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1 18 Boostng s a general strategy for learnng classfers by combnng smpler ones. The dea of boostng s to take a weak classfer that s, any classfer that wll do at least slghtly better than chance and use t to buld a much better classfer, thereby boostng the performance of the weak classfcaton algorthm. Ths boostng s done by averagng the outputs of a collecton of weak classfers. The most popular boostng algorthm s, so-called because t s adaptve. 1 s extremely smple to use and mplement (far smpler than SVMs), and often gves very effectve results. There s tremendous flexblty n the choce of weak classfer as well. Boostng s a specfc example of a general class of learnng algorthms called ensemble methods, whch attempt to buld better learnng algorthms by combnng multple smpler algorthms. Suppose we are gven tranng data {(x,y )} N =1, where x R K and y { 1,1}. And suppose we are gven a (potentally large) number of weak classfers, denoted f m (x) { 1,1}, and a -1 loss functon I, defned as I(f m (x),y) = { ffm (x ) = y 1 ff m (x ) y (1) Then, the pseudocode of the algorthm s as follows: for from 1 to N,w (1) = 1 for m = 1 to M do Ft weak classfermto mnmze the objectve functon: ǫ m = N =1 w(m) I(f m(x ) y ) w(m) where I(f m (x ) y ) = 1 f f m (x ) y and otherwse α m = ln 1 ǫm ǫ m for all do w (m+1) end for end for = e αmi(fm(x ) y ) After learnng, the fnal classfer s based on a lnear combnaton of the weak classfers: ( M ) g(x) = sgn α m f m (x) m=1 () Essentally, s a greedy algorthm that bulds up a strong classfer,.e., g(x), ncrementally, by optmzng the weghts for, and addng, one weak classfer at a tme. 1 was called adaptve because, unlke prevous boostng algorthms, t does not need to know error bounds on the weak classfers, nor does t need to know the number of classfers n advance. Copyrght c 15 Aaron Hertzmann, Davd J. Fleet and Marcus Brubaker 13
2 Fgure 1: Illustraton of the steps of. The decson boundary s shown n green for each step, and the decson stump for each step shown as a dashed lne. The results are shown after 1,, 3, 6, 1, and 15 steps of. (Fgure from Pattern Recognton and Machne Learnng by Chrs Bshop.) Copyrght c 15 Aaron Hertzmann, Davd J. Fleet and Marcus Brubaker 14
3 1.5 1 Tranng data 1.5 Classfed data.8.6 Loss on tranng set Exp loss Bnary loss f(x) = Σ α m f m (x) 5 6 Decson boundary Fgure : 5 steps of used to learn a classfer wth decson stumps. Copyrght c 15 Aaron Hertzmann, Davd J. Fleet and Marcus Brubaker 15
4 18.1 Decson stumps As an example of a weak classfer, we consder decson stumps, whch are a trval specal case of decson trees. A decson stump has the followng form: f(x) = s(x k > c) (3) where the value n the parentheses s 1 f the k-th element of the vector x s greater than c, and -1 otherwse. The scalar s s ether -1 or 1 whch allows one the classfer to respond wth class 1 whenx k c. Accordngly, there are three parameters to a decson stump: c R k {1,...K}, where K s the dmenson ofx, and s { 1,1} Because the number of possble parameter settngs s relatvely small, a decson stump s often traned by brute force: dscretze the real numbers from the smallest to the largest value n the tranng set, enumerate all possble classfers, and pck the one wth the lowest tranng error. One can be more clever n the dscretzaton: between each par of data ponts, only one classfer must be tested (snce any stump n ths range wll gve the same value). More sophstcated methods, for example, based on bnnng the data, or buldng CDFs of the data, may also be possble. 18. Why does t work? There are many dfferent ways to analyze ; none of them alone gves a full pcture of why works so well. was frst nvented based on optmzaton of certan bounds on tranng, and, snce then, a number of new theoretcal propertes have been dscovered. Loss functon vew. Here we dscuss the loss functon nterpretaton of. As was shown (decades after was frst nvented), can be vewed as greedy optmzaton of a partcular loss functon. We defne f(x) = 1 m α mf m (x), and rewrte the classfer as g(x) = sgn(f(x)) (the factor of 1/ has no effect on the classfer output). can then be vewed as optmzng the exponental loss: L exp (x,y) = e yf(x) (4) so that the full learnng objectve functon, gven tranng data {(x,y )} N =1, s E = e 1 y M m=1 αmfm(x ) (5) whch must be optmzed wth respect to the weghts α and the parameters of the weak classfers. The optmzaton process s greedy and sequental: we add one weak classfer at a tme, choosng t Copyrght c 15 Aaron Hertzmann, Davd J. Fleet and Marcus Brubaker 16
5 and tsαto be optmal wth respect toe, and then never change t agan. Note that the exponental loss s an upper-bound on the -1 loss: L exp (x,y) L 1 (x,y) (6) Hence, f exponental loss of zero s acheved, then the -1 loss s zero as well, and all tranng ponts are correctly classfed. Consder the weak classfer f m to be added at step m. The entre objectve functon can be wrtten to separate out the contrbuton of ths classfer: E = e 1 y m 1 j=1 α jf j (x ) 1 y α mf m(x ) (7) = e 1 y m 1 j=1 α jf j (x ) e 1 y α mf m(x ) (8) Snce we are holdng constant the frst m 1 terms, we can replace them wth a sngle constant = e 1 y m 1 j=1 α jf j (x ). Note that these are the same weghts computed by the recurson used by,.e., w (m 1) e 1 y α j f m 1 (x ). (There s a proportonalty constant that can be gnored). Hence, we have E = e 1 y α mf m(x ) (9) We can splt ths nto two summatons, one for data correctly classfed by f m, and one for those msclassfed: E = e αm + e αm (1) :f m(x )=y Rearrangng terms, we have E = (e αm e αm ) :f m(x ) y I(f m (x ) y )+e αm (11) Optmzng ths wth respect to f m s equvalent to optmzng w(m) I(f m (x ) y ), whch s what does. The optmal value for α m can be derved by solvng de dα m = : de = α m dα m Dvdng both sdes by ( e αm +e αm α m w(m) ), we have I(f m (x ) y ) α m e αm = (1) = e αm ǫm +e αm ǫm e αm (13) e αm ǫm = e αm (1 ǫm ) (14) α m +lnǫ m = α m +ln(1 ǫ m) (15) α m = ln 1 ǫ m ǫ m (16) Copyrght c 15 Aaron Hertzmann, Davd J. Fleet and Marcus Brubaker 17
6 E(z) 1 1 z Fgure 3: Loss functons for learnng: Black: -1 loss. Blue: Hnge Loss. Red: Logstc regresson. Green: Exponental loss. (Fgure from Pattern Recognton and Machne Learnng by Chrs Bshop.) Problems wth the loss functon vew. The exponental loss s not a very good loss functon to use n general. For example, f we drectly optmze the exponental loss over all varables n the classfer (e.g., wth gradent descent), we wll often get terrble performance. So the loss-functon nterpretaton of does not tell the whole story. Margn vew. One mght expect that, when reaches zero tranng set error, addng any new weak classfers would cause overfttng. In practce, the opposte often occurs: contnung to add weak classfers actually mproves test set performance n many stuatons. One explanaton comes from lookng at the margns: addng classfers tends to ncrease the margn sze. The formal detals of ths wll not be dscussed here Early stoppng It s nonetheless possble to overft wth, by addng too many classfers. The soluton that s normally used practce s a procedure called early stoppng. The dea s as follows. We partton our data set nto two peces, a tranng set and a test set. The tranng set s used to tran the algorthm normally. However, at each step of the algorthm, we also compute the -1 bnary loss on the test set. Durng the course of the algorthm, the exponental loss on the tranng set s guaranteed to decrease, and the -1 bnary loss wll generally decrease as well. The errors on the testng set wll also generally decrease n the frst steps of the algorthm, however, at some pont, the testng error wll begn to get notceably worse. When ths happens, we revert the classfer to the form that gave the best test error, and dscard any subsequent changes (.e., addtonal weak classfers). The ntuton for the algorthm s as follows. When we begn learnng, our ntal classfer s extremely smple and smooth. Durng the learnng process, we add more and more complexty to Copyrght c 15 Aaron Hertzmann, Davd J. Fleet and Marcus Brubaker 18
7 the model to mprove the ft to the data. At some pont, addng addtonal complexty to the model overfts: we are no longer modelng the decson boundary we wsh to ft, but are fttng the nose n the data nstead. We use the test set to determne when overfttng begns, and stop learnng at that pont. Early stoppng can be used for most teratve learnng algorthms. For example, suppose we use gradent descent to learn a regresson algorthm. If we begn wth weghtsw =, we are begnnng wth a very smooth curve. Each step of gradent descent wll make the curve less smooth, as the entres of w get larger and larger; stoppng early can prevent w from gettng too large (and thus too non-smooth). Early stoppng s very smple and very general; however, t s heurstc, as the fnal result one gets wll depend on the partculars n the optmzaton algorthm beng used, and not just on the objectve functon. However, s procedure s suboptmal anyway (once a weak classfer s added, t s never updated). An even more aggressve form of early stoppng s to smply stop learnng at a fxed number of teratons, or by some other crtera unrelated to test set error (e.g., when the result looks good. ) In fact, practoners often usng early stoppng to regularze unntentonally, smply because they halt the optmzer before t has converged, e.g., because the convergence threshold s set too hgh, or because they are too mpatent to wat. Copyrght c 15 Aaron Hertzmann, Davd J. Fleet and Marcus Brubaker 19
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