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1、Chapter4OPTIMIZATIONBYSTOCHASTICAPPROXIMATIONUp-tonowwehavebeenconcernedwithfindingrootsofanunknownfunctionobservedwithnoise.Inapplications,however,oneoftenfacestotheoptimizationproblem,i.e.,tofindingtheminimizerormax-inizerofanunknownfunctionItiswell
2、knowthatachievesitsmaximumorminimumvaluesattherootsetofitsgradient,i.e.,atalthoughitmaybeonlyinthelocalsense.ThegradientisalsowrittenasIfthegradientcanbeobservedwithorwithoutnoise,thentheoptimizationproblemisreducedtotheSAproblemwehavediscussedinprevi
3、ouschapters.Here,weareconsideringtheoptimizationproblemforthecasewherethefunctionitselfratherthanitsgradientisobservedandtheobservationsarecorruptedbynoise.ThisproblemwassolvedbytheclassicalKiefer-Wolfowitz(KW)algorithmwhichtookthefinitedifferencestos
4、erveasestimatesforthepartialderivatives.Tobeprecise,letbetheestimateattimefortheminimizer(maximizer)ofandletbetwoobservationsonattimewithnoisesandrespectively,wherearetwovectorsperturbedfromtheestimatebyandrespec-tively,onthecomponentofTheKWalgorithms
5、uggeststaking151152STOCHASTICAPPROXIMATIONANDITSAPPLICATIONSthefinitedifferenceastheobservationofthecomponentofthegradientItisclearthatwherethecomponentofequalsTheRMalgorithmwithdefinedaboveiscalledtheKWalgorithm.Itisunderstandablethatintheclassicalth
6、eoryforconvergenceoftheKWalgorithmratherrestrictiveconditionsareimposednotonlyonbutalsoonandBesides,ateachiterationtoformfinitedifferences,observationsareneeded,whereisthedimensionofInsomeproblemsmaybeverylarge,forexample,intheproblemofoptimizingweigh
7、tsinaneuro-networkcorrespondstothenumberofnodes,whichmaybelarge.Therefore,itisofinterestnotonlytoweakenconditionsrequiredforconvergenceoftheoptimizingalgorithmbutalsotoreducethenumberofobservationsperiteration.InSection4.1theKWalgorithmwithexpandingtr
8、uncationsusingrandomizeddifferencesisconsidered.Astobeshown,becauseofreplac-ingfinitedifferencesbyrandomizeddifferences,thenumberofobserva-tionsisreducedfromto2foreachiteration,andbecauseofinvolvingexpandingtruncationsinthealgorithmandapplying
