Online Population Size Adjusting Using Noise and Substructural Measurements

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This paper proposes an online population size adjustment scheme for genetic algorithms. It utilizes linkage-model-building techniques to calculate the parameters used in facetwise population-sizing models. The methodology is demonstrated using the dependency structure matrix genetic algorithm on a set of boundedly-difficult problems. Empirical results indicate that the proposed method is both efficient and robust. If the initial population size is too large, the proposed method automatically decreases the population size, and thereby yields significant savings in the number of function evaluations required to obtain high-quality solutions; if the initial population size is too small, the proposed scheme increases the population size on-the-fly and thereby avoiding premature convergence.

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Online Population Size Adjusting Using Noise and Substructural Measurements

  1. 1. Online Population Size Adjusting Using noise and Sub-Structural Measurements Tian-Li Yu, Kumara Sastry, David E. Goldberg Illinois Genetic Algorithms Laboratory Department of General Engineering University of Illinois at Urbana-Champaign http: //www-illigal. ge. uiuc. edu ksast uiuc. edu Supported by AFOSR F49620-03-1-D129. NSF/ DMR at MCC DMR-99-76550. NSF/ ITR at cpso DMR-0121695. DOE at FS-MRL DEFG02-91ER45439.
  2. 2. Motivation -: - Facetwise population-sizing models + Adequate supply of sub-structures [Goldberg et a/ , 2001] + Good decisions between competing substructures [Goldberg eta/ , 1992, Harik et a/ , 1997] + Accurate sub-structure identification [Pelikan eta/ , 2003] ~: ~ Linkage (sub-structure) learning methods + Implicit (e. g., LLGA, BOA) + Explicit (e. g., eCGA, DSMGA) «: - Utilize sub-structure information in estimating population size adaptively
  3. 3. O0 0 0.0 00 O 09 O 00 0 Outline Background Population-sizing models - Design structure matrix GA (DSMGA) Population-size estimation Population-size adjustment Summary & conclusions
  4. 4. Related Work ~: - Smith and Smuda (1993) A o Target selection loss L A o Adjust the population size by the factor of ~: » Harik and Lobo (1999)—Parameter| ess GA 9 eCGA (Lobo, 2000) o hBOA (Pelikan and Lin, 2004)
  5. 5. How Should I Size the Population Size? 1. Ensure an adequate supply of raw sub—solutions 4» Population size should be large enough to ensure at least one copy of all competing BBs in the population 2. Make decisions well among competing sub-solutions + Population size should be large enough to prefer the best BBs over worse ones with a high probability. 3. Identify and mix sub—solutions well air Population size should be large enough to ensure that the “innovation time" is less than “takeover or convergence time”. [Goldberg (2002). Design of Innovation]
  6. 6. Building-Block Supply -: - Ensure the presence of competing schema o Temporally through genetic operators o Spatially (Initial population) [Holland, 1975; Goldberg, 1989: Reeves, 1993; Goldberg, Sastry, 8. Latoza, 2001; Sastgg, O'Reilly, Goldberg, & Hill, 2003| Sub-structure size nzXk(k| ogX+lnm) # competing sub-components it components m‘ Population size, n 10' 10'- o 0 0000005’ to‘ Number of BBs, m od-‘o0°°° In
  7. 7. Gambler's Ruin Population Sizing Model -: - Combines decision-making and supply models [Harik. Cantu-Paz, Goldberg. and Miller, 1997] Signal between competing BBs Fitness variance of a BB Failure probability Sub-Component size (BB size) Error tolerance Signal-to-Noise ratio 19;). — Gambler'sruinrnode| ‘ 0 Empirical 140» # Competing sub-components 12° # Components (# BBs) 100, § . . so 40 20 . 0 200 400 600 B00 1000 Harik. Cantu-Paz. Goldberg. & Miller. ECJ 7(3) I999. Problem size, I
  8. 8. Population Sizing For Sub-Structure Identification ~: o Estimation of distribution algorithms [Pelikan, Goldberg, and Cantu-Paz, 2001; Sastry & Goldberg, 2001; Pelikan, Sastry, and Goldberg, 2002] Signal between competing BBs < Fitness variance of a BB ’ Sub-Component size (BB size) '60 -1 0 DSMGA Theory ijcn=0 24) X eCGA _ _ _Theerv ijcn=0 3'38) 100 . - I40 I20 it Competing sub~components < .9! Components (,3 BBs)< I .0 Bounds DSMGA/ EDA population sizing 40 Population Size NCCOCG (N 20 I 2 3 4 ‘ '3 T 8 Number of BB5 lm)
  9. 9. O0 0 0.0 00 O 00 Purpose: Online Population-Sizing Framework Develop an online population-sizing framework using sub-structural information Sub-structure identification o What are the components? Sub—structure quality estimation o What are the relative quality of sub-components? Sub-structure-based population sizing estimate o Ensure adequate supply of good sub-structures? o Ensure growth of good sub-structures?
  10. 10. Sub-Structure Identification: Use Linkage-Learning Techniques -: - Sub-structure identification o What are the components? -: - Use linkage-learning techniques o Demonstrate using DSMGA [Yu et a/ , 2003] o Can also use extended compact GA [Harik, 1999] : ~ Brief description of DSMGA next. ..
  11. 11. Design structure matrix (DSM) -: - GM powertrain development teams [McCord & Eppinger,1993] Engine Block A » Cylinder Heads B )amshaltlValve Train C Pistons 0 Connecting Rods E Crankshaft F Flywheel G Accessory Drive H Lubrication I Water F’Ump. "C0-oling J Intake Manifold K Exhaust L E. G.R. M Air Cleaner N A. l.R. 0 Fuel System P Throttle Body 0 EVAP R Ignition S E. C.M. T Electrical System U Engine Assembly V 0: No interaction 1: low interaction 2: med interaction 3: high interaction A iii 3 3 3 2 3 I 3 3 3 2 I I i. » IJVJ; -r‘;4-——”; -4'“ tdlelu run. ..i. o~. ai. ai. ¢i. .i. ;'. .aiu-. .i. i< lJ—— 4;-, .,. ... .
  12. 12. DSM clustering / Reordered PNORBCJKADIFGEOLMHSTUV‘ Fuel System P :11 Air Cleaner N Throttle Body 0 EVAP R Cylinder Heads B iamshafm/ alve Train 0 Water Pumpfcooling J Intake Manifold K Engine Block A Pistons D Lubrication I Crankshaft F Flywheel G Connecting Rods E A. l.R. 0 Exhaust L E. G.R. M Accessory Drive H Ignition S E. C.M. T Electrical System U Engine Assembly V Clusters 'J 'J '4) —I I) -— Id at ‘A —- ‘A Bus modular
  13. 13. DSMGA all : I‘ an fia pl’ #2 fir‘! I! :'. | an! Generation 0 Generation 5 Generation 10 WOUI Ill HIIH Gen5 ll ""lllTl’lTTTFlT‘l‘. Fl“. Gen10 II II lllllliVTl‘l_. 10 3-bit Trap Function
  14. 14. Online Adjustment of Population Size -: - Sub-structure identification -i Use linkage-teaming techniques = l= E. g., DSMGA, eCGA, BOA -: - Sub-structure quality estimation 4. What are the relative quality of sub-components? -: ~ Use fitness-estimation technique used in developing probabilistic endogenous fitness model 4» Pelikan er a/ 2004, Sastry et a/2004
  15. 15. Sub-Structure Quality Estimation ~: ~ Identify key sub-structures of the search problem -: - Estimate the fitness of sub-structure instances . Average fitness of all Average fitness of all Fitness of _ , _ _ , , _ , _ . ' evaluated individuals with - evaluated individuals in substructure instance _ _ the schema instance the population Fm. I. // H fr / / ~: ~ Used to estimate fitness of individuals [Sastry, Pelikan, & Goldberg, 2004. Pelikan, Sastry & Goldberg, 2004]
  16. 16. Online Adjustment of Population Size -: Sub-structure identification -i Use linkage-teaming techniques ~: - Sub-structure quality estimation 4» Schema fitness estimation ~: - Population-size estimation based on sub-structure info 4» Bounded by sub-structure identification + Estimate all parameters on-line using sub-structural measurements
  17. 17. Online Population Size Estimation Fitness variance of a BB it Components (# BBs) Sub-Component size (BB size) Signal between competing BBs Failure probability Error tolerance Signal-to-Noise ratio it Competing sub-components -: o Sub-structure models gives mand k, o: H 3 Bi: Schema of the i‘“ sub-structure tie. = ‘/3'2“ (H 3391 dz‘ = maxilf (H3891 — 5eC°”d—maXilf(HBB. Jl
  18. 18. How To Inject New Individuals? .0 (J1 «: ~ Aim: Increase the success rate of small initial population size ~: ~ Inject new individuals to ensure 4» BB supply + Diversity Proportion of correct 885 ~: ~ Issues to be addressed + Average fitness decreases + Convergence difficulty Estimated population size General on
  19. 19. Injection scheme: Initialize Early and Recombine Later ~: ~ Observations + Injection of new individuals 9 BB supply 9 early stage of GA run + Recombination 9 BB mixing 9 later stage of GA run ~: - Injection scheme, when l'1,+, >l1, + Early stage: Generate n, children by recombination, and inject (n, +, - n, ) new individuals. e Later stage: Simply generate n, +, children. z- When to switch? + The population-size estimation indicates large population is not needed.
  20. 20. Population-size adjustment on-the-fly 700 . ... ... ... ... ... ... ... -.. ... ... ... ... ... ... ... . . ... ... ... ... ... ... ... .-. ... .. ... ... ... ... .. . .-. ... Generation Similar behavior for different initial population sizes (except at extremely small initial population size).
  21. 21. Number of function evalutions nm Efficiency: Adaptation Doesn't Require Significantly Extra Evaluations -: ~ Large n0 does not cost too much. Generate new individuals by recombination 9000 : G O O C 7000 6000 r 5000 4000 3000 r 2000 r 1000 O FIX ' : F| X—trendline XADJ - - -ADJ-trendline 200 we aim Initial population size no son 1 000 Initialize early recombine later N 6003 . ... . . . . . . > c’; _ ""A J 3 E’ 5500 """ " “ --A 'ADJ+| NJECT ,9 ; : . . : :2: 5003 . ... . . . .3. . . . . . . . . . . . . . . . . . . . . .. w 4500 . ... . . . . ... .. .5 x 2 8 A 2 C 4003 ' ‘ ‘ ' ' ' ‘ ' ' ' ' " .2 | 1 : v. .. ‘F E E l 2 . . . . . . . . . . . . . . 15;. -.b. , v 3 A23. 1 E 3 Z 3000 ------ -A A ----- »— 2503 ' ' ' ' 0 200 400 600 300 In tial population size no
  22. 22. Success rate .9 .9 Robustness: Small Initial Population Sizes Yield Success ~: - Small no does not easily cause GA failure. Recovered from small initial population sizes. 1 : : I )’—‘ I 5 ' I 5 I 2| - 6 2 0.6 3 §0A 4 . . 6 . 0 50 100 150 200 Initial population size no 250 200 Initial population size no i" 50 100 150 250 ; --. {i--Ki3J -A-ADJ+| NJECT 300
  23. 23. What’s next? -z- Other linkage learning GEAs «: ~ Update rule 0 "? rt+1 E 9714+ (-1- . <¥)fit+1 o Pros: The change of the estimated population size are smoother. A more noise-robust scheme. o Cons: Another parameter to tune. -: - Optimal switch time o When do we have high enough confidence to switch from the individual injection scheme to the regular recombination? ~z» Real-world problems
  24. 24. Initial guess ozo Not seriously sensitive «to Prior knowledge (k=4~8, an antenna problem) ozo Doubling scheme + Start from small initial population + Keep doubling it until the parameter (m, /r, ... ) does not vary much + Use the final result as the initial guess of population size + Worst case: 4 times of what is needed, but only for the first generation.
  25. 25. Test Problem: Modified m-k Trap Functions -: - Linkage learning is critical to GA success ~: « Massively multimodal: 2"‘ distinct optima l| ll00000l00.. .|l0| I Jl II II i +4 ‘, ]L. l 0 J 3 I .00 0.75 , 0.50 0.25 0.00 ‘i Fitness fitness — 1.00 + L00 + 0.68 + + 0.00 [Ackley, 1987; Goldberg, 1987; Deb 8. Goldberg, 1993]

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