Online Boosting Algorithms for Multi-label Ranking

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1 Onlne Boosng Algorhms for Mul-label Rankng Young Hun Jung Deparmen of Sascs Unversy of Mchgan Ann Arbor, MI 4809 Ambuj Tewar Deparmen of Sascs Unversy of Mchgan Ann Arbor, MI 4809 arxv: v [sa.ml] 23 Oc 207 Absrac We consder he mul-label rankng approach o mullabel learnng. Boosng s a naural mehod for mullabel rankng as aggregaes weak predcons hrough majory voes, whch can be drecly used as scores o produce a rankng of he labels. We desgn onlne boosng algorhms wh provable loss bounds for mul-label rankng. We show ha our frs algorhm s opmal n erms of he number of learners requred o aan a desred accuracy, bu requres knowledge of he edge of he weak learners. We also desgn an adapve algorhm ha does no requre hs knowledge and s hence more praccal. Expermenal resuls on real daa ses demonsrae ha our algorhms are a leas as good as exsng bach boosng algorhms.. INTRODUCTION Mul-label learnng has mporan praccal applcaons (e.g., Schapre and Snger [2000]) and s heorecal properes connue o be suded (e.g., Koyejo e al. [205]). In conras o sandard mul-class classfcaons, mul-label learnng problems allow mulple correc answers. In oher words, we have a fxed se of basc labels and he acual label s a subse of he basc labels. Snce he number of subses ncreases exponenally as he number of basc labels grows, hnkng of each subse as a dfferen class leads o nracably. I s que common n applcaons for he mul-label learner o smply oupu a rankng of he labels on a new es nsance. For example, he popular MULAN lbrary desgned by Tsoumakas e al. [20] allows he oupu of mul-label learnng o be a mul-label ranker. In hs paper, we herefore focus on he mul-label rankng (MLR) seng. Tha s o say, he learner produces a score vecor such ha a label wh a hgher score wll be ranked above a label wh a lower score. We are parcularly neresed n onlne MLR sengs where he labeled daa arrve sequenally. The onlne framework s desgned o handle a large volume of daa ha accumulaes rapdly. In conras o a classcal bach learner, Ocober 24, 207 whch observes he enre ranng se, an onlne learner does no requre sorage of large amoun of daa n memory and can also adap o non-saonary n he daa because keeps updang s nernal sae as new nsances arrve. Boosng, frs proposed by Freund and Schapre [995], aggregaes mldly powerful learners no a srong learner. I has been used o produce sae-of-he-ar resuls n a wde range of felds (e.g., Korykowsk e al. [206] and Zhang and Wang [204]). Boosng algorhms ake weghed majory voes among weak learners predcons, and he cumulave voes can be nerpreed as a score vecor. Ths feaure makes boosng very well sued o MLR problems. The heory of boosng emerged n bach bnary sengs and became arguably complee (cf. Schapre and Freund [202]), bu s exenson o an onlne seng s relavely new. To he bes of our knowledge, Chen e al. [202] frs nroduced an onlne boosng algorhm wh heorecal jusfcaons, and Beygelzmer e al. [205] pushed he sae-of-he-ar n onlne bnary sengs furher by proposng wo onlne algorhms and provng opmaly of one. Recen work by Jung e al. [207] has exended he heory o mul-class sengs, bu her scope remaned lmed o sngle-label problems. In hs paper, we presen he frs onlne MLR boosng algorhms along wh her heorecal jusfcaons. Our work s manly nspred by he onlne sngle-label work (Jung e al. [207]). The man conrbuon s o allow general forms of weak predcons whereas he prevous onlne boosng algorhms only consdered homogeneous predcon formas. By nroducng a general way o encode weak predcons, our algorhms can combne bnary, sngle-label, and MLR predcons. Afer nroducng problem sengs, we defne an edge of an onlne learner over a random learner (Defnon ). Under he assumpon ha every weak learner has a known posve edge, we desgn an opmal way o combne her predcons (Secon 3.). In order o deal wh praccal sengs where such an assumpon s unenable, we presen an adapve algorhm ha can

2 aggregae learners wh arbrary edges (Secon 3.2). In Secon 4, we es our wo algorhms on real daa ses, and fnd ha her performance s ofen comparable wh, and somemes beer han, ha of exsng bach boosng algorhms for MLR. 2. PRELIMINARIES 2.. Problem Seng and Noaons The number of canddae labels s fxed o be k, whch s known o he learner. Whou loss of generaly, we may wre he labels usng negers n [k] := {,,k}. We are allowng mulple correc answers, and he label Y s a subse of [k]. The labels n Y s called relevan, and hose n Y c, rrelevan. A me =,,T, an adversary sequenally chooses a labeled example (x,y ) X 2 [k], where X s some doman. Only he nsance x s shown o he learner, and he label Y s revealed once he learner makes a predcon ŷ. As we are neresed n MLR sengs, ŷ s a k dmensonal score vecor. The learner suffers a loss (ŷ ) where he loss funcon wll be specfed laer n Secon 3.. In our boosng framework, we assume ha he learner consss of a booser and a fxed N number of weak learners. Ths resembles a manager-worker framework n ha booser dsrbues asks by specfyng losses, and each weak learner makes a predcon o mnmze he loss. Booser makes he fnal decson by aggregang weak predcons. Once he rue label s revealed, he booser shares hs nformaon so ha weak learners can updae her parameers for he nex example Onlne Weak Learners and Cos Vecor We wll keep he form of weak predcons h general n ha we only assume s a dsrbuon over [k]. Ths can n fac represen varous ypes of predcons. For example, a sngle-label predcon l can be encoded as a sandard bass vecor e l, or a mul-label predcon {l,,l n } by n n = e l. Due o hs general forma, our boosng algorhm can even combne weak predcons of dfferen formas. Ths mples ha f a researcher has a srong famly of bnary learners, he can smply boos hem whou ransformng hem no mul-class learners hrough well known echnques such as one-vs-all or one-vs-one. We exend he cos marx framework, frs proposed by Mukherjee and Schapre [203] and hen adoped n onlne sengs by Jung e al. [207], as a means of communcaon beween booser and weak learners. A eraon, booser compues a cos vecor c for he h weak learner WL whose predcon h wll suffer he cos c h. The cos vecor s unknown o WL unl produces h, whch s usual n onlne sengs. Oherwse, WL can rvally mnmze he cos. Fnally we assume ha weak learners can ake an mporance wegh as an npu. Ths s allowed for many onlne algorhms such as decson rees (Domngos and Hulen [2000]) or SVMs (Pedregosa e al. [20]) General Onlne Boosng Schema We nroduce a general algorhm schema shared by our boosng algorhms. α denoes he wegh for WL a eraon. We wll keep rack of weghed cumulave voes hrough s j := j = α h. Tha s o say, we can gve more cred for well performng weak learners by seng larger weghs. Furhermore, allowng negave weghs, we can avod poor learner s predcons. We call s j a predcon made by exper j. In he end, he booser makes he fnal decson by followng one of hese expers. The schema s summarzed n Algorhm. We wan o emphasze ha he rue labely s only avalable once he fnal predcon ŷ s made. Compuaon of weghs and cos vecors requres he knowledge of Y, and hus happens afer he fnal decson s made. To keep our heory general, we are no specfyng weak learners (lne 4 and 2). The specfc ways o calculae oher varables such as α, c, and wll depend on algorhms, whch wll be nroduced n he nex secon. Algorhm Onlne Boosng Schema : Inalze: α for [N] 2: for =,,T do 3: Receve example x 4: Gaher weak predcons h = WL (x ), 5: Record exper predcons s j := j = α h 6: Choose an ndex [N] 7: Make a fnal decson ŷ = s 8: Ge he rue label Y 9: Compue weghs α +, 0: Compue cos vecors c, : Weak learners suffer he loss c h 2: Weak learners updae he nernal parameers 3: Updae booser s parameers, f any 4: end for 3. ALGORITHMS WITH THEORETI- CAL LOSS BOUNDS An mporan facor for he performance of boosng algorhms s he performance of he ndvdual weak learners. For example, f weak learners are makng compleely random predcons, hey canno produce meanngful oucomes n accordance wh he booser s nenon. We deal wh hs maer n wo dfferen ways. One way s o defne an edge of a weak leaner over a 2

3 compleely random learner and assume all weak learners have posve edges. Anoher way s o measure each learner s emprcal edge and manpulae he wegh α o maxmze he accuracy of he fnal predcon. Even a learner ha s worse han random guessng can conrbue posvely f we allow negave weghs. The frs mehod leads o OnlneBMR (Secon 3.), and he second o Ada.OLMR (Secon 3.2). 3.. Opmal Algorhm We wll rgorously defne he edge of a weak learner. Recall ha weak learners suffer losses deermned by cos vecors. Gven he rue label Y, we resrc he booser s choce of cos vecors o C eor 0 := {c [0,] k maxc[l] mn c[r], mnc[l] = 0 and maxc[l] = }, l l where he name C0 eor s used by Jung e al. [207] and eor sands for edge-over-random. Snce booser wans weak learners o pu hgher scores a he relevan labels, coss a he relevan labels should be less han hose a he rrelevan ones. Resrcon o[0,] k makes sure ha he learner s cos s bounded. Along wh cos vecors, he booser also passes he mporance weghs w [0,] so ha he learner s cos becomes w c h. We also consruc a baselne learner ha has edge γ. Is predcon u Y γ s also a dsrbuon over[k] ha pus γ more probably for he relevan labels. Tha s o say, we can wre { u Y γ [l] = a+γ f l Y a f l / Y, where he value of a s deermned by he sze, Y. Now we sae our onlne weak learnng condon. Defnon. (OnlneWLC) For parameers γ, δ (0,), and S > 0, a par of an onlne learner and an adversary s sad o sasfy OnlneWLC (δ,γ,s) f for any T, wh probably a leas δ, he learner can generae predcons ha sasfy w c h w c u Y γ +S. γ s called an edge, and S an excess loss. Ths exends he condon made by Jung e al. [207, Defnon ]. The probablsc saemen s needed as many onlne learners produce randomzed predcons. The excess loss can be nerpreed as a warm-up perod. Throughou hs secon, we assume our learners sasfy OnlneWLC (δ,γ,s) wh a fxed adversary. Cos Vecors The opmal desgn of a cos vecor depends on he choce of loss. We wll use L Y (s) o denoe he loss whou specfyng where s s he predced score vecor. The only consran ha we mpose on our loss s ha s proper, whch mples ha s decreasng n s[l] for l Y, and ncreasng n s[r] for r / Y (readers should noe ha proper loss has a leas one oher defnon). Then we nroduce poenal funcon, a well known concep n game heory whch s frs nroduced o boosng by Schapre [200]: φ 0 (s) := LY (s) φ (s) := E l u Y γ φ (s+e l ). () φ (s) ams o esmae booser s fnal loss when more weak learners are lef unl he fnal predcon and s s he curren sae. I can be easly shown by nducon ha many arbues of L are nhered by poenals. Beng proper or convex s a good example. Essenally, we wan o se c [l] := φn (s + e l ), (2) where s s he predcon of exper. The proper propery nhered by poenals ensures he relevan labels have less coss han he rrelevan. To sasfy he boundedness condon of C0 eor, we normalze (2) o ge d [l] := c [l] mn r c [r] w, (3) [] where w [] := max r c [r] mn r c [r]. Snce Defnon s assumng w [0,], we should furher normalze w []. Ths requres he knowledge of w := max w []. Noe ha hs s unavalable unl we observe all he nsances, whch s fne because we only need hs value n provng he loss bound. Algorhm Deals The algorhm s named by OnlneBMR (onlne boos-by-majory for mul-label rankng) as s poenal deas sem from classcal boos-bymajory algorhm (Schapre [200]). In OnlneBMR, we smply se α =, or n oher words, he booser akes smple cumulave voes. Cos vecors are compued usng (2), and he booser always follows he las exper N, or = N. These daals are summarzed n Algorhm 2. Algorhm 2 OnlneBMR Deals : Inalze: α = for [N] 6: Se = N 9: Se he weghs α + =, [N] 0: Compue c [l] = φn (s + e l ), l [k] 3: No exra parameers o be updaed The followng heorem holds eher f weak learners are sngle-label learners or f he loss L s convex. 3

4 Theorem 2. (BMR, General Loss Bound) For any T and N δ, he fnal loss suffered by OnlneBMR sasfes he followng nequaly wh probably N δ: (ŷ ) N φ N (0)+S w. (4) Proof. From () and (2), we can wre φ N + (s ) = E Y l u = c uy γ φn γ = (s + e l ) = c (uy γ h )+c h c (u Y γ h )+φ N (s ), where he las nequaly s n fac equaly f weak learners are sngle-label learners, or holds by Jensen s nequaly f he loss s convex (whch mples he convexy of poenals). Also noe ha s = s + h. Snce boh u Y γ and h have l norm, we can subrac common numbers from every enry of c whou changng he value of c (uy γ h ). Ths mples we can plug n w []d a he place of c. Then we have φ N + (s ) φ N (s ) w []d uy γ w []d h. By summng hs over, we have φ N + (s ) w []d u Y γ φ N (s ) w []d h. OnlneWLC (δ, γ, S) provdes, wh probably δ, w [] w d h w Pluggng hs n (5), we ge φ N + (s ) w []d u Y γ +S. φ N (s ) Sw. (5) Now summng hs over, we ge wh probably Nδ (due o unon bound), N φ N (0)+S w = whch complees he proof. T φ 0 (sn ) = (ŷ ), Now we evaluae he effcency of OnlneBMR by fxng a loss. Unforunaely, here s no canoncal loss n MLR sengs, bu followng rank loss s a srong canddae (cf. Cheng e al. [200] and Gao and Zhou [20]): L Y rnk(s) := w Y ½(s[l] < s[r])+ ½(s[l] = s[r]), 2 where w Y = Y Y c s a normalzaon consan ha ensures he loss les n [0,]. Noe ha hs loss s no convex. In case weak learners are n fac sngle-label learners, we can smply use rank loss o compue poenals, bu n more general case, we wll use he followng hnge loss o compue poenals: L Y hnge (s) := w Y (+s[r] s[l]) +, where ( ) + := max(,0). I s convex and always greaer han rank loss, and hus Theorem 2 can be used o bound rank loss. In Appendx A, we bound wo erms n he RHS of (4) when he poenals are bul upon rank and hnge losses. Here we record he resuls. Table : Upper Bounds for φ N loss φ N rank loss hnge loss (0) and w (0) w e γ2 N 2 O( N ) (N +)e γ2 N 2 2 For he case ha we use rank loss, we can check N N w O( ) O( N). N = = Combnng hese resuls wh Theorem 2, we ge he followng corollary. Corollary 3. (BMR, Rank Loss Bound) For any T and N δ, OnlneBMR sasfes followng rank loss bounds wh probably N δ. Wh sngle-label learners, we have 2 N rnk (ŷ ) e γ 2 T +O( NS), (6) and wh general learners, we have rnk (ŷ 2 N ) (N +)e γ 2 T +2NS. (7) Remark. When we dvde boh sdes by T, we fnd he average loss s asympocally bounded by he frs erm. The second erm wll deermne he sample complexy. In boh cases, he frs erm decreases exponenally as N grows, whch means he algorhm does no requre oo many learners o acheve a desred loss bound. 4

5 Machng Lower Bounds From (6), we can deduce ha o aan average loss less han ǫ, OnlneBMR needs N = Ω( γ ln 2 ǫ ) weak learners and T = Ω( S ǫγ ) samples. A naural queson s wheher hese numbers are opmal. In fac he followng heorem consrucs a crcumsance ha maches hese bounds up o logarhmc facors. Throughou he proof, we consder k as a fxed consan. Theorem 4. For any γ (0, 2k ), δ,ǫ (0,), and S kln( δ ) γ, here exss an adversary wh a famly of weak learners sasfyng OnlneWLC (δ,γ,s), such ha o acheve asympoc error rae ǫ, an onlne boosng algorhm requres a leas Ω( γ ln 2 ǫ ) weak learners and a sample complexy of Ω( S ǫγ ). Proof. We nroduce a skech here and pospone he complee dscusson o Appendx B. We assume ha an adversary draws a label Y unformly a random from 2 [k] {,[k]}, and he weak learners generae snglelabel predcons w.r.. p [k]. We wll manpulae p such ha weak learners sasfy OnlneWLC (δ,γ,s) bu he bes possble performance s close o (6). Boundedness condons n C0 eor and he Azuma- Hoeffdng nequaly provde ha wh probably δ, w c [l ] w c p + γ w k + kln( δ ). For he opmaly of he number of weak learners, we le p = u Y for all. The above nequaly guaranees OnlneWLC s me. Then a smlar argumen of Schapre and Freund [202, Secon 3.2.6] can show ha he opmal choce of weghs over he learners s ( N,, N ). Fnally adopng he argumens n he proof of Jung e al. [207, Theorem 4], we can show EL Y rnk (ŷ ) Ω(e 4Nk2 γ 2 ). Seng hs value equal o ǫ, we have N Ω( γ ln 2 ǫ ), consderng k as a fxed consan. Ths proves he frs par of he heorem. For he second par, le T 0 := S 4γ and defne p = uy 0 for T 0 and p = u Y for > T 0. Then OnlneWLC can be shown o be me n a smlar fashon. Observng ha weak learners do no provde meanngful nformaon for T 0, we can clam any onlne boosng algorhm suffers a loss a leas Ω(T 0 ). Therefore o oban he ceran accuracy ǫ, he number of nsances T should be a leas Ω( T0 ǫ ) = Ω( S ǫγ ), whch complees he second par of he proof Adapve Algorhm Despe he opmal loss bound, OnlneBMR has a few drawbacks when s appled n pracce. Frsly, poenals do no have a closed form, and s compuaon becomes a major boleneck (cf. Table 3). Furhermore, he edge γ becomes an exra unng parameer, whch ncreases he runme even more. Fnally, s possble ha learners have dfferen edges, and assumng a consan edge can lead o neffcency. To overcome hese drawbacks, raher han assumng posve edges for weak learners, our second algorhm chooses he wegh α adapvely o handle varable edges. Surrogae Loss Lke oher adapve boosng algorhms (e.g., Beygelzmer e al. [205] and Freund e al. [999]), our algorhm needs a surrogae loss. The choce of loss s broadly dscussed by Jung e al. [207], and logsc loss seems o be a vald choce n onlne sengs as s graden s unformly bounded. In hs regard, we wll use he followng logsc loss: L Y log (s) := w Y log(+exp(s[r] s[l])). Noe ha hs loss s proper and convex. We wan o emphasze ha n he end he booser s predcon wll suffer he rank loss, and hs surrogae only plays an nermedae role n opmzng parameers. Algorhm Deals The algorhm s nspred by Jung e al. [207, Adaboos.OLM], and we wll call by Ada.OLMR. Snce nernally ams o mnmze he logsc loss, we wll se he cos vecor o be he graden of he surrogae: c := LY log (s ). (8) Nex we presen how o se he weghs α. Afer nalzng α equals o 0, we wll use onlne graden descen mehod, proposed by Znkevch [2003], o compue he nex weghs. Recall ha Ada.OLMR wans o choose α o mnmze he cumulave logsc loss: log (s +α h ). If we wre f (α) := LY log (s +αh ), we need an onlne algorhm ha chooses α sasfyng f (α ) mn α F f (α)+r (T), where F s some feasble se, and R (T) s a sublnear regre. To apply he resul by Znkevch [2003, Theorem ],f needs o be convex, andf should be compac. The former condon s me by our choce of logsc loss, and we wll use F = [ 2,2] for he feasble se. Snce he booser s loss s nvaran under he scalng of weghs, we can shrnk he weghs o f n F. Onlne, Logsc, Mul-label, and Rankng 5

6 Takng dervave, we can check f (α). Now le Π( ) denoe a projecon ono F: Π( ) := max{ 2, mn{2, }}. By seng α + = Π(α η f (α )) where η =, we ger (T) 9 T. Consderng ha s = s +α h, we can also wre f (α ) = c + h. Fnally, remans o address how o choose. In conras o OnlneBMR, we canno show ha he las exper s relably sophscaed. Insead, wha can be shown s ha a leas one of he expers s good enough. Thus we wll use classcal Hedge algorhm (cf. Freund and Schapre [995] and Llesone and Warmuh [989]) o randomly choose an exper a each eraon wh adapve probably weghs dependng on each exper s predcon hsory. In parcular, we nroduce new varables v, whch are nalzed as v =,. A each eraon, s randomly drawn such ha P( = ) v, and hen v s updaed based on he exper s rank loss: v + := v e LY rnk (s ). The deals are summarzed n Algorhm 3. Algorhm 3 Ada.OLMR Deals : Inalze: α = 0 and v =, [N] 6: Randomly draw s.. P( = ) v 9: Compue α + = Π(α f (α )) 0: Compue c = LY log (s ) 3: Updae v + = v e LY rnk (s ) Emprcal Edges As we are no mposng OnlneWLC, we need oher measures of he learners predcve powers o prove he loss bound. From (8), can be observed ha he relevan labels have negave coss and he rrelevan ones have posve cos. Furhermore, he summaon of enres of c s exacly 0. Ths observaon suggess a new defnon of wegh: w [] := w Y +exp(s l Y r/ Y [l] s [r]) = c [l] = c [r] = c. 2 l Y r/ Y (9) Ths does no drecly correspond o he wegh used n (3), bu plays a smlar role. Then we defne he emprcal edge: T γ := c h w. (0) Noe ha he baselne learner u Y γ has hs value exacly γ, whch suggess ha s a good proxy for he edge defned n Defnon. Now we presen he loss bound of Ada.OLMR. Theorem 5. (Ada.OLMR, Rank loss bound) For any T and N, wh probably δ, he rank loss suffered by Ada.OLMR s bounded as follows: rnk (ŷ 8 N2 ) γ T +Õ( γ ), () where Õ noaon suppresses dependence on log δ. Proof. We sar he proof by defnng he rank loss suffered by exper as below: M := rnk (s ). Accordng o he formula, here s no harm o defne M 0 = T 2 snce s0 = 0. As he booser chooses an exper hrough he Hedge algorhm, a sandard analyss (cf. [Cesa-Banch and Lugos, 2006, Corollary 2.3]) along wh he Azuma-Hoeffdng nequaly provdes wh probably δ, rnk (ŷ ) 2mn M +2logN +Õ( T), (2) where Õ noaon suppresses dependence on log δ. I s no hard o check ha +exp(a b) 2½(a b), from whch we can nfer w [] 2 LY rnk (s ) and w M, (3) 2 where w s defned n (9). Noe ha hs relaon holds for he case = as well. Now le denoe he dfference of he cumulave logsc loss beween wo consecuve expers: := = log (s ) LY log (s ) log (s +α h ) log (s ). Then he onlne graden descen algorhm provdes mn α [ 2,2] [ log (s +αh ) LY log (s )] +9 T. Here we record an unvarae nequaly: log(+e s+α ) log(+e s ) = log(+ eα +e s) +e s(eα ). (4) 6

7 We expand he dfference o ge = [ log (s l Y +αh ) LY log (s )] log +es r/ Y l Y r/ Y +e s =: f(α). [r] s [l]+α(h [r] h [l]) +e s [l] s [r] s [l] [r] (eα(h [r] h [l]) ) (5) We clam ha mn α [ 2,2] f(α) γ 2 w. Le us rewre w n (9) and γ n (0) as followng. w = γ = l Y r/ Y +e s [l] s [r] l Y r/ Y w h [l] h [r] +e s [l] s [r]. (6) For he ease of noaon, le j denoe an ndex ha moves hrough all uples of (,l,r) [T] Y Y c, and a j and b j denoe followng erms. a j = w +e s b j = h [l] h [r]. [l] s [r] Then from (6), we have j a j = and j a jb j = γ. Now we express f(α) n erms of a j and b j as below. f(α) w = j a j (e αbj ) e α j ajbj = e αγ, where he nequaly holds by Jensen s nequaly. From hs, we can deduce ha mn α [ 2,2] f(α) w e 2 γ γ 2, where he las nequaly can be checked by nvesgang γ = 0, and observng he convexy of he exponenal funcon. Ths proves our clam ha mn f(α) γ α [ 2,2] 2 w. (7) Combnng (3), (4), (5) and (7), we have γ 4 M +9 T. Summng over, we ge by elescopng rule log (sn ) 4 4 log (0) N γ M +9N T = N = γ mn M +9N T. Noe ha log (0) = log2 and LY log (sn ) 0. Therefore we have mnm 4log2 γ T + 36N T γ. Pluggng hs n (2), we ge wh probably δ, rnk (ŷ ) 8log2 γ T +Õ( N T γ +logn) 8 N2 γ T +Õ( γ ), where he las nequaly holds from AM-GM nequaly: cn T c2 N 2 +T 2. Ths complees our proof. Comparson wh OnlneBMR We fnsh hs secon by comparng our wo algorhms. For a far comparson, assume ha all learners are havng edge γ. Snce he baselne learner u Y γ has emprcal edge γ, for suffcenly larget, we can deduce ha γ γ wh hgh probably. Usng hs relaon, () can be wren as rnk (ŷ ) 8 Nγ T +Õ(N γ ). Comparng hs o eher (6) or (7), we can see ha OnlneBMR ndeed has beer asympoc loss bound and sample complexy. Despe hs sub-opmaly, Ada.OLMR shows comparable resuls n real daa ses due o s adapve naure. 4. EXPERIMENTS We performed an expermen on benchmark daa ses aken from MULAN 2. We parcularly chose four daa ses because Dembczynsk and Hüllermeer [202] already provded performances of bach seng boosng algorhms, gvng us a benchmark o compare wh. The auhors n fac used fve daa ses, bu mage daa se became no longer avalable from he source. Table 2 summarzes he basc sascs of daa ses, ncludng ranng and es se szes, number of feaures and labels, 2 Tsoumakas e al. [20],hp://mulan.sourceforge.ne/daases.hml 7

8 and hree sascs of he szes of relevan ses. The daa se m-reduced s a reduced verson of medamll obaned by random samplng whou replacemen. We kep he orgnal spl for ranng and es ses o provde more relevan comparsons. Table 2: Summary of Daa Ses daa #ran #es dm k mn mean max emoons scene yeas medamll m-reduced VFDT algorhms presened by Domngos and Hulen [2000] were used as weak learners. Every algorhm used 00 rees whose parameers were randomly chosen. VFDT s raned usng sngle-label daa, and we fed ndvdual relevan labels along wh mporance weghs ha were compued as max l c c [l]. Insead of usng all covaraes, he booser fed o rees randomly chosen 20 covaraes o make weak predcons less correlaed. All compuaons were carred ou on a Nehalem archecure 0-core 2.27 GHz Inel Xeon E processors wh 25 GB RAM per core. Each algorhm was raned a leas en mes 3 wh dfferen random seeds, and he resuls were aggregaed hrough mean. Predcons were evaluaed by rank loss. Noe ha he algorhm s loss was only recorded for es ses, bu kep updang s parameers whle explorng es ses as well. Snce VFDT oupus a condonal dsrbuon, whch s no of a sngle-label forma, we used hnge loss o compue poenals. Furhermore, OnlneBMR has an addonal parameer of edge γ. We red four dfferen values 4, and he bes resul s recorded as bes BMR. The resuls are summarzed n Table 3. Table 3: Average Loss and Runme n seconds daa bach 5 Ada.OLMR bes BMR emoons scene yeas medamll m-reduced OnlneBMR for m-reduced was esed 0 mes due o long runmes, and ohers were esed 20 mes 4 {.2,.,.0,.00} for small k and {.05,.0,.005,.00} for large k Two algorhms average losses are comparable o each oher and o bach seng resuls, bu OnlneBMR requred much longer runmes. Based on he fac ha bes BMR s performance s repored on he bes edge parameer ou of four rals, Ada.OLMR s far more favorable n pracce. Wh large number of labels, runme for OnlneBMR grows rapdly, and was even mpossble o run medamll daa whn a week, and hs was why we produced he reduced verson. The man boleneck s he compuaon of poenals as hey do no have closed form. 5. CONCLUSION In hs paper, we presened wo onlne boosng algorhms ha make mul-label rankng (MLR) predcons. The algorhms are que flexble n her choce of weak learners n ha varous ypes of learners can be combned o produce a srong learner. OnlneBMR s bul upon he assumpon ha all weak learners are srcly beer han random guessng, and s loss bound s shown o be gh under ceran condons. Ada.OLMR adapvely chooses he weghs over he learners so ha learners wh arbrary (even negave) edges can be boosed. Despe s subopmal loss bound, produces comparable resuls wh OnlneBMR and runs much faser. Onlne MLR boosng provdes several opporunes for furher research. A major ssue n MLR problems s ha here does no exs a canoncal loss. Forunaely, Theorem 2 holds for any proper loss, bu Ada.OLMR only has a rank loss bound. An adapve algorhm ha can handle more general losses wll be desrable. The exsence of an opmal adapve algorhm s anoher neresng open queson. Acknowledgmens We acknowledge he suppor of NSF under CAREER gran IIS and CIF References Rober E Schapre and Yoram Snger. Boosexer: A boosng-based sysem for ex caegorzaon. Machne learnng, 39(2-3):35 68, Oluwasanm O Koyejo, Nagarajan Naarajan, Pradeep K Ravkumar, and Inderj S Dhllon. Conssen mullabel classfcaon. In Advances n Neural Informaon Processng Sysems, pages , 205. Grgoros Tsoumakas, Elefheros Spyromros-Xoufs, Jozef Vlcek, and Ioanns Vlahavas. Mulan: A java 5 The bes resul from bach boosng algorhms n Dembczynsk and Hüllermeer [202] 8

9 lbrary for mul-label learnng. Journal of Machne Learnng Research, 2:24 244, 20. Yoav Freund and Rober E Schapre. A desconheorec generalzaon of on-lne learnng and an applcaon o boosng. In European conference on compuaonal learnng heory, pages Sprnger, 995. Marcn Korykowsk, Leszek Rukowsk, and Rafał Scherer. Fas mage classfcaon by boosng fuzzy classfers. Informaon Scences, 327:75 82, 206. Xao-Le Zhang and DeLang Wang. Boosed deep neural neworks and mul-resoluon cochleagram feaures for voce acvy deecon. In INTERSPEECH, pages , 204. Rober E Schapre and Yoav Freund. Boosng: Foundaons and algorhms. MIT press, 202. Yoav Freund, Rober Schapre, and N Abe. A shor nroducon o boosng. Journal-Japanese Socey For Arfcal Inellgence, 4(77-780):62, 999. Marn Znkevch. Onlne convex programmng and generalzed nfnesmal graden ascen. In Proceedngs of 20h ICML, Nck Llesone and Manfred K Warmuh. The weghed majory algorhm. In Foundaons of Compuer Scence, 989., 30h Annual Symposum on, pages IEEE, 989. Ncolo Cesa-Banch and Gábor Lugos. Predcon, learnng, and games. Cambrdge unversy press, Krzyszof Dembczynsk and Eyke Hüllermeer. Conssen mullabel rankng hrough unvarae loss mnmzaon. In Proceedngs of he 29h ICML, 202. Shang-Tse Chen, Hsuan-Ten Ln, and Ch-Jen Lu. An onlne boosng algorhm wh heorecal jusfcaons. ICML, 202. Alna Beygelzmer, Sayen Kale, and Hapeng Luo. Opmal and adapve algorhms for onlne boosng. ICML, 205. Young Hun Jung, Jack Goez, and Ambuj Tewar. Onlne mulclass boosng. In Advances n Neural Informaon Processng Sysems, 207. Indraneel Mukherjee and Rober E Schapre. A heory of mulclass boosng. Journal of Machne Learnng Research, 4(Feb): , 203. Pedro Domngos and Geoff Hulen. Mnng hgh-speed daa sreams. In Proceedngs of he sxh ACM SIGKDD nernaonal conference on Knowledge dscovery and daa mnng, pages ACM, F. Pedregosa, G. Varoquaux, A. Gramfor, V. Mchel, B. Thron, O. Grsel, M. Blondel, P. Preenhofer, R. Wess, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perro, and E. Duchesnay. Sck-learn: Machne learnng n Pyhon. Journal of Machne Learnng Research, 2: , 20. Rober E Schapre. Drfng games. Machne Learnng, 43(3):265 29, 200. Wewe Cheng, Eyke Hüllermeer, and Krzyszof J Dembczynsk. Bayes opmal mullabel classfcaon va probablsc classfer chans. In Proceedngs of he 27h ICML, 200. We Gao and Zh-Hua Zhou. On he conssency of mullabel learnng. In Proceedngs of he 24h annual conference on learnng heory, pages , 20. 9

10 Appendx A. SPECIFIC BOUNDS FOR OnlneBMR We wll begn hs secon by nroducng a random walk framework o compue poenals. Suppose X := (X,,X k ) s a random vecor ha racks he number of draws of each label among..d. random draws w.r.. u Y γ. Then accordng o (), we may wre φ (s) = E (s+x). Ths framework wll appear frequenly hroughou he proofs. We sar from rank loss. Lemma 6. Under he same seng as n Theorem 2 bu wh poenals bul upon rank loss, we may bound φ N (0) as followng: φ N (0) e γ2 N 2. Proof. For smplcy, we wll drop n he proof. Le X N be he aforemenoned random vecor. Then we may wre he poenal by φ N (0) = EL Y rnk(x N ) w Y E½(X r X l ) = w Y P(X r X l 0). Fx l Y and r / Y. By defnon of u Y γ, we have a := u Y γ [l] = u Y γ [r]+γ =: b. Now suppose we draw wh probably a, wh probably b, and 0 oherwse. Then P(X r X l 0) equals he probably ha he summaon of N..d. random numbers s non-negave. Then we can apply Hoeffdng s nequaly o ge P(X r X l 0) e γ2 N 2. Snce w Y s he nverse of he number of pars (l,r), hs proves our asseron. Lemma 7. Under he same seng as n Theorem 2 bu wh poenals bul upon rank loss, we can show ha, w O( N ). Proof. Frs we fx and. We also fx l Y and r Y c. Then wre s := s +e l and s 2 := s +e r. Agan we nroduce X N. Then we may wre c [r ] c [l ] = φ N (s 2 ) φ N (s ) = E[ rnk (s 2 + X N ) rnk (s + X N )] w Y f(r,l), r/ Y l Y where f(r,l) := E[½(s 2 [r]+x r s 2 [l]+x l ) ½(s [r]+x r > s [l]+x l )]. Here we nenonally nclude and exclude equaly for he ease of compuaon. Changng he order of erms, we can derve f(r,l) P(s [l] s [r] X r X l s 2 [l] s 2 [r]) 3max n P(X r X l = n), where he las nequaly s deduced from he fac ha (s [l] s [r]) (s 2 [l] s 2 [r]) {0,,2}. Usng Berry-Esseen heorem, s shown by Jung e al. [207, Lemma 0] ha max n P(X r X l = n) O( N ), whch mples ha c [r ] c [l ] O( ). N Snce l and r are arbrary, and he bound does no depend on, he las nequaly proves our asseron. Now we provde smlar bound when he poenals are compued from hnge loss. Lemma 8. Under he same seng as n Theorem 2 bu wh poenals bul upon hnge loss, we may bound φ N (0) as followng: φ N (0) (N +)e γ2 N 2. Proof. Agan we wll drop n he proof and nroduce X N. Then we may wre he poenal by φ N (0) = EL Y hnge(x N ) = w Y E(+X r X l ) + N = w Y P(X r X l n) n=0 w Y (N +)P(X r X l 0). We already checked n Lemma 6 ha whch concludes he proof. P(X r X l 0) e γ2 N 2, Lemma 9. Under he same seng as n Theorem 2 bu wh poenals bul upon hnge loss, we can show ha, w 2. 0

11 Proof. Frs we fx and. We also fx l Y and r Y c. Then wre s := s +e l and s 2 := s +e r. Agan wh X N, we may wre c [r ] c [l ] = φ N (s 2 ) φ N (s ) where = E[ hnge (s 2 + X N ) hnge (s + X N )] = w Y f(r,l), r/ Y l Y f(r,l) := E[(+(s 2 + X N )[r] (s 2 + X N )[l]) + (+(s + X N )[r] (s + X N )[l]) + ]. I s no hard o check ha he erm nsde he expecaon s always bounded above by 2. Ths fac along wh he defnon ofw Y provdes ha c [r ] c [l ] 2. Snce our choce of l and r are arbrary, hs proves w [] 2, whch complees he proof. Appendx B. COMPLETE PROOF OF THEOREM 4 Proof. We assume ha an adversary draws a label Y unformly a random from 2 [k] {,[k]}, and he weak learners generae sngle-label predcons w.r.. p [k]. Any boosng algorhm can only make a fnal decson by weghed cumulave voes of N weak learners. We wll manpulae p such ha weak learners sasfy OnlneWLC (δ,γ,s) bu he bes possble performance s close o (6). As we are assumng sngle-label predcons, h = e l for some l [k] and c h = c [l ]. Furhermore, he bounded condon of C0 eor ensures c [l ] s conaned n [0, ]. The Azuma-Hoeffdng nequaly provdes ha wh probably δ, w c [l ] w c p + 2 w 2 2 ln( δ ) w c p + γ w 2 2 k w c p + γ w k + kln( δ ) + kln( δ ), (8) where he second nequaly holds by arhmec mean and geomerc mean relaon and he las nequaly holds due o w [0,]. We sar from provdng a lower bound on he number of weak learners. Le p = u Y for all. Ths can be done by he consran γ < 4k. From he condon of C0 eor ha mn l c[l] = 0,max l c = along wh he fac ha Y {,[k]}, we can show ha c (u Y γ u Y ) γ k. Then he las lne of (8) becomes w c u Y + γ w + kln( δ ) k (w c u Y γ γw k )+ γ w k w c u Y γ +S, + kln( δ ) whch valdaes ha weak learners ndeed sasfy OnlneWLC (δ, γ, S). Followng he argumen of Schapre and Freund [202, Secon 3.2.6], we can also prove ha he opmal choce of weghs over he learners s ( N,, N ). Now we compue a lower bound for he booser s loss. Le X := (X,,X k ) be a random vecor ha racks he number of labels drawn from N..d. random draws w.r.. u Y. Then he expeced rank loss for he booser can be wren as: EL Y rnk (X) w Y P(X l < X r ). Adopng he argumens n he proof by Jung e al. [207, Theorem 4], we can show ha P(X l < X r ) Ω(e 4Nk2 γ 2 ). Ths showsel Y rnk (X) γ 2 ). Seng hs value Ω(e 4Nk2 equal o ǫ, we have N Ω( γ ln 2 ǫ ), consderng k as a fxed consan. Ths proves he frs par of he heorem. Now we move on o he opmaly of sample complexy. We record anoher nequaly ha can be checked from he condons of C0 eor : c (u Y 0 uy γ ) γ. Le T 0 := S 4γ and defne p = uy 0 for T 0 and p = u Y for > T 0. Then for T T 0, (8) mples w c [l ] w c u Y 0 + γ w + kln( δ ) k w c u Y γ +γ(+ k ) w + kln( δ ) w c u Y γ +S. (9) where he las nequaly holds because w T 0 = S 4γ.

12 For T > T 0, agan (8) mples T 0 T w c [l ] w c u Y γ w k =T 0+ + kln( δ ) w c u Y w c u Y γ + k + k γt 0 + kln( δ ) w c u Y γ +S. (20) (9) and (20) prove ha he weak learners ndeed sasfy OnlneWLC (δ, γ, S). Observng ha weak learners do no provde meanngful nformaon for T 0, we can clam any onlne boosng algorhm suffers a loss a leas Ω(T 0 ). Therefore o ge he ceran accuracy, he number of nsancest should be a leasω( T0 ǫ ) = Ω( S ǫγ ), whch complees he second par of he proof. 2