Building Probabilistic Models For Natural Language
Transcription
Building Probabilistic Models For Natural Language
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'& * ) - ) 067 P; 49 ) I) :+ P; ) average over ten runs at each size, up to 50,000 sentences single run at each size 11.5 11.5 11 11 10.5 cross-entropy of test data (bits/token) cross-entropy of test data (bits/token) 10.5 10 9.5 9 TIPSTER bigram WSJ bigram 8.5 TIPSTER trigram WSJ trigram 10 Brown bigram Brown trigram 9.5 9 8.5 8 TIPSTER bigram 7.5 8 WSJ bigram TIPSTER trigram 7 WSJ trigram 7.5 100 ) 6.5 100 1000 10000 sentences of training data (~25 words/sentence) 1000 10000 100000 sentences of training data (~25 words/sentence) 1e+06 1e+07 g+ ;T / +:#$ 6) ( B.2 (7 ! ( (7 ! 9. b 8 J " / #$ ( B.> * 3 . 9 < + / S 4 2 / #$ - -,-,- - -,-,- -,-,- average over ten runs at each size, up to 50,000 sentences single run at each size, up to 10,000,000 sentences 7 7 plus-one difference in test cross-entropy from baseline (bits/token) difference in test cross-entropy from baseline (bits/token) plus-one 6 5 plus-delta 4 3 2 church-gale 1 0 6 5 4 2 1 church-gale 0 katz, interp-held-out, interp-del-int, new-avg-count, new-one-count (see below) -1 100 plus-delta 3 katz, interp-held-out, interp-del-int, new-avg-count, new-one-count (see below) -1 100 1000 10000 sentences of training data (~25 words/sentence) 1000 1e+06 1e+07 single run at each size, up to 10,000,000 sentences 0.02 0.04 0 0.02 difference in test cross-entropy from baseline (bits/token) difference in test cross-entropy from baseline (bits/token) average over ten runs at each size, up to 50,000 sentences 10000 100000 sentences of training data (~25 words/sentence) -0.02 interp-del-int -0.04 -0.06 katz -0.08 -0.1 interp-held-out -0.12 new-one-count -0.14 interp-del-int 0 -0.02 katz -0.04 -0.06 interp-held-out -0.08 new-one-count -0.1 -0.12 new-avg-count -0.14 new-avg-count -0.16 100 ) - -,-,- 1000 10000 sentences of training data (~25 words/sentence) -0.16 100 ; 1000 10000 100000 sentences of training data (~25 words/sentence) 1e+06 1e+07 g+ $Tj *( W + Y R 8', I 6) 36 7( ! b " #$ 7 * = " # & / + ) ( B. ; .4 7(! ( 7(! 9. < / / .#$ ( B * *+ 59 < .+:/ : 2 / #$ )% ( B = =+" ( ( * X & (/g ( B. *$g = / = 6 ! < & / ! & / = - -,-,- -,-,- average over ten runs at each size, up to 50,000 sentences single run at each size, up to 10,000,000 sentences 5 plus-one 4.5 difference in test cross-entropy from baseline (bits/token) difference in test cross-entropy from baseline (bits/token) 5 4 3.5 3 plus-delta 2.5 2 1.5 1 church-gale 0.5 0 4.5 4 3.5 plus-one 3 2.5 2 1.5 1 0 katz, interp-held-out, interp-del-int, new-avg-count, new-one-count (see below) -0.5 100 katz, interp-held-out, interp-del-int, new-avg-count, new-one-count (see below) -0.5 100 1000 10000 sentences of training data (~25 words/sentence) 1000 average over ten runs at each size, up to 50,000 sentences 10000 100000 sentences of training data (~25 words/sentence) 1e+06 1e+07 single run at each size, up to 10,000,000 sentences 0.02 0.02 0 -0.02 difference in test cross-entropy from baseline (bits/token) difference in test cross-entropy from baseline (bits/token) plus-delta church-gale 0.5 church-gale interp-del-int -0.04 -0.06 -0.08 -0.1 interp-held-out new-one-count -0.12 new-avg-count 0 -0.04 -0.08 church-gale -0.1 new-one-count -0.12 katz katz new-avg-count 1000 10000 sentences of training data (~25 words/sentence) ) :+ - -,-,- interp-held-out interp-del-int -0.06 -0.14 100 church-gale -0.02 -0.14 100 ; 1000 10000 100000 sentences of training data (~25 words/sentence) 1e+06 1e+07 g+ ;T ( + R 8', I 6) * 6 7( ! < " .#$ 7 * = " # & / + ) ( B. ; .4 7(! ( 7(! 9. < / / .#$ ( B * *+ 59 < .+:/ : 2 / #$ )% ( B = =+" ( ( * X & (/g ( B. *$g = / = 6 ! < & / ! & / = bigram model trigram model 0.02 0.02 0 difference in test cross-entropy from baseline (bits/token) difference in test cross-entropy from baseline (bits/token) - -,-,- -,-,- church-gale -0.02 interp-del-int -0.04 -0.06 -0.08 -0.1 interp-held-out -0.12 new-avg-count new-one-count katz -0.14 -0.16 0 -0.02 -0.04 -0.06 katz -0.08 interp-del-int -0.1 -0.12 interp-held-out -0.14 new-one-count new-avg-count -0.18 100 ) -0.16 100 1000 10000 sentences of training data (~21 words/sentence) g+ T ( ( = " # & + + ' k" ( 1000 10000 sentences of training data (~21 words/sentence) # ) 6 7( ! b " #$ 7 trigram model 0.04 0 0.02 church-gale -0.02 -0.04 difference in test cross-entropy from baseline (bits/token) difference in test cross-entropy from baseline (bits/token) bigram model 0.02 interp-del-int interp-held-out -0.06 -0.08 -0.1 new-one-count -0.12 new-avg-count -0.14 katz -0.16 ) - interp-del-int -0.04 katz -0.06 -0.08 -0.1 -0.14 new-one-count 1000 10000 100000 sentences of training data (~25 words/sentence) new-avg-count -0.18 100 1e+06 1000 10000 100000 sentences of training data (~25 words/sentence) 1e+06 =9 + ( # ) 36 7( ! < " #$ g+ 2 T ( - *( e% + _1( , *! 7 = " # & + katz interp-held-out -0.12 -0.16 church-gale -0.18 100 0 -0.02 6 */ : *# ( * +!7 6 7(! /:9. + * +*#$ 6 " ! b . 9. b 6 / / #$ * >I # PV H :( B. < */ 3+ 9 &X ." # +! .+ / 6 7 & D;O "Y * R 8',= I 0c ( B & (/ g g + : # / ( B. & 7( Lb #*3+ = / ( * ( 6: ! < & / ! & / + _ " ! j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bigram model trigram model 40 40 35 35 predicted by ML predicted by ML actual 30 30 average corrected count average corrected count actual 25 20 15 25 20 15 10 10 5 5 0 0 0 5 10 15 ) 20 original count 25 30 35 0 # 2(2 7(! ("#(*# 1# # '(8&.( 6 5 10 *( 15 20 original count + 2 bigram model 25 7 30 35 (*. + trigram model 40 40 35 predicted by ML 35 actual predicted by ML 30 30 average corrected count average corrected count actual 25 20 15 25 20 15 10 10 5 5 0 0 0 5 ) ) 10 15 20 original count 25 30 35 0 # 2 7(! (#(# # # ( &.( 6 ,+ 5 . *( 10 15 20 original count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g H,`8`!G5;E%<4"A IH " :;[K H]GEHI$F, ;5<Z=8<1O? e*HI<4M%A3=8D7"<4H iA HI$D!A,"T?!;5<1M9A,=8D7"O? a ) ) P; bigram model trigram model 2.2 2.2 interp-baseline church-gale interp-held-out katz new-avg-count new-one-count 2 1.8 expected count / actual count 1.8 expected count / actual count interp-baseline church-gale interp-held-out katz new-avg-count new-one-count 2 1.6 1.4 1.2 1.6 1.4 1.2 1 1 0.8 0.8 0.6 0.6 0 5 10 15 20 count 25 30 35 0 5 10 15 20 count # 2 # 7(! # #( # # (%7 # ( ( " &.( 2 ( .+ 6 bigram model 30 *( 35 + trigram model 2.2 2.4 interp-baseline church-gale interp-held-out katz new-avg-count new-one-count 2 interp-baseline church-gale interp-held-out katz new-avg-count new-one-count 2.2 2 expected count / actual count 1.8 expected count / actual count 25 1.6 1.4 1.2 1.8 1.6 1.4 1 1.2 0.8 1 0.6 0.8 0 5 10 15 20 count 25 30 35 0 5 10 15 # 2 # 7(! # #( # # (%7 # ( ( " &.( %(* 3+ 6 20 count 25 *( 30 35 + bigram model trigram model 0.4 church-gale interp-held-out katz new-avg-count new-one-count 0.1 difference in test cross-entropy from baseline (bits/token) difference in test cross-entropy from baseline (bits/token) 0.15 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 church-gale interp-held-out katz new-avg-count new-one-count 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0 5 10 15 20 count 25 30 35 0 5 10 15 20 count 25 # 2 6 7( !( #$ 6 # # # ( 7 * # ( (* " (&.( 2 %(* 3+ 6 + 30 bigram model ( trigram model 0.1 0.4 0 difference in test cross-entropy from baseline (bits/token) difference in test cross-entropy from baseline (bits/token) 35 church-gale interp-held-out katz new-avg-count new-one-count -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 church-gale interp-held-out katz new-avg-count new-one-count 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0 5 10 15 20 count 25 30 35 0 5 10 15 20 count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bigram model trigram model 1 1 r<=inf r<=inf 0.8 r<=2 r<=5 r<=1 0.6 r<=10 cumulative fraction of total entropy cumulative fraction of total entropy 0.8 r<=15 r=0 0.4 0.2 0 100 r<=10 r<=2 r<=5 r<=15 0.6 r<=1 r=0 0.4 0.2 1000 10000 100000 sentences of training data (~25 words/sentence) 1e+06 0 100 1e+07 1000 # 2 + # 9 !7 7 * # # # (7 ! ! +&.( *( 3+ 10000 100000 sentences of training data (~25 words/sentence) 1e+06 1e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bigram model trigram model 14 14 n_1=8 n_1=8 12 desired counts devoted to zero count n-grams desired counts devoted to zero count n-grams 12 10 8 6 n_1=6 4 n_1=1 n_1=2 n_1=4 2 10 8 n_1=4 6 n_1=6 n_1=2 4 n_1=1 2 n_1=0 n_1=0 0 0 0 2 4 6 8 10 12 N = total counts in distribution 14 16 18 # 2 7(! ( # # * + # #( .&.( *( 3+ 0 ) 2 4 6 8 10 12 N = total counts in distribution 14 16 18 ( " ! # # ("7 # ) 14 desired counts devoted to zero count n-grams 12 10 n_1=8 8 n_1=6 6 n_1=4 4 n_1=2 2 n_1=1 0 n_1=0 0 2 4 # 2 (7 ! ( # # * + .# :& 6# * + 6 8 10 12 N = total counts in distribution ) 14 16 18 ( " ! # # ("7 # ) ! # # ) ( "9 #!# # *3+ !1# # *& #+ 95 */ & *! &$ 7( 3+3 ( % 8 6 7 6 * 6 7( !( #$9 E/ ! 9( * "% 7(!=.E/ !* .+ :& !4&( *( & 0# ! 7( 0( !( #$" 6 7( !9 ; !( #$ ! "# # # # / !"# # (* ; # 3 ( & + # ! 7( ! ( #! 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(*+! 3+ # 8& # # ( +*# # ( !03 3+D" . .6 # / ! 6 7 3 /# / *3# ( #. 5 / / ! # # # * * # 5 # *# 8( 3 3+ 3 ! 6 * "#(*# # # ! # # 7 # / 5 #(*# # # ! # # ( 7 + ! 6 4#(*# # # ) # & # # ( 3 # # 8( # 1 ( ! 1# # ( ( (& &. 1 * + 6 ) ( 4< ;56D %";5A3D 87a Ca 3e!YX=!< H,D "A]$%$Z"A%A3=8D7"<;5D#H9A,D!;[";5A3D8H,G!;5<"1$;8=C";EA,D"$ ' 766 LAI$TH " 76 e8H,<4A3`8`iA,< "A9A3=!D0"<Z;5D "F3G5A 8 H,G FI$H,&!;5<"1$;8 =C";5A3Dba !;5<#A ! <%$, HI";5A3D ;5< "H 9D H , H,D7"H,F A!];5D # "I'i) ! $ # GEH,D!F3=*HIF% &>A&9G5;5D8F =!8Dbe %' '( A3<9D9G) H,D* =8H,D!F!e &8 ( A,<9D+9G)_e H Pa # bigram model trigram model 1000 1000 100 100 shape predicted by linear interpolation 10 shape predicted by linear interpolation 10 1 corrected count corrected count 1 0.1 church-gale 0.01 0.1 church-gale 0.01 0.001 0.001 0.0001 0.0001 1e-05 1e-05 1e-06 1e-13 1e-06 1e-12 1e-11 1e-10 1e-09 1e-08 bucketing value 1e-07 1e-06 1e-07 1e-14 1e-05 1e-13 # % (# # # + !8# # & ( + # 1e-12 1e-11 1e-10 1e-09 bucketing value 1e-08 1e-07 1e-06 1e-05 *# 8( %( (3 & # # ( &( bigram model trigram model 10 10 r=10 r=10 6 8 shape predicted by linear interpolation corrected count corrected count 8 r=5 4 6 shape predicted by linear interpolation r=5 4 r=3 r=3 2 2 r=2 r=2 r=1 0 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 bucketing value 1e-07 1e-06 r=1 1e-05 0 1e-14 1e-13 1e-12 # 2 % (# # # 7 * # 5 # # & &.( * ( + 1e-11 # 1e-10 1e-09 bucketing value 1e-08 1e-07 1e-06 1e-05 *# ( ( ( & # # ( trigram model 0.02 0 0 difference in test cross-entropy from baseline (bits/token) difference in test cross-entropy from baseline (bits/token) bigram model 0.02 -0.02 -0.04 interp-del-int -0.06 -0.08 interp-held-out -0.1 -0.12 -0.02 interp-del-int -0.04 -0.06 -0.08 -0.1 interp-held-out -0.12 -0.14 -0.14 100 -0.16 100 1000 10000 sentences of training data (~25 words/sentence) # = #+ 7(!* # + ! 6 * # & / + &.( . *( 1000 10000 sentences of training data (~25 words/sentence) ', (7 ! 6 6 7( ! 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J" o§ ) W:TS] GD9B di À oNAP4;<9:=]4;<rP~<9_dQJ|]R=< :T698`8;:TS]8;BH<C6rD9BH¥d GBH4;>I_ L" AP4;F 698GBqe?AVD@:3AP4;F L] ;4;SAPRTSPPD@:T698;:g4 698GB ND@<C66¶'¡F;]AP:=4;<E8;:=RTBp:=4u698GB`NAVD@65OPgl<C|BEBH>b8 FG]AP:g4'BAVD9Bq] G6C|BED9PD@tBHFcJ^_ |OSPD@AP tdF;BHR=<tJN G6h'BqsVAP<C69RT_ ] G6C|BED9PD@698;Be»AVD@:ZAP4;F L] ;4GSnAPRTSPPD@:T698;qi ]NAVD@:=4GSu698GBk6¶'sPBED@<@:T]4;<pP] ;DqAPRTSPPD@:T698;qB U4;F698;AV6`698GBkBIo6CBH4;FGBHFAPRTSPPDb:T698; |BED9PD@<t<9:TS]4N: U>EAP4^69RT_¡J|BE6C6CBEDh698NAP4c698GBqJUAP<9:=>`APRTSPPD@:=698;]4c698;B 4GS]Rg:=<98olR=:=PBAVD969: U>E:=APR36CBIo6E AP4;F|BED9PD@<MAVD9S]:=4;APRgRT_pJKBE6C6CBED*PD*t]<C6m¡]4698;B ¤ lRg:TPB\AVD@69: U>E:=APR?6CBId6Ei 4698GB\NAVD965OP <C|BEBH>@8FG]AP:=4jG698;B*6lnAPRTSPPD@:=698;<r|BED9PD@ APR=t]<C6Z:=FGBH4^69:=>EAPR=RT_PDt]<96i 54hW:TS] GD@B di À o'BmF;:=<CUR=AH_\A<9APtURTBPj698GBZSPD@APAVDr:=4;F; N>IBHFt:=4698GB 4GS]Rg:=<98olR=:=PBAVD969: U>E:=APR FG]AP:g4ji©8GB US] GD9BZFN:=<CNR=A_d<698GBMBIdNAP4N<9:T]4;<'P698GBm<C_oJ|]R=< :T698h698GBm8;:TS]8GBH<C6';D9PJNAVJU:=R=:T69:TBH< L"bZZ8;:=>@8:=<n;D9P|PD969:T]4NAPR6C8G D9BH¥d GBH4^69RT_698GBD9BHF; N>I69:T]4 :=< ;<CBHFji 54 <C]tB <CBH4;<9BP^698GBH<9BM<C_oQJK]Rg<LAVD9BM698GBZt]<96'D9BH¥d GBH4X69R=_\o>E>E GD9Db:=4GS<C_oQJK]Rg<Ei 'AP>@8D9>IPD9D@BH<C|]4;F;<L6CA F;: |BED9BH4^6Z<C_oQJK]ROUR=:=<96CBHF:=4FGBH>ID9BHAP<9:g4GShD9BH¥d GBH4;>I_q<C69AVD969:g4GShD9]2698GBQ]<C6ZD9BH¥d GBH4^6M<C_oQJK]R?:=4 698GBtSPD@APhAVDHiQ©8GBBId;D@BH<9<9:T]4F;:=<9NR=AH_PBHFBIo;D9BH<9<9BH<7APR=R~|]<9<9:TJNR=BQ<C6CD@:g4GS]<M698GB<C_oJ|]R~BIdUAP4;F;< 6C £ :T6FG^BH<4GP6rD9+B NBH>I6698GBmAP>I69 ;APRKSPD@APAVDrD@ NRTBH<L:=698h698;AV6r<C_oJ|]RU]4698GBMR=BE65l8;AP4NFh<9:=F;BPiW;PD BIGAPtNRTBPP698GB3t]<96~D9BH¥d GBH4^6W<9_dQJ|]R^:=4\698GBSPD@APhAVDBIdNAP4NF;<~6C*698GB3<C6CD@:=4GS]< ¬ ¬ nAP4;F ¬ %V§ « ¬ ;AP4;F698GB N698olt]<C6MD9BH¥^ ;BH4X6m<C_oJ|]R?BIdNAP4NF;<M6C698;B\<C6CDb:=4GS]< ¨ *« , ?) « o§ ¨ « ¨ @¬ Q) « o§ ¨ « ;BE69>Vi G>IBE;63PDr698GB*] GD9698p<C_oQJK]ROdAPR=RKP~698GBH<CB*<C_oJ|]R=<BIoNAP4;Fp6Ct<C6CD@:g4GS]<698;AV6 AVD9B>I]4N<C69:T69 GBH4^69<ZAP>E>IPDbF;:=4GS6Ch698GBPD@:=S]:=4;APR|SPD@APhAVDHi'©8d ;<EU:=4`698;:=<3F;]AP:=4`] GDMAPRTSPPD@:T698N:=< AVJNRTB*6Cn>EAV;69 GD9B<9]tB*PW698GB7<C6CD@ ;>I69 ;D9B*;D9BH<CBH4^6:=4p698;B*PD@:TS]:=4;APR|SPDbAPAVDHi 54`W:TS] GD9B di ;B7F;:=<CUR=AH_698;B*SPD@APAVD:=4NF; ;>IBHFnJ^_698GB*e»AVD@:jAP4NF L] ;4GSAPRTSPPD@:=698;:T698 4;:=4;BM4G]4^6CBED@:g4;APRU<9_dQJ|]R=<r:=4698GB 4GS]R=:g<98olR=:TPBMAVD969: U>E:gAPRUFG]hAP:=4ji B7F;:=<9NR=AH_t]4NRT_h698G]<9BMDb ;RTBH< :T698;D9PJUAVJN:=R=:T6¶_`AVJKsPB i GY £ D@ NRTB;D@PJNAVJN:=R=:=69:TBH<AVD9Bt<98G4:=4NAVD9BH4^698GBH<CBH<Ei 4;R=:TPB\:=4~:=S] GD9B di À 8GBED9B\B\R=:g<C6 ¨ , , « A<C_oJ|]R»BIdNAP4NF;<6CG:g4q698;:=< NS] GD@BB\R=:g<C6698GB ¬ (' « § );§ )« , § A<9_dQJ|]RLBIoNAP4;F;<:T698jiq©8N:=<7SPD@APhAVD\F;^BH<\AD@BHAP<C]4;AVJNRTB 5PJuP3SPD9] GU:=4GSq6CPSPBE698GBED <9:=h:=R=AVDW6CBED@:=4NAPRd<9_dQJ|]R=<Ei 'BEsPBEDHo:T6LFGdBH<LRTBH<@<'BHRgRGAV6LD9BH>IPS]4;: H:g4GS78;:TS]8GBED9lRTBEsPBHRG<C6CD@ ;>I69 ;D9BH<:=4 698GBQSPD@APAVDHi ]<C6m<C_oJ|]R=<:g4q698GB\:=4;F; ;>IBHFqSPD@APAVDmFGh4GP6MAV69>@8BHR=RZ:T698`698GB\<C_dQJ|]R=< Y ( ! 6 ! 6 6 ( 9 6;: 6 ( 9 6;: 6 ! 6 ! 6;: 9 ( 6 6 ! 6;( 6 ! " : 9 6 6;: 6 ( 6 6;: 9 6 ! ( : ! ! C ( ! ?:1 B i " Q) B i " ¨ § i ][ " § § l§ i ]À " ¬ %V§ « i Y " ¬ « i Y " *, i Y . " |¨ B i Y « " . i " i " i " ¡¨ « i Qi " " ¬ § i " i Y " ¬ i ][ " i " «*, i "i " ¨ « i " i@¬ " i Y " ) i Y " X G§ §§ i i " j " ¨ i O§ " i " |¨ l§ i " ,&, . i Y " « . i YH[ " i " i Y " «*, i À " ¨ « i " @¬ i " ) i À " ¡¨ « i ][ " ¬ § i ]À " ¬ i YH[ " i " i " i " i " ¡¨ i Y= " § i Q " i ][ " « i " ¬ i " ¬ i " i " i " i " i " 6 ( 6 6 E E ! 9 9 6;: 9 6;: ! E E E 9 ¨ %P§ i ]À " E ( ( E ?:1 ! E 6;: 9 ( 9 ! ( : : E i " " i i " W:TS] GD9Bdi D@APAVDZ:=4;F; ;>IBHFpZ:T698`e»AVD@:jAP4NF L] ;4GShAPR=SPPD@:T698; BH4X6CD@P^_ 4GGi 69:=tB ¤ lR=:=PB AVD969: K>E:=APR JU:T69< PD@F^" NAVD@AP< ¢<9BH>=" |OSPD@APtdF;BHR i ÀoY Y e»AVDb:AP4;F L] ;4;S [ iÀ JNAP<9:g>*APRTSGi ND@<C6NAP<9< Y JNAP<9:g>*APRTSGi |]<C65ONAP<9< i À ©~AVJNRTB di $Z ;QJ|BED3PWNAVD@APtBE6CBED@<ZAP4;Fp6CD@AP:=4N:=4GS69:=tB*PWBHAP>@8APRTSPPD@:T698N :=4698GBhPD@:TS]:=4;APRWSPDbAPAVDHih;PDBIGAPtNRTBP?698GBh<C_oJ|]R SPD9] GN<76CPSPBE698GBED\4;] ;4;<m:T698k4;] ;4;< ]R=RTBHFpJd_AsPBED9J`69AVo:=4GSt4GAVD@S] ;tBH4^69<Ei ZBH4;>IBPG'BQ<CBEB*698;AV63] GDAPRTSPPDb:T698;ª:=<3>ERTBHAVD@R=_hJ|BE6C6CBED 698;AP4`698;Be»AVD@:jAP4NF L] ;4GShAPR=SPPD@:T698; AV6>EAVN69 GD@:=4GStD@BHRTBEs¦AP4^6<C6CD@ N>I69 GD9BPi 54©~AVJNRTB di KBF;:=<CUR=AH_pAh<9APNRTB*P698GB4d ;QJ|BEDPNAVD@APtBE6CBEDb<AP4;FBIdBH>E G69:=]469:=B ]4 A BH>E<969AV69:T]4 "rAP<9<Co>E:=AV6CBHFn:T698BHAP>b8pAPRTSPPD@:T698;qi Bm>@8;^]<CBmq6C\_o:TBHR=FhAVN;D9HG:=AV6CBHR=_ BH¥d ;:TsVAPRTBH4X63|BED9PD@AP4N>IB*PDBHAP>b8APR=SPPD@:T698;`iL©8GB L« o¨ D9 D9BEBED@<36Ch698;B7AP:=4`SPD@APhAVD :=4;FN ;>I69:T]4`N8NAP<CBP] GDmAPRTSPPD@:=698;`NAP4;F698GB ¬ o¨ D9 D9BEBED@<Z6Ch698GB\ 4N<9:=FGBI m ;69<9:=FGBm|]<C65 NAP<9<Hi $ZP69:=>IB3698;AV6W] GDAPRTSPPD@:=698; ;D9oF; ;>IBH<~AM<@:TS]4;: U>EAP4^69RT_7tPD@Br>I]NAP>I6LtoFGBHRd698;AP4\698GBr|OSPD@AP toFGBHRvj8N:=RTBDb ;4;4;:=4;Sn<9:TS]4N: U>EAP4^69RT_q¢AP<C6CBED7698;AP4698GBe?AVD@:WAP4;F L] ;4GSqAPR=SPPD@:T698; BEsPBH4k698;] GS]8 J|P698¡APRTSPPDb:T698;<*BHNRT_k698;B 4N<9:=FGBI m ;69<9:=FGBtAPRTSPPD@:=698;`i`WAVD96P3698;:g<7F;:=<9>ID@BENAP4;>I_k:=<7F; ;B6C 698GB¢AP>I6L698;AV6'BZD9BH¥^ N:TD9B3A7<9APRgRTBEDL4^ NJ|BEDPj4GBEf4G]4^6CBED@:g4;APRG<C_oJ|]R=<W6C\AP>@8;:TBEsPBBH¥^ ;:=s¦APRTBH4^6 |BED9PD@AP4;>IBPoJN G6rB78;AHsPB7APRg<Ct] N4;F698;AV63] GD3|]<C65ONAP<9<3>I]4^sPBED9SPBH<ZtPD9B7¥d ;:=>9oRT_BEsPBH4qS]:TsPBH4 698GB<@APtB74^ NJ|BED3P4G]4^6CBED@:=4;APRj<9_dQJ|]R=<Ei 54`W:TS] GD9BH< di AP4;F di [ ;BF;:=<CUR=AH_698;B*BIdBH>E G69:=]4`69:=tB*AP4;F`tBHPD9_ N<9AVSPB*P?698;B7AP:=4 SPD@APhAVDQ:=4;FN ;>I69:T]4APRTSPPD@:T698N ]4sVAVD@:T] ;<*APt] N4X69<P6CDbAP:=4;:=4GS`F;AV69Aoi P698Pr698GBH<CBSPDbAVN8;< Y E E ( E E E E C E E ! ! !! !! & ( 900 800 execution time (seconds) 700 600 500 400 300 200 100 0 0 5000 10000 15000 20000 25000 training data (tokens) 30000 15000 20000 25000 training data (tokens) 30000 35000 40000 W:TS] ;D9Bdi L dBH>E G69:=]4`69:=tB*sPBED@<@ ;<36CD@AP:=4N:=4GStF;AV69A<9:5EB 16 14 memory usage (Mbytes) 12 10 8 6 4 2 0 0 5000 10000 35000 W:TS] GD@Bdi [ BHtPD9_` ;<9AVSPB7sPBEDb<9 ;<36CD@AP:g4;:=4GStF;AV69A<@:EB Y ( 40000 AVD9B4;BHAVD@RT_nRg:=4GBHAVDHi w m GDtAPRTSPPDb:T698;>I]4;<9:g<C6CBH4X69R=_¡] G6C|BED9PD@tBHFu698GBqe?AVD@:rAP4;F L] ;4GSAPRTSPPDb:T698;:=4u698GBH<CB`BIo|BED@: tBH4^69<Ei m4GB|BED@<C|BH>I69:TsPBP'698N:=<MD9BH<9 ;RT6m>EAP4kJ|B69AVPBH4¡4GP69:g>E:=4GS698;AV6*J|P698kAPRTSPPD@:T698;h<* ;<CB698GB 54;<9:=FGBI * G69<9:=FGBMAPRTSPPD@:T698N AP<A U4;APRj<C6CBE»iLZ<3698;B* 4;<@:=FGBI m G69<@:=FGBZAPR=SPPD@:T698; :=<3Ah8;:=R=R l>ERg:=JU:=4GS APRTSPPD@:=698;`j:T6*>EAP4J|B:g4X6CBED9ND9BE6CBHFAP< U4;F;:g4GSh698GBt4GBHAVD@BH<C67RTo>EAPR~h:=4;:=\ ;:g4698;B<CBHAVD@>@8k<9NAP>IB 6C`698GB:=4N:T69:=APR?SPD@APAVDHi\54698GBe»AVD@:WAP4;F L] N4GS`DbAPtBE'PD@U698N:=<m:=4;:T69:=APR?SPD@APhAVD:=<m>@8G]<9BH4 BH<9<CBH4^69:=APR=R=_D@AP4;FG]hRT_Pi 4] GD3D@APtBEPD9KG698;:=<3:=4N:T69:=APR;SPDbAPAVD:=<r>b8G]<CBH4q:=4^6CBHR=R=:=SPBH4X69RT_n ;<9:=4GSA rAH_PBH<9:gAP4<9BHAVD@>@8ji©8^ ;<HU:T6:g<4GP6M<9 GD9;Db:=<9:=4GS\] GDMAPRTSPPD@:T698;] G6C|BED9PD@<698;Be»AVD@:?AP4;F L] N4GS APRTSPPD@:=698;`i GD@]AtF;: |BED9BH4^63|BED@<C|BH>I69:TsPBPo698GBAP:g4`tBH>@8;AP4N:=<9ªP~698GBe»AVDb:jAP4;F L] ;4GShAPR=SPPD@:T698;ªPD <CBHRTBH>I69:g4GS`SPD@APAVDD@ ;RTBH<*:=<m698;Bh8;:=R=RTl>ER=:=QJN:=4GS<CBHAVDb>@8Pr698GBt54;<9:gFGBI m G69<9:gFGBQAPRTSPPDb:T698;`it B so:TBEª698;:=<:=4a6CBED@<PZAtsPBq<CBE6E698;BJUAP<9:=>tsPB`PZ698GB`e»AVD@:rAP4;F L] ;4GSAPR=SPPD@:T698; :=<\6C AP"F 9 ;<C6tD@ NRTB`;D9PJNAVJU:=R=:T69:TBH<Hiu54] GDnAPRTSPPD@:T698;q'B ;<CBAkD@:=>b8ctsPB<CBE6698NAV6n>IPD9D9BH<9K]4NF;<6C <CBHAP4^69:=>EAPR=R=_tBHAP4;:=4;SP ;R?4GP69:T]4;<Hi B8;AHsPBttsPBH<*698;AV6*>EAP4BIdUR=:=>E:T69RT_`>ID9BHAV6CB4;BE Db ;RTBH<mAP4;F 4G]4^6CBED@:=4NAPRj<C_dQJ|]R=<3 ;4;Rg:TPBMe?AVD@:AP4;F L] N4GSGUtsPBH<698;AV6BId;D@BH<9<>I]4;>IBE;69<Z<9 ;>b8qAP<>ER=AP<@<9:=4GSG <C|BH>E:=APR=5: HAV69:T]4jNAP4;FSPBH4GBED@APR=5: HAV69:T]4ji 8;:=RTB7J|P698APRTSPPD@:T698;h<BHtNR=_pSPD9BEBHFG_<9BHAVD@>@8GBH<MAP4;F>EAP4 698d ;<J|B:g4X6CBED9ND9BE6CBHFAP< U4NF;:=4GSt4GBHAVD9Jd_qRTo>EAPR?h:=4;:=AoN] GDZD@BH<9 ;RT69<ZFGBHt]4;<C6CDbAV6CB698;AV6MJ^_q N<9:=4GS AuD@:=>b8GBEDsPB<CBE6p698;:=<>I]4;<C6CDbAP:=4X6p:=<p\ ;>@8 RTBH<9<<CBED@:=] ;<Ei ©8GBED9B8;AHsPBkJ|BEBH4 <CBEsPBED@APRMD9BH<@ ;RT69< FGBHt]4N<C6CD@AV69:=4GS698GBh<CBEsPBED@:=6l_Pr698GBRTd>EAPR:=4;:=hAp;D9PJNRTBH PDQ698GB54;<9:=FGBI * G69<9:=FGBtAPRTSPPDb:T698; r8GBH4u§ ¨ , g YH[P[ £ FGB AVDb>9PBH4j YH[P[ "bi 54¡6CBEDb<PB n>E:TBH4;>I_P698GBAPRTSPPD@:T698N<QFN: KBED\<9:TS]4;: U>EAP4X69R=_J|BH>EAP ;<9BhP698GBF;: |BED9BH4^6rAH_o< D@ ;R=BH<AVD9B\<CBHRTBH>I6CBHFjiZ 4698GBQe?AVD@:»AP4NF L] ;4;SnAPRTSPPD@:=698;`N]4;B\<C69AVD@69<ZZ:T698qAR=AVD9SPBSPDbAPAVD*AP4;F ;<CBH<698;B54;<9:gFGBI m G69<9:gFGBhAPRTSPPD@:=698;6Ck;D@ ;4;BnAHrAH_ ;4^rAP4X6CBHFDb ;RTBH<\J^_u<CBE6C69:=4GS698GBH:=D\ND9PJNA¦ JN:=Rg:T69:TBH<\6Cu4;BHAVD EBED9Gi ©8N:=<tAV;;D9]AP>b8<@>EAPRTBH<h|dPD@RT_aJKBH>EAP N<CB698GBSPD@APAVD'PD@PBHFf:=698c:=< <9 GJU<C69AP4X69:gAPR=RT_QRgAVD9SPBED698;AP4698GBFGBH<9:=D9BHFt69AVD9SPBE6'SPD@APAVDHdAP4;Fh N<9:=4GS*R=AVD9SPBSPD@APAVD@<'8;AP<LASPD9BHAV6 >I]tN ;69AV69:T]4;APRoBIoKBH4N<CBPi? 6:=<:=tND@AP>I69:=>EAPRd6C7 ;<CB3tPD9Br698;AP46CBH4N<WPK4G]4^6CBED@:g4;APR^<9_dQJ|]R=<~:T698 698GB*e»AVDb:UAP4;F L] ;4;SAVN;D9]AP>@8pJ|BH>EAP ;<9BZ698GB*4d ;QJKBEDP»SPDbAPAVDrD@ ;RTBH<:=<r>E GJN:g>:=4698GBm4^ NJ|BED P4G]4^6CBED@:=4;APR|<C_oQJK]Rg<Ei 54>I]4X6CD@AP<96E] GD*APRTSPPD@:T698NJ|BES]:=4N<:=698Ap<9hAPR=R»SPDbAPAVD*AP4;FkAPF;F;<ZD@ ;R=BH<M:g4;>ID9BHtBH4^69APR=RT_Pi ©8GB3PD9o:=4GSmSPD@APhAVD]4;RT_\>I]4X69AP:g4;<D@ ;R=BH<WKBEDb>IBH:TsPBHFAP<PD9698^_AP4NFt:=<698d ;<L4GP6L ;4N4GBH>IBH<9<9AVD@:gRT_ R=AVD9SPBPiW54t] GDAPR=SPPD@:T698;`PB>EAP4hJN ;:=R=FQSPD@APhAVD@<:T698t8d ;4;F;D9BHF;<~Pj4G]4X6CBEDb:=4;APRo<C_oJ|]R=<HXAP4;F <C69:=RgRU698;B7BIdBH>E ;69:T]4`69:=tB*PW698GBQAPRTSPPD@:T698; :g<3FG]:=4;AV6CBHFqJ^_698GB754;<9:=F;BI m G69<9:=F;B|]<C65ONAP<9<Hi m ;6CKBED@PD@h:=4GS|OSPDbAP tdF;BHR=<W:=4\698GB ND@<C6W6lFG]hAP:=4;<WFGBHt]4;<96CD@AV6CBH<W698;AV6] GDLAPRTSPPDb:T698; :=<'AVJNRTB6C\69AVPBmAPFGs¦AP4^69AVSPBmP698GBZSPD@APAV69:=>EAPRK<C6CD@ N>I69 GD9BND9BH<CBH4^6':=4nF;AV69AoiLGPDrBIoAPNRTBPo ;4;R=:TPB " ? 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