2: X2.-. ' - ' +X£~ Xe} -: Parse program tree inorder oz» Primitive first encountered is expressed. Parsedprimitives {X 1e 5'1-I XL X4» X1~ 5'2} -41: K R xx‘ ‘ A “:5. ‘«"’: :.‘ Exprcsscd primitives r H: X, » — — I {X1, 3'2, X4} oz» Fitness: # of expressed primitives Xi ~: - Similar to OneMax in GAs
  17. 17. GP-Hard Problem: Deceptive Trap °: ’ X1): (1$ ‘X2: X2: ' ' ' 3 X59 ~: ~ Expression mechanism: Same as ORDER »: BB identification and exchange is critical § Fitness value . ¢ ° . ° 8 % d . c 8 Expressed primitives Number of ones. u {X1 , }{2, X4}
  18. 18. GP-Easy Problem: Competent GP vs. Simple GP : '0" simple GP: O(l2'9‘) . -0- eCGP: O(l2‘86) 4 10; Average number of function evaluations I O # :5 5 i 10 15 210 30 410 Problem size -: ~ Simple GP & eCGP: Cubic scale-up
  19. 19. GP-Hard Problem: eCGP vs. Simple GP - I I T 6 I I l ‘ 7 -0- Simple GP: o(i°-9‘) I -6- eCGP: O(l3‘18) 10 Average number of function evaluations L 6 12 is 24 310 36 42 Problem size ~: - Simple GP: Could not solve problems > 24 terminals «: » eCGP: Cubic scale-up
  20. 20. 0:0 0 0.0 Future Work Handle more complex variable interactions o Overlapping, and hierarchical building blocks Apply eCGP to different class of problems o Symbolic regression o Study scale-up behavior Convergence-time and population-sizing analsyis o Theoretical and Empirical Extend eCGP to handle other variable types o Ephemeral random constants (ERCS) o Automatically defined functions (ADFs)
  21. 21. Summary & Conclusions ~: » Developed a competent genetic programming + Probabilistic model building GP 4: Combine of eCGA and PIPE + Polynomial scale-up on a GP-hard problem »: » Advantageous when linkage-learning is critical o: ~ Do such problems exist in GP domain? + Building-block identification & exchange is critical + Broader issue of problem difficulty in GP = i= Context, content, and structure + Optimization vs. System identification

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