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- 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. 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. 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. 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. 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. 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. 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. 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. .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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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