Dependency of GPA-ES Algorithm Efficiency on ES Parameters Optimization Strength

In herein presented work, the relation between number of ES iterations and convergence of the whole GPA-ES hybrid algorithm will be studied due to increasing needs to analyze and model large data sets. Evolutionary algorithms are applicable in the areas which are not covered by other arti cial intelligence or soft computing techniques like neural networks and deep learning like search of algebraic model of data. The di erence between time and algorithmic complexity will be also mentioned as well as the problems of multitasking implementation of GPA, where external in uences complicate increasing of GPA e ciency via Pseudo Random Number Generator (PRNG) choice optimization. Hybrid evolutionary algorithms like GPA-ES uses GPA for solution structure development and Evolutionary Strategy (ES) for parameters identi cation are controlled by many parameters. The most signi cant are sizes of GPA population and sizes of ES populations related to each particular individual in GPA population. There is also limit of ES algorithm evolutionary cycles. This limit plays two contradictory roles. On one side bigger number of ES iterations means less chance to omit good solution for wrongly identi ed parameters, on the opposite side large number of ES iterations signi cantly increases computational time and thus limits application domain of GPA-ES algorithm.


Introduction
snresing mount of dt to e proessed fores needs for improving e0ieny of existE ing lgorithmsF hile rti(il neurl networks nd deep lerning tehnology n e resoned rther s dt representtion or interpoltionD rell nd pproximtion toolD evolutionry lE gorithms re ple to trnsform dt into models whih n e understood nd nlyzed y humnsD serhed for optimD nd solved mny next tsks on the se of trining dtF sn the re of model development y evolutionry lgorithms lled symoli regressionD qeneti rogrmming elgorithms @qeAD enlyti roE grmming I nd relted tehniques re usedF ryrid evolutionry lgorithms like qeEi P uses qe for solution struture development nd ivolutionry trtegy @iA for prmeters identi(tion re ontrolled y mny prmeE tersF he most signi(nt re sizes of qe popE ultion nd sizes of i popultions relted to eh prtiulr individul in qe popultionF here is lso limit of i lgorithm evolutionE ry ylesF his limit plys two ontrditory rolesF yn one side igger numer of i iterE tions mens less hne to omit good solution for wrongly identi(ed prmeters Q nd it ws the min ide of qeEi hyrid lgorithm deE velopmentF yn the opposite side lrge numer of i itertions signi(ntly inreses omputE tionl time nd thus limits pplition domin of qeEi lgorithmF sn this studyD the reltion etween numer of i itertions nd onvergene of the whole qeEi hyrid lgorithm will e studiedF he di'erene etween time nd lgorithmi omE plexity will e lso mentioned s well s the prolems of multitsk nd multithred impleE see eFgF PF mll qe popultions fores evoE lutionry pressure nd in the se of spei( onditions it might speed up evolutionF yn the opposite sideD in the smll popultions there is inresed risk of stuk in lol optim on the ple of glol oneF ery lrge popultions ring prolems with smll speed nd low e0ieny of evolution frequentlyF rolem is tht preise mening of terms smll or ig popultion deE pends on (tness funtion lndspeF ixtremely smll qe popultions lso ring ig dispersion of needed evolutionry yles nd thus lso of omputtionl timeF sn the se of i populE tionsD nlogil resons re vlid only if there re eliminted rndom in)uenes of tsk swithE ing nd other soures etF st is possile to ept results of experiments onluding tht lrge popultions might e reE pled y igger numer of genertions nd vie versF fut suh resons re out ility to (nd solutionF hen the numer of (tness funE tion evlutions @omputtionl omplexityA or omputtionl time is evlutedD dependenies etween them nd popultion sizes re not the sme nd smll popultions re more e0ient s it will e presented in the next setionsF here is lso prmeter representing numer of i lgorithm evolutionry ylesF his numE er plys two ontrditory rolesF he igger numer of i itertions mens less hne to omit good solution for wrongly identi(ed prmE eters QE W nd it ws the min ide of qeE i hyrid lgorithm development ut the lrge numer of i itertions signi(ntly inreses omputtionl time nd thus limits pplition domin of qeEi lgorithmF tndrd setEup of this lgorithm in this study exept results presented on pigF P is RH popultions of i lgorithm for eh qe individul ut in the lst experiment series there re studied in)uE enes of i popultion numer nd popultion sizes onto required numer of qe yles @nd thus whole qeEi lgorithm numer of iterE tionsAF iquivlent of pure qe is none i popE ultion yleF he experiments pulished in the next setion desrie in)uene of this prmeE terF revious study of xqs in)uene onto qe dynmis pointed tht results of experiE ments re similr nd di'erenes etween di'erE   opertor is pplied ut this omplited miroE dynmis study is not the min sujet of this workF mll numer of individuls in i popultion represents nlogy of smll popultions in qe prt of lgorithmF elso there they n ring fster onvergene in verge @from the viewE point of timeAD ut euse the numer of genE ertions is (xedD it nnot orrespond to lrger numer of genertionsF hus the qulity of results @resulting (tness funtion mgnitudesA must e worse for smll i popultions due to illEidenti(ed prmetersF sn the superior qe it might use worse reognition etween good nd wrong strutures ginst originl ides of qeEi designF es the onlusionD if the numE er of i popultion deresesD superior qe will need more popultion to hieve the omE prle qulity resultsF he signi(nt question is if the fster is to derese of i popultions or to inrese of qe onesF per IH ws foused to reltions etween qe nd i popultion sizes of omposed qeE i lgorithmF xow the in)uene of i popuE ltion size nd i popultion numer limit is studied pplying modi(ed methodology of exE perimentsF ery low limit of genertion numer for onstnt optimizing emedded i prt of the lgorithm ws not studied yetF his modi(E tion fouses to elimintion in)uenes ' might not e deterministi euse it s in)uE ened y omputer network tr0 etF resented experiments eliminte some of these soures of nonEdeterminism y the simplest wy !y elimintion of multiEthred exeutionF ih tsk ws running i for prmeter optiE miztion s singleEthred one without ny omE munition with othersF

3.
Experiments and obtained data vorenz ttrtor system served s test se for experiments with symoli regression of di'erE entil equtions desriing this dynmi system on the se of preEomputed dt setF vorenz ttrtor system is desried y @IA nd @PAF      gomputing time ws di'erentF here ws smller vrine of results nd the optim were etween S nd SH popultions nd smllest popE ultions of IH individulsF uh results prtilly defers to ove desried expettions nd they re used y omputtionl omplexity of i lgorithm whih depends qudrtilly on the numer of individuls ut the improvement of lrger popultion to onvergene is of the lower orderF resented dt points tht qeEi lgorithm ehvior orresponds to expettions in ompuE ttionl omplexityF ime omplexity of lgoE rithms orresponds less nd it is given y used hrdwre propertiesF xon looking tht pigF I presents derese of qe e0ienyD smll popultions nd higher numers of itertions re interesting wy of efE (ieny inresing exept extremely low mgniE tudesF woreoverD with deresing of popultion memers numer nd with inresing of ount of evolutionry yles there inreses dispersion of needed itertions nd thus even if the verge numer is smllD some runs n e extremely longF resented experiments lso points tht omE puttionl time is ontroversil mesure of lE gorithm e0ieny !from the lgorithm omE plexity viewpoint numers of individuls n e repled y mount of itertions ut qe nd i lgorithms hs inomprle omplexity nd their implementtion might hve di'erent e0E ienyF i works fsterD thus in time spe omE prison the results displyed on pigsF SEV re not signi(nt nd do not opy results from pigsF PER representing numers of itertions @evolutionry ylesAF resented study uses reltively simple proE lem s se studyF sn the futureD when more omputtionl time will e otinedD there will e need to repet herein presented experiments on more omplex prolems where the re of highest e0ieny is expeted fr from the lowest popultion sizes nd itertion numers of iF U vooD pF tFD vimD gF pFD 8 wihlewizD F @idsFAF @PHHUAF rmeter setting in evoluE tionry lgorithms @olF SRAF pringer iE ene 8 fusiness wediF V eevesD gF F @IWWQD tuneAF sing qeE neti elgorithms with mll opultionsF sn sgqe @olF SWHD pF WPAF W iszzD eFD 8 ouleD F @PHHTD tulyAF qeE neti progrmmingX yptiml popultion sizes for vrying omplexity prolemsF sn roeedings of the Vth nnul onferene on qeneti nd evolutionry omputtion @ppF WSQEWSRAF egwF IH frndejskyD F @PHIQAF mll popultions in qeEi lgorithmF snX wtousekD FD @edFA wixhiv PHIQF IWth snterene gonferene on oft gomputing wixhiv PHIQF frnoD QIEQTF II ssevD yF fFD uuznetsovD F FD 8 wosekE ildeD iF @PHIIAF ryperoli hoti ttrE tor in mplitude dynmis of oupled selfE osilltors with periodi prmeter moduE ltionF hysil eview iD VR@IAD HITPPVF IP qhiselinD wF F @IWWRAF he imginry vmrkX look t ogus history9 in shoolooksF he extook vetterD S@RAF IQ fldwinD tF wF @IVWTAF e new ftor in evoE lutionF he merin nturlistD QH@QSRAD RRIERSIF

About Authors
Tomas BRANDEJSKY otined hh from tehnil yernetis t gzeh ehnil niversity in rgue nd he now is essoiE ted rofessor t niversity of rduieF ris reserh interest fouses to re of softE omputingD espeilly to intervl vlued fuzzy sets nd geneti progrmmingF "This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited (CC BY 4.0)."

304 c 2019
Journal of Advanced Engineering and Computation (JAEC)

Fig. 1 :
Fig. 1: Count of the ES population improvements for whole GPA population evaluated from the best to the worst individual.

Fig. 2 :
Fig. 2: Average number of iterations of GPA-ES algorithm depending on PRGNs inuencing GPA and ES part of the algorithm.
16, β = 4, ρ = 45.91 @PA he limit of qe lgorithm numer ws set to IHHHH ylesD the termintion ondiE tion ws set to sum of error squres less thn IHE V for pplied SWW smples of trining dtD while the numer of i lgorithm itertions hs vryE ing etween mgnitudes IDIH nd IHHD s well s the sizes of i popultions ws vrying mgniE tudes IHD RH nd IHHF ize of qe popultion ws TR individulsF ixperiments were repeted IHHHH times for di'erent seeds of used xqsF sn the presented study we use two vrile pE rmetersD numer of i lgorithm itertions nd size of i popultionsF hey determine numE er of evlutions of (tness funtion oth in i nd omposed qeEi lgorithmsF tndrd g nd gCC funtion rnd@A served s xq in herein presented experimentsF o nlyze omE puttionl e0ieny of qeEi lgorithmD oth numer of needed qe yle itertions nd omE puttionl time re mesured @omputtionl time n e repled y nother r indepenE dent mesureD numer of (tness funtion iterE tionsAF esults of experiments re presented on the following pigF QEVF

Fig. 3 :
Fig. 3: Number of iterations of encapsulating GPA for x variable depending on ES.

Fig. 4 :
Fig. 4: Number of iterations of encapsulating GPA for y variable depending on ES cycle limit and ES population size.

Fig. 5 :
Fig. 5: Number of iterations of encapsulating GPA for z variable depending on ES cycle limit and ES population size.

Fig. 7 :
Fig. 7: Computing time of whole GPA-ES for y variable depending on ES cycle limit and ES population size.

Fig. 8 :
Fig. 8: Computing time of whole GPA-ES for z variable depending on ES cycle limit and ES population size.
(xed prmeters of IHHH di'erent initil xqs seed mgnitudesF izes of popultions were IHHH individuls oth in qe nd relted i popuE ltionsF he numer of i lgorithm yles ws (xed nd equl to RH @it ws not shorten in (tE ness vlue of the est individul in the popuE ent omintions of xqs re on the similr mgnitude s oserved noiseF hese (rst experiments were exeuted with 306 c 2019 Journal of Advanced Engineering and Computation (JAEC) rry pseudoErndom genertor of PREit numE ers nd exvPR is the sme type produE ing RVEit numersD exh denotes stndrd gGgCC rnd@A funtion nd wIWWQU repreE sents wersenne wister pseudoErndom generE tor of QPEit numers with stte size of IWWQU itsF hetil dynmis of qeEi system is omE