Formalizing Triggers : A LearningModel for Finite Spaces

I n a r ecent seminal paper , Gi bs on and Wexl er (, GW) t ake i mpor t ant s t eps t o formal i zi ng t he not i on of l anguage l ear ni ng i n a ( ni t e) s pace whos e gr ammar s ar e char act er i zed by a ni t e number of parameters. One of t he i r ai ms i s t o char act er i ze t he compl exi t y of l ear ni ng i n s uch s paces . For exampl e , t hey demons t r at e t hat even i n ni t e s paces , conver gence may be a pr obl ems i nce i t i s pos s i bl e under s ome s i ngl es t e gr adi ent as cent met hods t o r emai n at a l ocal maxi mum. Fr omt he s t andpoi nt of l ear ni ng t heor y, however , GWl eave open s ever al ques t i ons t hat can be addr es s ed by a mor e pr ec i s e f ormal i zat i on i n t erms of Mar kov s t r uct ur es ( a pos s i bl e f ormal i zat i on s ugges t ed but l e f t unpur s ued i n a f oot not e of GW) . I n t hi s paper we expl i c i t l y f ormal i ze l ear ni ng i n a ni t e par amet er s pace as a Mar kov s t r uct ur e whos e s t at es ar e par amet er s et t i ngs . Sever al i mpor t ant r es ul t s t hat f ol l owdi r ect l y f r omt hi s char act er i zat i on, i nc l ude cor r ect ed ver s i on of GW's cent r al conver gence pr oof ; ( 2) an expl i c i t f ormul a f or cal cul at i ng t he t r ans i t pr obabi l i t i es between hypot hes es and t he exi s t ence of \pr obl ems t at es " i n addi t i on t o l ocal maxi ma; ( 3 an expl i c i t cal cul at i on of t he t i me needed t o conver ge , i n t erms of number of ( pos i t i ve) exampl es ; ( 4 t he conver gence and compar i s on of s ever al var i ant s of t he GWl ear ni ng pr ocedur e , e . g. , r andomwal k; ( 5) bat chand PACs t yl e l ear ni ng bounds f or t he model .  This report describes researchdone within the Center for Biological andComputational Learning in the Department of Brain andCognitive Sciences, and at the Arti cial Intel l igence Laboratory. This research is supported byNSF grant 9217041-ASC andARPAunder the HPCCprogram. Correspondence by e-mai l could be directed to pn@ai .mit.eduor berwick@ai .mit.edu. 1 Introducti on: The Tri ggeri ng Model as a Markov structure Recent l y, Gi bs on and Wexl er (  , GW) have begun t o f ormal i ze t he not i on of l anguage l ear ni ng i n a ( ni t e) s pace whos e gr ammar s ( and l anguages ) ar e char act er i zed by a ni t e number of par amet er s or 1di mens i onal Bool eanval ued ar r ays , n l ong. Agrammar i n t hi s s pace i s s i mpl y a par t i cul ar nl engt h ar r ay of 0' s and 1' s ; hence t her e ar e 2 n pos s i bl e gr ammar s ( l anguages ) . One of Gi bs on and Wexl er ' s ai ms i s t o es t abl i s h t hat under s ome s i mpl e hi l l c l i mbi ng l ear ni ng r egi mes , namel y, s i ngl es t ep gr adi ent as cent , s ome l i ngui s t i cal l y nat ur al , ni t e , s paces ar e unl ear nabl e , i n t he s ens e t hat pos i t i veonl y exampl es l ead t o local maxima|i ncor r ect hypot hes es f r omwhi ch a l ear ner can never es cape. Mor e br oadl y, t hey wi s h t o s how t hat l ear nabi l i t y i n s uch s paces i s s t i l l an i nt er es t i ng pr obl em, i n t hat t her e i s a s ubs t ant i ve l ear ni ng t heor y concer ni ng f eas i bi l i t y, conver gence t i me, and t he l i ke , t hat mus t be addr es s ed beyond t r adi t i onal l i ngui s t i c t heor y and t hat mi ght even choos e between ot herwi s e adequat e l i ngui s t i c t heor i es . I n t hi s paper , we choos e as a conveni ent s t ar t i ng poi nt t hei r Tr i gger i ng Lear ni ng Al gor i t hm(TLA) t o f ocus our i nves t i gat i on of par amet er l ear ni ng. Our cent r al r es ul t i s t hat t he per f ormance of t hi s al gor i t hmi s compl et e l y model ed by a Mar kov chai n. The r emai nder of t he cur r ent paper i s devot ed t o expl or i ng t he bas i c cons equences of t hi s f act . Let us r s t r evi ew t he GWmodel and t he TLA. Fol l owi ng Gol d [ 2] t he bas i c f r amewor k i s t hat of i dent i cat i on i n t he l i mi t . The l ear ner ( chi l d) s t ar t s out i n an ar bi t r ar y st at e= some s et t i ng of t he n par amet er val ues . The l ear ner ( chi l d) r ece i ves a ( count abl y i n ni t e) s equence of pos i t i ve exampl e s ent ences dr awn f r oms ome t ar get l anguage, L t. Af t er each pr es ent at i on, t he l ear ner can ei t her ( i ) s t ay i n t he s ame s t at e ; or ( i i ) move t o a new hypot hes i s s t at e , us i ng t he al gor i t hmgi ven bel ow. I f af t er s ome ni t e number of exampl es t he l ear ner conver ges t o t he cor r ect t ar get l anguage (= par amet er s et t i ngs ) and never changes s t at e , t hen i t has cor r ect l y i dent i ed t he t ar get l anguage; ot herwi s e , i t does not conver ge . I n addi t i on, i n t he GWmodel t he l anguage l ear ner obeys two f undament al cons t r ai nt s : ( 1) t he si ngl e-val ue const rai nt|the l ear ner can change onl y 1 par amet er val ue at a t i me; and ( 2) t he greedi ness const rai nt|i f , t he l ear ner i s gi ven a pos i t i ve exampl e i t cannot r ecogni ze ( accept ) , and i f t he l ear ner changes one par amet er val ue and nds t hat i t can accept t he exampl e , t hen t he l ear ner r et ai ns t hat new par amet er val ue . Fi nal l y, we al s o r ecal l GW' s de ni t i on of a l ocal t ri gger (mi nor not at i onal changes as i de) : gi ven val ues f or al l par amet er s but one, a l ocal t ri gger f or val ue v of par amet er s p i, pi( v ) , i s a s ent ence s f r omt he t ar get gr ammar G T s uch t hat s i s gr ammat i cal i p i( v ) =v . GWthen s t at e t hei r TLAas f ol l ows : [ I ni t i al i ze ] St ep 1. St ar t at s ome r andompoi nt i n t he ( ni t e) s pace of pos s i bl e par amet er s et t i ngs , s pec i f yi ng a s i ngl e hypot hes i zed gr ammar wi t h i t s r es ul t i ng ext ens i on as a l anguage; [ Pr oces s i nput s ent ence ] St ep 2. Rece i ve a pos i t i ve exampl e s ent ence s i at t i me t i ( exampl es dr awn f r om t he l anguage of a s i ngl e t ar get gr ammar , L(Gt) ) , f r om a uni f orm di s t r i but i on on t he l anguage (we s hal l be abl e t o r e l ax t hi s di s t r i but i onal cons t r ai nt l at er on) ; [ Lear nabi l i t y on er r or det ect i on] St ep 3. I f t he cur r ent gr ammar par s es ( gener at es ) s i, t hen go t o St ep 2; ot herwi s e , cont i nue. [ Si ngl es t ep gr adi ent as cent ] Se l ect a s i ngl e par amet er at r andom, uni f orml y wi t h pr obabi l i t y 1=n, t o i p f r om i t s cur r ent s et t i ng, and change i t ( 0 mapped t o 1, 1 t o 0) i t hat change al l ows t he current sent ence t o be anal yzed; ot herwi s e go t o St ep 2; Of cour s e , t hi s al gor i t hm never hal t s i n t he us ual s ens e . GWai mt o s howunder what condi t i ons t hi s al gor i t hm conver ges \i n t he l i mi t "|that i s , af t er s ome number , n; of s t eps , wher e n i s unknown, t he cor r ect t ar get par amet er s et t i ngs wi l l be s e l ect ed and never be changed. Thei r cent r al c l ai mi s s t at ed as t hei r Theor em 1 ( p. 7 i n t hei r manus cr i pt ) . 1 Theorem 1 As l ong as t he probabi l i t y i s al ways great er t han a l ower bound b (b > 0) t hat t he l earner wi l l 1) encount er a l ocal t ri gger for some i ncorrect l yset paramet er P , and 2) t hen reset P accordi ngl y t o t he t arget val ue, i t t urns out t hat t he t arget grammar can al ways be l earned usi ng t he Tri ggeri ng Learni ng Al gori t hm. 1.1 The Markov formulation Fr omt he s t andpoi nt of l ear ni ng t heor y, however , GW l eave open s ever al ques t i ons t hat can be addr es s ed by a mor e pr ec i s e f ormal i zat i on of t hi s model i n t erms of Mar kov chai ns ( a pos s i bl e f ormal i zat i on s ugges t ed but l e f t unpur s ued i n f oot not e 9 of GW) . We can pi ct ur e t he hypot hes i s s pace , of s i ze 2 , as a s et of poi nt s , each cor r es pondi ng t o one par t i cul ar ar r ay of par amet er s et t i ngs ( l anguages , gr ammar s ) . Cal l each poi nt a hypot hesi s st at e or s i mpl y st at e of t hi s s pace . As i s convent i onal , w de ne t hes e l anguages over s ome al phabet as a s ubs et of . One of t hemi s t he t ar get l anguage ( gr ammar ) . We ar bi t r ar i l y pl ace t he ( s i ngl e) t ar get gr ammar at t he cent er of t hi s s pace . Si nce by t he TLAt he l ear ner i s r es t r i ct ed t o movi ng at mos t 1 bi nar y val ue i n a s i ngl e s t ep, t he t heor et i cal l y pos s i bl e t r ans i t i ons between s t at es can b dr awn as ( di r ect ed) l i nes connect i ng par amet er ar r ays ( hypot hes es ) t hat di er by at mos t 1 bi nar y di gi t ( a 0 or a 1 i n s ome cor r es pondi ng pos i t i on i n t hei r ar r ays ) . Rec l l t hat t hi s i s t he s ocal l ed Hammi ng di st ance . We may f ur t her pl ace wei ght s on t he t r ans i t i ons f r om s t at e i t o s t at e j cor r es pondi ng t o t he nonzer o b ' s ment i oned i n t he t heor em above; t hes e cor r es pond t o t he pr obabi l i t i es t hat t he l ear ner wi l l move f r om hypot hes i s s t at e i t o s t at e j . I n f act , as we s hal l s how bel ow, gi ven a di s t r i but i on over L(G) , we can f ur t her car r y out t he cal cul at i on of t he act ual b ' s t hems el ves . Thus , we 1Note that the notion of \trigger" does not enter into the statement of the TLAor the constraints the TLAemploys, bu only into the statement of the theorem. 1 can pi ct ur e t he TLA l ear ni ng s pace as a di r ect ed, l abe l ed gr aph V wi t h 2 n ver t i ces . 2 Mor e pr ec i s e l y, we can make t he f ol l owi ng r emar ks about t he TLAs ys t emGW