!!! dglvda. ax rcin ly gezipe dcinll `ean

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1 ax rcin ly gezipe dcinll `ean :qxew 'qn '` cren,fryz,'` xhqnq ezaq oeiq,uiaexhpw dix` :mivxn :dpigad jix`z zery 3 :dpigad jyn xeq` :xfr xneg :zeaeyg zeigpd.zery 3 ogand jyn qteha creind mewnl zeaeyzd z` ewizrd okn xg`l.dheih zxagna dligz ogand z` exzt.cenrd aba zetqez e` dheihd zxagn z` oeayga `iaz `l ogand zwica.zeaeyzd.sc lk y`xa ogapd xtqn z` enyx.zel`yd lk lr zeprl yi.miknzqn mz` zeprh eli` lr wiecna oiivl yi j`,lebxzdne dzikdn ze`vez hhvl xzen.oze` gikedl yi,ziad ilibxzn ze`vez lr jnzqdl mivex mz` m`.zwnepn zeidl zaiig daeyz Question Grade Out of total I(1-8) 24 II(1) 10 II(2)a 7 II(2)b 3 II(2)c 4 II(3)a 7 II(3)b 7 II(4)a 7 II(4)b 4 II(4)c 5 II(5) 8 II(6)a 7 II(6)b 7 Total 100 :wcead yeniyl!!! dglvda 1

2 zexvw zel`y :oey`x wlg dpekpd daeyzd z` siwdl yi dl`y lka.zecewp 3 dl`y lk lwyn.zexvw zel`y likn df wlg.cewipa dkfz `l oekp wenip `ll daeyz.gikedl jxev oi`.miizy e` zg` dxeya xvw wenip siqedle biezn mbcn S D m idie,x {0, 1} lrn zebltzd D idz.ziteq ze`nbec zveaw X idz.1.s mbcnd lr ERM beqn dcinl mzixebl` ly hltd ĥ idi.s -a driten X -a dcewp lky jk.d zgz zg` zieez wx lawl dleki x X dcewp lk m"m` err(ĥ, D) = 0 :miiwzn oekp `l oekp dcxtdl ozip didy mbcny okzii,pca zervn`a biezn mbcn lr cnin zcxed miyer xy`k.2 oekp `l oekp.dcxedd xg`l zix`pil cixt izlal jetdi cnind zcxed iptl zix`pil jk dwixhnd z` dpype ilnihte`d clustering d z` gwip m`,k-means clustering d zhiya.3,(iepiy `ll ex`yii miwgxnd x`yy cera) elcbi mipey clusters a zecewp zebef oia miwgxndy dpzyi `l cxii dlri ixewnd clustering d xear dxhnd zivwpet ly jxrd if`,inewn menipinl miribne gradient descent zhiy i"r dxenw divwpet xrfnl miqpn xy`k.4 oekp `l oekp.ilaelb menipin mb edf gxkda 2

3 m H lceba S D m mbcn,h ziteq zexryd zgtyn mr ERM zervnà micnel xy`k.5 oekp `l oekp.d zebltzdd lr cnlpd beeqnd ly dkenp dìby gihan z` z`ven cinz EM y èd gradient ascent zhiy ipt lr EM zhiy ly zepexzid cg`.6 oekp `l oekp.gradient ascent l cebipa ilaelbd menihte`d hold- -d lceb,soft-svm-a λ-d z` xegal zpn lr holdout set zervnà validation miyer xy`k.7 oekp `l oekp.xzei lecb mbcnd cnin m` xzei lecb zeidl jixv llk jxca out set ix`pia xehwe ici lr zebveind oal-xegy rava zepenz od ze`nbecd m`,zigpen dcinl ziraa.8 mixegy miikp` mieew yi m` wxe m` 1 ìd dpenz ly zieezde,milqwitd ikxr z` liknd dtwz dpi` dtwz Naive Bayes -d zgpd ik miiekiqd aex if`,dpenza 3

4 zelibx zel`y :ipy wlg zivwpet z` xrfnl miywan ep`,s = ((x 1, y 1 ),..., (x m, y m )) biezn mbcn ozpda ('wp 10).1 m f(w) = l( w, x i, x i ) + λ w 2 2, i=1 :dxhnd zira z` e`ha.l(y, y ) := 2y 3y ici lr xcben l oneqnd loss function -d xy`k.zihxcphq dxev zlra (QP) zireaix zipkezk divfinihte`d 4

5 ,S = (X i, Y i ) mbcn ozpda.àd oteà mbcn ielz (kernel) oirxb xicbdl ywan wipte 'text.2 i"r ψ S : R d { 1, 0, 1} feature map -d z` xicbn èd,y i { 1, 1},X i R d,i m { Y i, x = X i S ψ S (x) = 0, x / S. egipd.s izexixy mbcn lr ψ S zgz γ ilniqwnd hard-svm margin -d z` eayg ('wp 7) (`) hard-svm -d ly dxcbdd :zxekfz.zepey zeieez mr miinrt driten dpi` `nbec s` ik ìd ixnbl mbcnd z` cixtn xy` w R d xear margin γ(w) := 1 R min i m w, x i w, R := max x i. i zeikeaiq ik qxebd,dzika epgqipy htynde (`) sirqa margin -d mqg xe`l ('wp 3) (a) ze`nbec O( 1 ) γ 2 ik oreh wipte 'text,o( 1 ) γ 2 dpid γ margin mr dcxtd ly dxwna mbcnd exiaqd.ely oirxbd zervnà D dnibcd zebltzd lr dkenp dìby biydl ick zewitqn.dfd dxwna swz `l l"pd htynd recn,lykp (a) sirqa dllkd mqg gikedl oeiqpdy zexnly oreh qiwppeax'v 'text ('wp 4) (b) ɛ, δ > 0 lkly oreh èd.wipte 'text ly oirxbl miaeh mirevia gihany xg` mqg yi 1 δ zegtl zexazqda,m 0 zegtl lceba mbcn ozpday jk m 0 = m 0 (ɛ, δ) reaw yi okzii m`d.dnibcd zebltzdl qgia zegt e` ɛ dìby biyi hard-svm -d mzixebl`.mkzaeyz z` exiaqd?wcev qiwppeax'v 'texty 5

6 izla lhe lb ly mipega`d."ìxa" e` "dleg"k ltehn oga`l miqpn,lhe lb,mi`tex ipyy gipp.3 dleg e` ìxa zeidl leki izin`d avnd.ltehnd ly izin`d avnd ozpda dfa df miielz.ivg-ivg zexazqda zexazqdd dn,onfdn 80% oekp lh ly oega`de onfdn 90% oekp lb ly oega`d m` ('wp 7) (`) daeyzd z` xi`ydl ozip?edylk oezp ltehn ly oega`d iabl enikqi mi`texd ipyy.zixtqn daeyz zzl jxev oi`,zihnzix` dgqepk ly mipega`d z` alyl dvexe l"pd zeiexazqdd lk z` zrcei dwlgnd zldpn ('wp 7) (a) yeniy eyr?lhe lb ly zeaeyzd z` alyl dilr cvik.wiecn xzei oega` zzl n"r lhe lb "dlegk ltehnd z` oga` lb" rxe`nd z` onqn G xyà G, T {0, 1} mixehwicpià ly ilnihte`d dhlgdd llk xear iehia epz.lh xear mi`znd rxe`nd z` onqn T -e.t -e G zervnà dwlgnd zldpn 6

7 .zg` zvayn lcebk heaexd ly elceb,gy gel lr jldzn heaex :àd i`xw`d jildza ehiad.4 m`.dl`ny,dpini,dhnl,dlrnl :mipeeikd zrax`n cgà cg` crv rp heaexd,aeaiq lka oal :zerahn ipy heaexl.geld lr eze` xi`yiy oeeika ff cinz èd,geld zty lr `vnp èd oteà minrt 2 zvaynd rava rahn lihn èd,zvayn lr zgep heaexdy mrt lk.xegye.rahnd zlhda "ur" el `viy minrtd xtqnl deeyy dnvera xe` hlet fè,ielz izla mipnqnd,s = {b, w} miieagd miavnd mr,(hmm) ieagd iaewxnd lcena ehiad ('wp 7) (`) t onfa heaexd ly avnd S t S idi.dxegy e` dpal zvayn lr `vnp heaexd m`d xarnd zeiexazqd z` enyx.t onfa hlt heaexdy xe`d znver Y t {0, 1, 2} idze xear PY t = y S t = s hltd zeiexazqd zè s, s S xear PS t = s S t = s.y {0, 1, 2}, s S gipp.dn`zda,"ur" lr elti oalde xegyd rahndy zeiexazqdd p b, p w eidi ('wp 4) (a) z` zrcl epl xyt`n df cvik exiaqd.s 1 = b ik mircei ep`,heaexd ly mieqn jeliday.t lkl S t 7

8 ,y 1, y 2,..., y T zeitvzd zxcq lr qqeand p b xear Maximum Likelihood Estimator egqp ('wp 5) (b).mcewd sirqa yeniy eyr :fnx.s 1 = b did oey`xd ieagd avnd ik dgpda 8

9 idz.y i {0, 1} sqepae x i X xy`k,biezn mbcn S = ((x 1, y 1 ),..., (x m, y m )) idi ('wp 8).5 hltd ĥi idi.h i Y X,zexryd zegtyn zxcq H 1,..., H n dpiidz.x Y lrn zebltzd D,ĥi lkl ik gihadl mivex ep`.s mbcnd lr H i xear ERM mzixebl` ly err(ĥi, D) inf h H i err(h, D) + ɛ i. z` lirln meqgl zpn lr PAC inqga eynzyd P i {1,..., n} s.t err(ĥi, D) inf h H i err(h, D) ɛ.. H i zenvera ielz didi mqgd 9

10 ze`nbec 4 mr mbcn S = ((x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ), (x 4, y 4 )) idie,y = R,X = R 2 eidi.6 x 1 = (0, 1), y 1 = 1 x 2 = ( 1 2, 1 2 ), y 2 = 1 x 3 = (0, 1), y 3 = 1 x 4 = (0, 1), y 4 = 1. xy`k,zebiezn,absolute loss -l qgia l"pd mbcnd xear zilnihte`d iefigd zivwpet z` eayg ('wp 7) (`) z` xzet xy` h : R 2 R z`ivn i"r m Minimize h:r 2 R h(x i ) y i. i=1.h(x) = w, x,ix`pil cixtnk z`fd divwpetd z` ebivd,t 1 lkle i lkl x i = (0, 1) -y jk S = ((x 1, y 1 ),..., (x m, y m )) mbcna ehiad ('wp 7) (a) qgia h : R 2 R zeilnihte`d zeivwpetd sqe` z` ex`z.y 2t = t -e y 2t 1 = t divfnipind ziral ilnihte` oexzt zepzepy zeivwpetd zveaw,xnelk,absolute loss -l.oeayga z`f zgwl mkly oexztd lr,m a dielz dveawd,al eniy.` sirqa 10

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