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Greedy Function Approximation: A Gradient Boosting …

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February24,1999AbstractFunctionapproxima tionisviewedfromtheperspectiveofnumerica loptimizationinfunctionspace, { {descent\ Boosting "paradigmisdevelopedfor additiveexpansionsbasedonany cal-gorithmsarepresentedforleast{squares ,least{absolute{deviation,andHuber{Mloss func-tionsforregression,andmulti{classlo gisticlikelihoodforclassi ,andtoolsforinterpretingsuch\TreeBoost" ,highlyrobust,interpretableproceduresfor regressionandclassi cation, ,andFriedman,Hastie, \output"or\re-sponse"variableyandasetofr andom\input"or\explanatory"variablesx=fx 1; ; \training"samplefyi;xigN1ofknown(y;x){va lues,thegoalisto ndafunctionF (x)thatmapsxtoy,suchthatoverthejointdist ributionofall(y;x){values,theexpectedval ueofsomespeci edlossfunction (y;F(x))isminimizedF (x)=argminF(x)Ey;x (y;F(x))=argminF(x)Ex[Ey( (y;F(x))jx]:(1)Frequentlyemployedlossfun ctions (y;F)includesquared{error(y F)2andabsoluteerrorjy Fjfory2R1(regression),andnegativebinomia llog{likelihood,log(1+e 2yF),wheny2f 1;1g(classi cation).)}}}}}}}}}}}

1987), MARS (F riedman 1991), w a v elets (Donoho 1993), and supp ort v ector mac hines (V apnik 1995). Of sp ecial in terest here is the case where these functions

  Boosting, Approximation, Derating, A gradient boosting

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