Variants of Pegasos. December 11, 2009
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1 Inroducon Varans of Pegasos SooWoong Ryu December, 009 Youngsoo Cho Developng a new SVM algorhm s ongong research opc. Among many exng SVM algorhms, we wll focus on Pegasos. Pegasos uses alernang eraon schemes, whch are by alernang sochasc graden descen seps and projecon sep. The Pegasos can solve a ex classfcaon problem from Reuers Corpus Volume wh 800,000 ranng examples n 5 seconds. We wll examne he algorhms robusness by applyng he algorhm o raher smple ex classfcaon problem (namely, he one gven n he cs9 homework. We also presen hree possble varans of Pegasos; namely, undo mehod, wo-ou mehod and probably mehod. We wll examne con and pro of each mehod. Algorhm descrpon. Pegasos The Pegasos s a subgraden based algorhm ha mnmzes he followng unconsraned srcly convex objecve funcon. λ f( ω = ω + l( ω;( x, y ( m ( x, y S where l( ω;( x, y = max{0, y ω, x } ( However, does no use eq( as an objecve funcon a each eraon. Insead, uses λ f( ω; A = ω + l( ω;( x, y (3 k ( x, y A where a se A S whose sze s k and S, a ranng se. The Pegasos consss of wo major seps n updang ω :. subseepes graden updae ω + and. projecon sep. Specfcally, n each eraon, he Pegasos looks for a seepes graden search drecon gven A and ses he learnng rae o beη = λ. Then, projecs he updaed ω + no he ball, B = { ω : ω }. The reason for projecon s ha we already know he opmal sqr ( λ soluon,ω, resdes n B.. A Varan of Pegasos A key pon n a varan of pegasos s how o deal wh oulers n ranng se, S. There are oulers n S. These oulers resul n ncrease n objecve funcon value n some eraon snce pegasos chooses A..d. from S. To preven hs, hree mehods are used;. Undo mehod. wo-ou mehod and 3. probably mehod. A modfcaon from pegasos s appled a he end of eraon when ω + s compued. In he undo mehod, wh compuedω +, he new objecve funcon value, f ( ω +, s also compued. If f ( ω+ > f ( ω hen we do no use compuedω +. Insead we le ω + o be equal oω. If f ( ω+ f ( ω, hen we keep
2 ω + as updaed. In he wo-ou mehod, he ranng se, S, s updaed a each eraon. If f ( ω+ > f ( ω, hen we do no use compue ω + and also append a flag o each daa, ( x, y n A + and coun he number of flags n each daa n A +. If he number of flags n a daa becomes wo (.e n f ( x, y =, he daa s aken away from S +. Le D + o be a se of hose daa ha are aken away from S +. Then, we oban S+ = S+ D +. A he eraon of +, S + s used as a ranng se. Agan, f f ( ω+ f ( ω, hen we keep ω + as updaed as n undo mehod. The probably mehod s smlar o he wo-ou mehod. The dfference s n how o updae S. In he probably mehod, for each daa, ( x, y, he probably of beng sampled o A a eraon (le s call ha probably p( x, y s se o n he begnnng. As he eraon proceeds, hese probables are updaed by examnng he objecve funcon value. If f ( ω+ > f ( ω, hen we decrease p( x, y for each ( x, y n A. The varan algorhm s shown below. INPUT: S, λ, T, k INITIALIZE: Choose w s.. FOR =,,3,..., T Choose A w λ S, where A = k. Se A + = {( x, y A : y w, x < } η =, w = ( η λ w + + yx η Se + k ( x, y A IF f ω+ > Se ω = + ELSE ( f ( ω ω FOR =,,3,..., Sze( A n ( x, y = n ( x, y + f f IF n f ( x, y = Se D = D {( x, y } + + ω = mn{, } w / λ Se + + w + λ S = S D, OUTPUT: ω + 3 Resul Frs of all, we downloaded he Pegasos source code from hp:// shas/code/ndex.hml. There are hree sample ses of ranng daa and one es daa se: One ranng se of sze 000, wo oher ses of musch smaller sze and a es se of sze 600. One neresng characersc of hs daa se s ha l norm of every daa s one. Ths pus all he daa ono he surface of spheres. We needed o look for more general daa ses whch do no exhb such a characersc. Thus, we use he ranng se and es ses gven for cs9 homework #. The sze of ranng se s 44 and es se has sze of 800. One of he neresng characerscs of he Pegasos s ha we have a conrol on he sze of A, whch s k. If k =, hen Pegasos s very smlar o sochasc graden mehod. If k = S,
3 hen resuls n a modfed graden-descen algorhm. 3. undo mehod Wha he undo mehod acually dose n subgraden based algorhm s o preven he ascen drecon. Please see fg.( and (. In he begnnng of he eraons, he undo mehod seems o do much beer han he pegasos. However, as he eraon reaches he pon close o he convergence regon, he pegasos does beer han undo mehod. Ths resul s analogs o he comparson beween seepes descen algorhm and conjugae graden mehod. Alhough he seepes descen may exhb beer performance n he begnnng of eraons, he one ha wns he performance s conjugae graden mehod. The lesson here s ha allowng ascen drecon somemes can be poson a ha momen, bu can be medcne. As descrbed n algorhm descrpon secon, we combned he undo, wo-ou, and probably mehod and he resul s ha he undo mehod s domnan n choosng he search drecon. In fg.(3 and (4, all he hree mehods (.e. undo mehod, undo-probably combned mehod, and undo-probably-wo ou combned mehod exhb almos dencal performance. 3. wo-ou mehod Snce we ve found ha he undo mehod should no be used, we decded no o use n boh wo-ou mehod and probably mehod. Thus, n wo-ou mehod, alhough f ( ω+ > f ( ω n some, we keep he w+ as updaed. Only hng we change s o updae S every eraon by elmnang daa whose flag s equal o wo. The resul s almos dencal o he orgnal pegasos mehod. (.e. see he fg.(5 and (6. We srongly suspec ha he reason for he almos dencal resul s ha he ranng se we have (.e. he sze of abou 000 s oo small o ake an advanage of wo-ou mehod. To ake an advanage of wo-ou mehod, we need enough eraon o ge rd of oulers. The average number of daa aken away from he ranng se unl convergence s around 80. Among 80 daa, here may be good daa n. I s because akng away he daa whose number of flag s wo s oo cruel. Thus, we ook away he daa whose number of flag s four. Then, however, he number of daa aken away from he ranng se s very small (.e. zero, one or wo unl he convergence. The reason ha we have hope n hs mehod s ha even wh our ranng se, he wo-ou mehod exhbs slghly beer performance han he pegasos rgh before he convergence occur (.e. see fg.(7. However, hs s smply our hope and need o be proven wh bgger ranng se n he fuure. 3.3 probably mehod As n case of he wo-ou mehod, he probably mehod exhbs almos dencal resul o pegasos (.e. see fg.(8 and (9. Ths also leaves us a hope ha he probably mehod may 3 works beer han pegasos when he ranng se sze s much bgger. Fgure. Comparson beween pegasos and undo mehod Fgure. Comparson beween pegasos and undo mehod
4 Fgure3. undo mehod s domnan n choosng search drecon Fgure4. undo mehod s domnan n choosng search drecon Fgure5. comparson beween pegasos and wo-ou mehod Fgure6. comparson beween pegasos and wo-ou mehod Fgure7. zoomed n comparson beween pegasos and wo-ou mehod
5 Fgure8. comparson beween pegasos and probably mehod Fgure9. comparson beween pegasos and probably mehod 4 Concluson Developng a new algorhm s a very hard job. Afer readng many papers and geng very deep nsgh abou he algorhm, hen fnally someone can develop wha s so called noble algorhm. Throughou he projec, we have red many oher algorhms whch have no been presened n hs repor. However he resuls were all pessmsc. Inally, for example, we red o exrac only suppor vecors from ranng se o ncrease he convergence rae. However, due o he presence of oulers, when we exrac he suppor vecors (SVs, oulers were also exraced wh SVs. Ths caused even worse es error han pegasos gave. In hs repor, we presened hree dfferen mehods as a varan of Pegasos: undo mehod, wo-ou mehod and probably mehod. The undo mehod aced lke seepes descen mehod. I may converge o a good soluon, bu n oher mes, would no converge o a good soluon. Boh wo-ou mehod and probably mehod gve a way of fndng oulers n ranng se. Takng away daa ha ges wo flags durng he smulaons, we may have hgh rsk of gvng away a good daa. Ths resul can be fxed eher by ncreasng wo o four or fve. However o accomplsh hs, we need a bgger ranng se. The probably mehod also shows he poenal o be used n bgger ranng se. 5 Reference. Shalev-Shwarz, S., Snger, Y. and Srebro, N. (007. Pegasos: Prmal Esmaed sub-graden Solver for SVM. Proceedngs of he 4 h Inernaonal Conference on Machne Learnng.. Joachms, T. (999. Makng Large-Scale SVM Learnng Praccal. MIT Press Cambrdge, MA, USA 3. Joachms, T. (006. Tranng Lnear SVMs n Lnear Tme. Proceedngs of he h ACM SIGKDD nernaonal conference on Knowledge dscovery and daa mnng
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