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Evaluation Relaxation as an Efficiency-Enhancement Technique: Handling Variance and Bias

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This study develops a decision-making strategy for deciding between fitness functions with differing bias values. Simple, yet practical facetwise models are derived to aid the decision-making process. …

This study develops a decision-making strategy for deciding between fitness functions with differing bias values. Simple, yet practical facetwise models are derived to aid the decision-making process. The decision making strategy is designed to provide maximum speed-up and thereby enhance the efficiency of GA search processes. Results indicate that bias can be handled temporally and that significant speed-up values can be obtained.

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  • 1. Evaluation Relaxation as an Efficiency-Enhancement Technique: Handling Variance and Bias Kumara Sastry, and David E. Goldberg {ksastry, deg}@uiuc. edu Illinois Genetic Algorithms Laboratory University of Illinois at Urbana-Champaign Urbana, IL 61801 http: //www-i1liga1.ge. uiuc. edu Genetic and Evolutionary I’ O 5.. " r °EB'. '3#’. t.‘§. '.‘Z'. ' E0033 " I July 9-13, 2oo2. New York, NY
  • 2. Evaluation Relaxation as an EET 1 BEICKQFOUTICI . Competent GAS (Goldberg, 1999) — Solve hard problems quickly, reliably, and accurately — Subquadratic number of function evaluations . Costly function evaluations - Subquadratic function evaluations can be high - f = 100, Time/ Eval. = 1 s. Need 2.5 hours . Efficiency—enhancement techniques (EETs) - Evaluation relaxation is one such EET In. .. ll u-l ILL . ..»un-Io-.5.. a. . ... ... ... _.. ..m . ... ... . . ... ... ... ... ... .. . ‘i? f¢. ‘1‘: IE002: K Sastry. D. E Goldberg . """‘. ... ... '.'. ‘.‘: '.‘. '7.'. ... .. Ji. :Iu.1-E: w—--mm "'
  • 3. Evaluation Relaxation as an EET 2 . Evaluation Relaxation — Replace accurate, but costly function evaluation — Use approximate, but cheap function evaluation — Approximation introduces error . Practical guidelines are lacking — Which, When and How? — What's the speed-up? gt ~: ::; ':': :aooa; ‘ K Sastry, o. E Goldberg
  • 4. Evaluation Relaxation as an EET 3 Overview . Evaluation relaxation . Decomposition of evaluation relaxation . Problem statement . Key assumptions . Part 1: Handling Variance . Part II: Handling Bias . Conclusions _, t ~: :;: ;': ;aoo2§‘ K Sastry. D. E Goldberg
  • 5. Evaluation Relaxation as an EET Evaluation Relaxation 0 Use approximate, but low-cost fitness function . Approximation introduces error . Previous work: — Grefenstette & Fitzpatrick (1985) — Aizawa & Wah (1993, 1994) — Miller & Goldberg (1995, 1996, 1997) — Jin, Olhofer, & Sendhoff (2001) - Smith, Dike, & Stegmann (1996) - Sastry, Goldberg, & Pelikan (2001) - Albert & Goldberg (2001, 2002) . ... ._. . . .-. ... ... ... . , , *: ::: _': ':; aooa; K Sastry, D. E Goldberg . ""‘. ... ..'. '.‘. ‘:': .": .'". f. ’” : .": .-1:. -.'. ‘**. .. ADl.1¢: Vuvuaxhv
  • 6. Evaluation Relaxation as an EET 5 Decomposition of Evaluation Relaxation . Fitness function approximation — Endogenous: Error introduced by the GA * Eg. , fitness inheritance — Exogenous: Error in fitness functions * Eg. , response surface methodology 0 Components of error: (Geman et al, 1992) — Variance: Changes the fitness value — Bias: Changes the optimal solution K Sastry, D. E Goldberg ic- Bi 85 gill; till! I ‘ll: I
  • 7. Evaluation Relaxation as an EET 6 Problem Statement 0 Two fitness functions — High cost, but low error — Low cost, but high error . Which fitness function to use? — Spatial vs. temporal decision making — Strategy that yields maximum speed-up — Utilize facetwise models * Convergence-time and population-sizing models gt ~: ::; ':': :aooa; ‘ K Sastry, o. E Goldberg
  • 8. Evaluation Relaxation as an EET 7 0 Non—over| apping population of fixed size . Generationwise selectorecombinative GAs — Uniform crossover, tournament selection . Binary encoding and fixed string length 0 Stationary fitness functions . models for OneMax domain . Tight linkage or usage of a competent GA , .t '"‘: i;". i."'3E'; 'E'c'J"r3'zW§“ K Sastry. D. E Goldberg J5Ilfl¢: ‘Ouv'uhV
  • 9. Evaluation Relaxation as an EET 8 Part 1: Handling Variance . Noisy OneMax function — OneMax function + Gaussian noise 0 Two fitness functions: f1. and f2 — fl: High cost (c1), and low noise variance (airl). — f2: Low cost C2, and high noise variance (ofvz). c. > C2, 0%. ] < 0%, ? — Higher the accuracy, costlier the evaluation. . Averaging eliminates variance effects , _£ m‘:7o“': ':: ':‘f': k§i'5i5i鑧‘ K Sastry, D. E Goldberg
  • 10. Evaluation Relaxation as an EET 9 Convergence—Time Model - I . Selection—intensity based model (Bulmer, 1980) — Mllhlenbein & Schlierkamp-Voosen (1993); Thierens & Goldberg (1993); Back (1994); Miller & Goldberg (1996) a Gaussian fitness distribution: F, ~lf(; i., of+o,2,. _.) 0 Prop. of correct BB3: (Miller & Goldberg, 1996) I Pt+i ‘Pt: VPt(1‘Pt) pew/5 — Elongation factor, p, _.(t) = ‘/1 + / of) 0 Exact solution is complex 3‘? ‘ ill I gt *: ::; ':': :aooa; K Sastry, o. E Goldberg . A 1:1 I ii I I
  • 11. Evaluation Relaxation as an EET 10 Convergence—Time Model — II . Prop. of correct BB5: (Miller & Goldberg, 1996) I Pt+1—Pl= j/ Pi(1‘I7i) p. .-/7 - Constant elongation factor: pc(t) = /)c(t = 0) 0 Convergence time 2 tcon" : + UNI ‘ 2I 03 . Convergence-time ratio 2 2 2 2 _ teem. (UNI) _ (00 + UNI) — 2 2 '2 tcouv (aN, _,) 00 + 0N2 tea‘ I ll . . -. .,. .,. ... .,. ._, , . ... ... .. . .-. ... . kg , *; ““'* , ,,, ,: ;aooa; K Sastry, D. E Goldberg . ""‘. ... ..'. '.‘. .". ':. ' . ... .. ' : .v-. olnai. uC -. ... u.d. -our ti
  • 12. Evaluation Relaxation as an EET 11 Convergence—Time Model Verification TDUWIIIVIOHI SIIO. S I 2 Tournament $510. 5 I 3 I I _‘ ,3 as} 0.3 ‘E 0.3 E E o 5 heavy 0.6 2 .5. E 9 u 100 E‘ 3 0.4 - zoo 0.4 , § - aoo . 4oo Toxrnarnent size. s = 5 0.! DJ Convurgonco limo ratio. l Convergence lime ratio. l =50 .100 -200 -300 -400 R1:-Vvuht and tvuIiII. v'U_g I311! . , .7 """". ,,. ,.. ... .Aeooa; K. Sastry. D. E. Goldberg m. """. ... ... """. ... ... ... ... g‘.1I. I‘£ Ami): -II: -o--mm
  • 13. Evaluation Relaxation as an EET PopuIation—$izing Model 0 Gambler's ruin population-sizing model — Harik. Cantu-Paz. Goldberg & Miller (1997) — Building-block supply + Decision-making model . External noise (Miller & Goldberg, 1997) x/77 d 2"’_1ln(a) 03 + 0%. ’, 12 I 2 2 2 E n _ n. (UN! ) _ (00 + (TM) "T ‘ 2 ‘ 2 2 71 (UN! ) 00 + UN2 gt K Sastry, D. E Goldberg M, ., ,, ., nu-. .-uuounuo.
  • 14. Evaluation Relaxation as an EET 1.3 PopuIation—Sizing Model Verification Population size ratio. n’ Population size ratio. n’ Tournament size. 5 = 2 Tournament size. 5 = 3 1 . E. .9 0.8 > E 0.6 - r: I 5% he Theory - Q8 i I so 0.4 - § 0 4 - I " I = 100 - ‘ I = 200 "5 i: > l - 300 0.2 0 2 0 0.2 0.4 0.6 0 B 1 0 0.2 0.4 0.6 0.8 1 Tournzimr. -nt s-zc. s = 4 Tournament size. s = 5 . ; ‘ ‘ ' , o 1 ‘ ‘ ‘ o. fi, ~e~. *.‘c‘. ‘3 . - 5,299’ 0.8- $_g. e"‘ “’ o a 5939’ B9 U 0- 5 P"l"' 3 33 0.6 A : ;.§. >’* 5 o 5 3?. cl ‘7 g 191/ 1 Theory "'r (" ~ . " / = 50 0.4-)5/’ g 04 3/ x I=10O lg - : I = 200 L I - 300 02 . U 2 . . . . . 0 0.2 0 0 2 0.4 0.6 0.8 1 a: K Sastry, D E C-‘. c,i| (lt)<; rq :5’; : r.“. L . I-D-n nu. .. u u. «nu. .ip. r-. . win. .. it 4.4 i. .._. ... . ll--1 or mo. .. --p-4..
  • 15. Evaluation Relaxation as an EET Number of Function Evaluations 0 Convergence-time, and population-size ratios 1 5 tc_.1' : 717' = ( ) . No. of function evaluations ratio ( 03 + 0%. ‘ 03 + air) 2 2 00 + 0 N1 05 + 0%. ? n'fe, vr : nrtc, r ) -A-a. -.. .«ir up--—. ia. a_. r -——¢. n--I I-u—m-u luvtlru and tvuulhy (#305131 E - , .. ... ..-. 2002; 1.: l! mu ‘¢—v'uhV K Sastry, D. E Goldberg im-h a-i, I inv. x.~»unI-45.4. 14 iwamu «run. a I rs. ..-'u-nu u
  • 16. Evaluation Relaxation as an EET Number of Function Evaluations 2 Tournament size. 5 s 2 ‘- Tournament sue. s = 3 r 1 _ , _ :3 1 , 0' o‘ »= .. .= E . ll‘ g 0.8 g 0.8 .9 .9 S 0.. S _ em, .5 0.4 -. ;. I = so 3 : I133 ‘=3 3 °‘2 : » I = 300 5 3 . ;;~ I - 400 3 g 0 . . . . g Z 0 0.2 0.4 20.6 0.8 1 2 “fl ‘UN Tournament size. 5 = 4 Number ol luncliorr evaluations ratio. rt“ Number oi function evaluations ratio. rt“ K Sastry, D E C-‘. c,i| (lt)<; r:; .14} 15
  • 17. Evaluation Relaxation as an EET 16 Optimal Decision for Handling Variance . Total computational cost ratio 08 + 0%“ 0(2) + 0%, ? C1 C1 CL0t, 'r : :nfe,1' : (T ( »2 12 o If C2/C1 > (03 +0fi,1)/ (05 + 012%), then use f1 . If C2/c1<(a3+0,2V1)/ (c7g+ a§,2), then use f2 o If C2/C1 = (a3+01?V1)/ (08+0}2V2), then use fl or f2 . ,.-. .. -. . tW, l|J(', V‘hC. ‘ . ... t.. .. . ... ... ._q , _, - _‘““‘“°w. ,, ,, _: _aooa; K Sastry. D. E. Goldberg . """'. ... ..'. '.‘. ?_: “."""'». .:. ... ..
  • 18. Evaluation Relaxation as an EET Verification of Optimal Decision Function evaluation cost iatio. clfcl Function evaluation cost ratio. oz-’c‘ 0.8’ 0.6’ 0.4 0.2- 0.8 0.6’ 0.4 ~ 0.2 0 | =100 El I-fix) l: r | =300 C: - I=400 0'’ Tournament size. 5 = 2 Theory (J I: 50 Choose 12 0.2 0.4 0.6 0.8 1 N0 ol lunclion evaluations ratio fl‘. r Tournament size. 5 - 4 / Theory" Cl I-50 8 . - I-t00 Choosel‘ D l=20O D | =300 0.2 0.4 0.6 0.8 1 No. o1funi: tion evaluations ratio n__ ' K S. i~. »t"y'. D E (_~‘. ~:, i|Lll)<: r:_i Function evaluation cost tatia. :2/c‘ Function evaluation coat ratio. czlc‘ 0.8 0.6 0.4 0.2 Tournament size. s = 3 Theory ch l=50 ’ x l=100 Chooser‘ Qfi D l-200 , ._. . 1:300 .3 l=400 Chooselz , -D J9 0 O2 04 06 DB 1 No of lunct-on evaluations ratio. rt“ Tournament size. 5 - 5 0 02 0.4 0.6 0.3 1 No. 0! tunchon evaliiiations ratio. rt“ on-. _.. in. .. . .u u. -u~. .i. ... . . 1.7
  • 19. Evaluation Relaxation as an EET 18 Model to Predict Speed-Up . Optimal vs. Na'i've decision — Use low-variance, high-cost fitness function 2 2 , 00-l-0N Ct.0t. v' < 17.5 : ('1 a"+a«'V'2 1 otherwise an-gr. -an nu-—. i.i—ry . ... ... . -. . gfi 7‘&. “". ... ‘.‘2_aooai" K Sastry. D. E. Goldberg . """. ... ... .'. '.‘. T.': ‘."""". ... ... .. ugI. A&I. |M. A .3 l) mu Vow -in hV ""'”-'-""“'
  • 20. Evaluation Relaxation as an EET 19 Verification of Speed-Up Tournament size. s - 2 Tournament size. s - 3 8 7 in II 6 ‘T ‘-T if $ 5 l E‘ 3 2 ’ . —. - : . ‘ ' ’ 1 . . > 7 0 0.2 0.4 0.6 0.8 t 0.4 , 2 Tournament size. 3 - 4 Tournament size. 3 - 5 Spaad—up. I1‘ speed-up. vi‘ .6 0.5 1 o 0.2 04 ofuof. - 2 hlcvillt o--l~tu; -nil; ---.3, float. -1-: ‘$? ilo¢—'y , - .7 ’*""m, ,_, ___eoua; K. Sastry, D. E. Goldberg . '*"'. _._. _.. ._1_, _.I1_1._. .*“. ... u. i». ;uui-. s.. n. a, -ainvlzu--tam
  • 21. Evaluation Relaxation as an EET Aside: Sampling Fitness Functions . Noisy fitness function f’ = f+N(0,afV) . Sampling fitness function 1 , f71a(S): : 3 J. =1 . Effect of sampling — Variance: 0%, /n, ., - Cost: (1 + 71.5.13 , j£ K Sastry, D. E Goldberg 20
  • 22. Evaluation Relaxation as an EET Optimal sampling . Total computational cost cm oc (a + nsfi) (03 + 0%, /72.3) . Optimal sampling rate act oi. 8n, 2 n‘ (EON L3 06 . Same result as Miller & Goldberg (1997) - Does not account for Convergence-time 0 | | i. :|- — Uses BB decision—making model K Sastry, D. E Goldberg 21
  • 23. Evaluation Relaxation as an EET Handling Variance: Summary . Decision-making strategy: — Fitness functions with differing variances — Utilized facetwise models — Provides maximum speed-up . Key parameters: — Fitness-variance and fitness-cost ratios 0 Variance can be handled spatially K Sastry, D. E Goldberg 22
  • 24. Evaluation Relaxation as an EET Ingredients of Error . Variance: Changes the fitness value — Sampling eliminates effects of variance . Bias: Changes the optimal solution — Bias cannot be averaged out — Can be handled temporally * Use high-bias function initially * Then switch to low—bias function K Sastry, D. E Goldberg nu. -._-ii ll'Il—oIQulI'v -——a. u-a i-—. .- 23 lubrntl nun. -i»s. ..4-nan in. -ii M-l tit. an--—upu. .n. .A.
  • 25. Evaluation Relaxation as an EET 24 Bias in Fitness Functions . . , . . . . . . . . . 12 _V_. _1»~, __ 1 12 on « on _ 1 / ' Fr0'fll VIAO O 0 Films: who oo '4 ll 04. V‘. '‘', ‘ - 04- 5| - . ll , ".1 , l oz '35 « oz : ‘ l . _ Aoolnle Ilneee iumsiion "7-. Acairue Dian luvrcltun l ‘ * Ftussliirttloii niilti bias 2 — Fhnenllialiiiiwillitiian ', l 9 I If ' 0 0 ll 0.! 0.2 0.) (IA 0.5 00 07 ea 09 I 0 (ti 0.? OJ (H 05 06 Q7 03 09 1 Decisionimiulilevum Dccisionvuiiableveue (a) Bias in optimal value (b) Bias in optimal solution Irish: and cumin. “ -n-sc-ii -ion-I--—-i K. Sastry. D. E. Goldberg m. """__ : '.’. ;.’I. .'_; .‘. ‘.. '.. ".r"'. ... .. -u. ~»~‘>anu-'o-3.. .. uni): -II: -o--mm
  • 26. Evaluation Relaxation as an EET Part 11: Handling Bias Two fitness functions: f1. and f2 — fl: High cost (C1), and low bias (bl) — f2: Low cost ((32), and high bias (b2). (31 > C2, ()2 > b1 - Higher the accuracy, costlier the evaluation. Use f2 initially Switch to fl later Key parameter: Switching time K Sastry, D. E Goldberg {ii l ail? I in ialli 3 l 3% 25
  • 27. Evaluation Relaxation as an EET Weighted OneMax function Weights associated With each bit K f(X) = Z: 'i1.‘, j.’I. T,: ‘lili 6 {-1, +1} i=1 Different bias values = Different weights OneMax properties - Unimodal, uniformly scaled - Analytical tractability Bias does not affect population sizing K Sastry, D. E Goldberg I fill. falls: 3 i ll 26
  • 28. Evaluation Relaxation as an EET GA Run: Initial stages (it <15) . High-bias, low—cost function fl . Gaussian fitness distribution £Pt‘(€‘€1‘b) €pi(1-pi) lt. f2.~t 2 Ufzlt . SeIection—intensity based model 1 1 + , I t pl = — sin — 2 / Z , l K Sastry, D. E Goldberg we ‘¢—w'uhV """""' I igllf ‘ll: 27
  • 29. Evaluation Relaxation as an EET GA Run: At Switching Time (it = t5) . Proportion of correct building blocks l1+si= i(": )l 7 1 Pt. 2 0 Switch from fl to fl — f— b BBS: pt” — b BBS: 1 —pl, ,‘ . Corrected proportion of correct BB5 b b I 17;, - (1— 2? Pt, + , l K Sastry, D. E Goldberg V , ____M L. ‘,. ‘.‘. ‘.3i-. .'ii. ‘.‘.1 ‘ll: 28
  • 30. Evaluation Relaxation as an EET GA Run: Later Stages (2% > ts) . Proportion of correct BBS - Unbiased and biased portions are normally distributed — Overall fitness distribution is normal I t—t. , , _ l1_C0S (; )+2s1n 1 (l/ pl‘ l/ E . Convergence time / F t-conv : ls ‘l’ T 1 2 P1 I t. -4 l [W — 2sin_1( p . ... ... . . .-. ... ... -.. K Sastry, D. E Goldberg ""_"""""""‘ L7.'. ‘.-¢'T.5.‘. .nl. A. 29 iwmu 4 inn. 0 I rs-. .r--nu st.
  • 31. Evaluation Relaxation as an EET Run—Duration Model Verification — I 30 . ° to . ° az Proponton of comet 885 . ° . ° 0 l . ° to 9 ca 9 l Proponton of correct 88: . ° 0! Ruvnlu and zv-am-uv-ay (M31511 m I ‘ ' _ Ixlrvlnvvxg I Asaunlz-u--mm K. Sastry. D. P 9 Proportion 0! correct B85 9 . ° 0 20 40 60 P P Proponionolcorvectaavs P 9 . ° 0 1o 20 so No. 0! gomralions. l E. Goldberg m‘ mus: -an -to-t—I—a—r to--no-It---I loun- duooutvs-o-rt-cc «mg an-Llu. —. m.»Iao-nun
  • 32. Evaluation Relaxation as an EET 31 Run—Duration Model Verification - II 8Ias. b-10 Bias b-50 Expt , : . . 55_ . Expl #1 E><ot.2XO 3% Exp(.2XO — Theory _ 4 5o_ — Theocy Cauv Lurv Convergence time. I Convergence Iime. I 5 no 15 20 Swntching time. I‘ Swilcm ng Iime. I_ Problem we. I = 50 Problem size‘ I = 300 . - «"4 . . . V "I c- Exp! , § 75' . ~. Expl. axo A __3 __. , — Thu ui ui 7°‘ . § . § In D U U C C § § 3 E D O 0 0 0 to 20 30 40 Swnlchlng lime‘ ls Swllc7WgIime.1: , —E rum . .u u. -«nun. ..-. . K S. a~. .I"y'. D E G~: ,:lLIl)<: r:; HA, » ""
  • 33. Evaluation Relaxation as an EET When Should We Switch? . Total evaluation cost Ctot. = 77’ lc2ts + Cl (tconv — ts)i . Optimal switching time actot = 0 3t_. , * t: 4 , {3(1 — .3) 7'3 = z 1 — — 2 tconx 1 7T — C7. — 1 — Bias proportion (3 = b/ £ — fl is used alone: t. ,., ,W,1 = 1n/ F/(21) K Sastry. D. E. Goldberg Dnfitnnvl up--—. i-a—vy -——¢. n--I I-—m-u 32 luau-a-vnc-. oI»—. .a'u-nu s IVE‘. «nu « uv. x.~»unn-.54.‘.
  • 34. Evaluation Relaxation as an EET Optimal Switching Time Coslralio. C, = 1.5 05 ‘tr 1 rim“ D D 0.1 0,2 5 ccmn-. l 0 0.1 02 0 - I#“_J'3": 'Ca': H)QCV—_k: ":l¢l? “ 0 . 3 Bras propomon Cost ratio. 0' - 0.3 Exprt I- 100 0.4 0.5 3.5 Bess propomon K Sashy. D E Gr_1|(lb<; r<; ‘:1 I - - i3'~tr3cl'b-: -o«: ~ca3r. nv 0 D. 1 0 ,2 0.3 0.4 0.5 Baas proportion Cost raiio. C, = 2.5 Then I! Ex’ 50 a Exptrl-I00 Cost ratio. 0' - 5.0 ""2. __ 0 Exp ‘ I - 50 var Expr I - 100 0.1 0 2 0.3 Bess propon-on ; ‘. :¢&_‘L 33 nu. .. u Ilptliult-trio . m-. .. u. .. ...4 non. - nn a I. .h. .A.
  • 35. Evaluation Relaxation as an EET Model for Predicting Speed-Up . Optimal vs. Na'I've decision — Use high-cost, low-bias function c, . ns = T [ti-onvll — (Cr _ — Cost proportion: c. ,. = C1/C2 — Valid when (3,. > (1 — 23)“ K Sastry. D. E. Goldberg , III.
  • 36. Evaluation Relaxation as an EET Verification of Speed-Up 1 2 cl Theory 1 .15 b = .“ _ "'5, g I 1 Q I AV E 1.05 w _. . 1 0 95 D“ 1 5 7 Theory 0 Ex riment ~ ‘"‘3!a«; ruo~Iooooa£~c»~: >o: :Ia)or1: Costralio, c’: 1.5 L) Ex arimenl 0:2 of: 0.4 0.5 Braspropon-on, il Coszra. Iiu. c'-3.5 0 2 ole 8-as proportion. ll K Sastry, D E Speed—up. I]! speed-up. 11 Cost ratio. 1:’ = 2.5 C-‘. r_; |(ll)e: r<; 0‘.2 0:3 Baas propomon. ii Cost ratio. o’ - 5.0 0.1 0.1 0 2 0.3 Baas propon-on. ii all 04 “.5 Theory . '11 Ex erimeni 0. — Theory 0 Ex erimem ‘ 5 0.5 I‘ :1‘. 35 nu. .. u -rp<nu. .Ip. .-. . u. ... ... .u~.4i. ,.. . I». y. .. all-.1: . ~.. In. .. nli a . ... ... ..i. .. ... A. -u I hlydn
  • 37. Evaluation Relaxation as an EET Handling Bias: Summary . Decision-making strategy: — Fitness functions with differing bias values — Utilized facetwise models — Provides maximum speed-up . Key parameters: — Bias proportion and fitness-cost ratios . Bias can be handled temporally , _£ m‘:7o“'i': :': ‘i': i§i'5i5i鑧‘ K Sastry, D. E Goldberg 36
  • 38. Evaluation Relaxation as an EET Conclusions . Evaluation-relaxation can yield good speed-up . Error has two parts: Variance, and bias . Handle variance spatially — Fitness-variance and fitness-cost ratios . Handle bias temporally — Use high-bias, low-cost function initially — Switch to low-bias, high-cost function later — Bias proportion and fitness-cost ratios K Sastryy D. E Goldberg . ... .. ... ... ... ,.. ... .., , . =.. =.m. .- : ,. .. . LLMI; -vu 3003: in u 3:: Vn'uhv 37
  • 39. Evaluation Relaxation as an EET Acknowledgments a Air Force Office of Scientific Research, Air Force Materiel Command, USAF, F49620-00-1-0163. o National Science Foundation, DMI-9908252. . ... ... .. . ... ... ... ._. K Sastry. D. E. Goldberg ""‘_""""""""' i. .'. ‘I'? .:An-. :i. ‘.: -4.‘. 38 iwin-a-vnc-. oI»—. .ru-nu s