We present the algorithm first, then derive it later. Assume access to a dataset {(x i, y i )} n i=1, where x i R d and y i { 1, 1}.

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1 CS 189 Introducton to Machne Learnng Sprng 2018 Note 26 1 Boostng We have seen that n the case of random forests, combnng many mperfect models can produce a snglodel that works very well. Ths s the dea of ensemblethods. However, random forests treat each member of the forest equally, takng a pluralty vote or an average over ther outputs. The dea of boostng s to combne thodels (typcally called weak learners n ths context) n a more prncpled manner. The key dea s as follows: to mprove our combned model, we should focus on fndng learners that correctly predct the ponts whch the overall boosted model s currently predctng naccurately. Boostng algorthms mplement ths dea by assocatng a weght wth each tranng pont and teratvely reweghtng so that mspredcted ponts have relatvely hgh weghts. Intutvely, some ponts are harder to predct than others, so the algorthm should focus ts efforts on those. These deas also connect to matchng pursut. In both cases, our overall predctor s an addtve combnaton of peces whch are selected one-by-one n a greedy fashon. The algorthm keeps track of resdual predcton errors, chooses the drecton to move based on these, and then performs a sort of lne search to determne how far along that drecton to move. 1.1 AdaBoost There arany flavors of boostng. We wll dscuss one of thost popular versons, known as AdaBoost (short for adaptve boostng), whch s a method for bnary classfcaton. Its developers won the prestgous Gödel Prze for ths work. 1.2 Algorthm We present the algorthm frst, then derve t later. Assume access to a dataset {(x, y )} n =1, where x R d and y { 1, 1}. 1. Intalze the weghts w = 1 n for all = 1,..., n tranng ponts. 2. Repeat for m = 1,..., M: (a) Buld a classfer G m : R d { 1, 1}, where n the tranng process the data are weghted accordng to w. (b) Compute the weghted error = msclassfed w w. Note 26, UCB CS 189, Sprng All Rghts Reserved. Ths may not be publcly shared wthout explct permsson. 1

2 (c) Re-weght the tranng ponts as w w 1 em 1 f msclassfed by G m otherwse (d) Optonal: normalze the weghts w to sum to 1. We frst address the ssue of step (a): how do we tran a classfer f we want to weght dfferent samples dfferently? One common way to do ths s to resample from the orgnal tranng set every teraton to create a new tranng set that s fed to the next classfer. Specfcally, we create a tranng set of sze n by samplng n values from the orgnal tranng data wth replacement, accordng to the dstrbuton w. (Ths s why wght renormalze the weghts n step (d).) Ths way, data ponts wth large values of w arore lkely to be ncluded n ths tranng set, and the next classfer wll place hgher prorty on such data ponts. Suppose 1 that our weak learners always produce an error < 1. To make sense of the formulas 2 we see n the algorthm, note that for step (c), f the -th data pont s msclassfed, then the weght w gets ncreased by a factor of 1 (more prorty placed on sample ), whle f t s classfed correctly, the prorty gets decreased. AdaBoost does have a practcal weakness n that ths aggressve reweghtng can cause the classfer to focus too much on certan tranng examples f the data contans outlers or a lot of nose, the boostng algorthm s generalzaton performance may suffer as t overfts to a few challengng examples. We have not yet dscussed how to make a predcton on test ponts gven our classfers G 1,..., G M. One concevablethod s to use logstc regresson wth G m (x) as features. However, a smarter choce that s based on the AdaBoost algorthm s to set and classfy x by α m = 1 2 ln ( 1 em ) M h(x) = sgn α m G m (x) Note that ths choce of α m (derved later) gves hgh weght to classfers that have low error: As 0, 1 em, so α m. As 1, 1 em 0, so α m. m=1 We now proceed to demystfy the formulas n the algorthm above by presentng a matchng pursut nterpretaton of AdaBoost. Ths nterpretaton s also useful because t generalzes to a powerful technque called Gradent Boostng, of whch AdaBoost s just one nstance. 1 Ths s a reasonable thng to ask. A classfer wth error 1 s even worse than the trval classfer whch predcts the 2 class wth thost total weght wthout regard for the nput x. Note 26, UCB CS 189, Sprng All Rghts Reserved. Ths may not be publcly shared wthout explct permsson. 2

3 1.3 Dervaton of AdaBoost Suppose we have computed classfers G 1,..., G m 1 along wth ther correspondng weghts α k and we want to compute the next classfer G m along wth ts weght α m. The output of our model so far s F m 1 (x) := m 1 =1 α G (x), and we want to mnmze the rsk: α m, G m = arg mn L(y, F m 1 (x ) + αg(x )) =1 for some sutable loss functon L(y, ŷ). Loss functons we have prevously used ncludean squared error for lnear regresson, cross-entropy loss for logstc regresson, and hnge loss for SVM. For AdaBoost, we use the exponental loss: L(y, ŷ) = e yŷ Ths loss functon s llustrated n Fgure 1. Observe that f yŷ > 0 (.e. ŷ has the correct sgn), the loss decreases exponentally n ŷ, whch should be nterpreted as the confdence of the predcton. Conversely, f yŷ < 0, our loss s ncreasng exponentally n the confdence of the predcton. Fgure 1: The exponental loss provdes exponentally ncreasng penalty for confdent ncorrect predctons. Ths fgure s from Cornell CS4780 notes. Pluggng the exponental loss nto the general optmzaton problem above yelds α m, G m = arg mn e y (F m 1 (x )+αg(x )) =1 = arg mn e y F m 1 (x ) e y αg(x ) =1 The term := e y F m 1 (x ) s a constant wth respect to our optmzaton varables. We can splt out ths sum nto the components wth correctly classfed ponts and ncorrectly classfed ponts: α m, G m = arg mn =1 e y αg(x ) Note 26, UCB CS 189, Sprng All Rghts Reserved. Ths may not be publcly shared wthout explct permsson. 3

4 = arg mn = arg mn = arg mn y =G(x ) e α e α + =1 (e α e α ) y G(x ) y G(x ) y G(x ) e α ( ) + e α + e α =1 y G(x ) To arrve at ( ) we have used the fact that y G m (x ) equals 1 f the predcton s correct, and 1 otherwse. For a fxed value of α, the second term n ths last expresson does not depend on G. Thus we can see that the best choce of G m (x) s the classfer that mnmzes the total weght of thsclassfed ponts. Let = y G m(x ) w(m) w(m) Once we have obtaned G m, we can solve for α m. Dvdng ( ) by the constant n obtan α m = arg mn (1 )e α + e α α =1 w(m) We can solve for thnmzer analytcally usng calculus. Settng the dervatve of the objectve functon to zero gves Multplyng through by e α yelds 0 = (1 )e α + e α = e α + (e α + e α ) 0 = 1 + (1 + e 2α ) Addng one to both sdes and dvdng by, we have 1 = 1 + e 2α.e. e 2α = 1 1 = 1 Takng natural log on both sdes and halvng, we arrve at α m = 1 2 ln ( 1 em as clamed earler. From the optmal α m, we can derve the weghts: w (m+1) = exp ( y F m (x ) ) = exp ( y [F m 1 (x ) + α m G m (x )] ) ) = exp ( y G m (x )α m ), we Note 26, UCB CS 189, Sprng All Rghts Reserved. Ths may not be publcly shared wthout explct permsson. 4

5 ( = exp y G m (x ) 1 ( ) ) 1 2 ln em = = = exp ln ( 1 em em 1 ( 1 em ) 1 2 y G m(x ) y G m(x ) Here we see that thultplcatve factor s 1 Ths completes the dervaton of the algorthm. ) 1 2 y G m(x ) when y = G m (x ) and 1 otherwse. As a fnal note about the ntuton, we can vew these α updates as pushng towards a soluton n some drecton untl we can no longer mprove our performance. More precsely, whenever we compute α m (and thus w (m+1) ), for the ncorrectly classfed entres, we have y G m(x ) Dvdng the rght-hand sde by n the correctly classfed entres, we have y =G m(x ) w(m+1) n =1 w(m) w (m+1) = =1 w(m) y G m(x ), we obtan 1 1 em em = (1 ) = (1 ) 1 = (1 ). Smlarly, for Thus these two quanttes are the same once we have adjusted our α, so thsclassfed and correctly classfed sets both get equal total weght. Ths observaton has an nterestng practcal mplcaton. Even after the tranng error goes to zero, the test error may contnue to decrease. Ths may be counter-ntutve, as one would expect the classfer to be overfttng to the tranng data at ths pont. One nterpretaton for ths phenomenon s that even though the boosted classfer has acheved perfect tranng error, t s stll refnng ts ft n a max-margn fashon, whch ncreases ts generalzaton capabltes. 1.4 Gradent Boostng AdaBoost assumes a partcular loss functon, the exponental loss functon. Gradent boostng s a more general technque that allows an arbtrary dfferentable loss functon L(y, ŷ). Recall the general optmzaton problem wust solve when choosng the next model: mn L(y, F m 1 (x ) + αg(x )) =1 Note 26, UCB CS 189, Sprng All Rghts Reserved. Ths may not be publcly shared wthout explct permsson. 5

6 Here G should no longer be assumed to be a classfer; t may be real-valued f we are solvng a regresson problem. By a Taylor expanson n the second argument, L(y, F m 1 (x ) + αg(x )) L(y, F m 1 (x )) + L ŷ (y, F m 1 (x )) αg(x ) We can vew the collecton of predctons that a model G produces for the tranng set as a sngle vector g R n wth components g = G(x ). Then the overall cost functon s approxmated to frst order by L L(y, F m 1 (x )) + α ŷ (y, F m 1 (x )) g =1 =1 }{{} ŷl(y,fm 1 (X)),g where, n abuse of notaton, F m 1 (X) s a vector wth F m 1 (x ) as ts th element, and ŷl(y, ŷ) s a vector wth L(y ŷ, ŷ ) as ts th element. To decrease the cost n a steepest descent fashon, we seek the drecton g whch maxmzes ŷl(y, F m 1 (X)), g subject to g beng the output of somodel G n thodel class we are consderng. 2 Some comments are n order. Frst, observe that the loss need only be dfferentable wth respect to ts nputs, not necessarly wth respect to model parameters, so we can use non-dfferentable models such as decson trees. Addtonally, n the case of squared loss L(y, ŷ) = 1 2 (y ŷ)2 we have L ŷ (y, F m 1 (x )) = (y F m 1 (x )) so ŷl(y, F m 1 (X)) = y F m 1 (X) Ths means the algorthm wll follow the resdual, as n matchng pursut. 2 The absolute valuay seem odd, but consder that after choosng the drecton g m, we perform a lne search to select α m. Ths search may choose α m < 0, effectvely flppng the drecton. The key s to maxmze thagntude of the nner product. Note 26, UCB CS 189, Sprng All Rghts Reserved. Ths may not be publcly shared wthout explct permsson. 6

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