Machine Learning. Inference and Learning in GM. Eric Xing , Fall Lecture 18, November 10, b r a c e

Transcription

1 Min Lrning Fll 2015 Inrn n Lrning in GM ri Xing r Ltur 18 Novr Ring: p. 8.B ook ri MU

2 Inrn n Lrning W now v opt rprsnttions o proility istriutions: BN BN M sris uniqu proility istriution Typil tsks: Tsk 1: How o w nswr quris out? W us inrn s n or t pross o oputing nswrs to su quris Tsk 2: How o w stit plusil ol M ro t D? i. W us lrning s n or t pross o otining point stit o M. ii. But or Bysin ty sk pm D wi is tully n inrn prol. iii. Wn not ll vrils r osrvl vn oputing point stit o M n to o inrn to iput t issing t. ri MU

3 ppros to inrn xt inrn lgorits T liintion lgorit Bli propgtion T juntion tr lgorits ut will not ovr in til r pproxit inrn tniqus Vritionl lgorits Stosti siultion / spling tos Mrkov in Mont rlo tos ri MU

4 oo w: Qury: By in oposition w gt Mrginliztion n liintion g g B D F G H nïv sution ns to nurt ovr n xponntil nur o trs Wt is t proility tt wks r lving givn tt t grss onition is poor? g g 4 ri MU

5 Qury: N to liint: BDFGH Initil tors: oos n liintion orr: HGFDB Stp 1: onitioning ix t vin no i.. on its osrv vlu i.. : Tis stp is isoorpi to rginliztion stp: B D F G H g ~ p ~ p ~ B D F G Vril liintion 5 ri MU

6 Qury: B N to liint: BDFG Initil tors: Stp 2: liint G oput B D F G H g g 1 g g g p B D F g xpl: Vril liintion 6 ri MU

7 Qury: B N to liint: BDF Initil tors: Stp 3: liint F oput B D F G H xpl: Vril liintion g g p B D 7 ri MU

8 B D Qury: B N to liint: BD Initil tors: Stp 4: liint oput B D F G H xpl: Vril liintion g g p B D 8 ri MU

9 Qury: B N to liint: BD Initil tors: Stp 5: liint D oput B D F G H xpl: Vril liintion g g p B 9 ri MU

10 Qury: B N to liint: B Initil tors: Stp 6: liint oput B D F G H xpl: Vril liintion p g g B 10 ri MU

11 Qury: B N to liint: B Initil tors: Stp 7: liint B oput B D F G H xpl: Vril liintion g g p 11 ri MU

12 Qury: B N to liint: B Initil tors: Stp 8: Wrp-up B D F G H xpl: Vril liintion g g ~ p p p p ~ p p ~ 12 ri MU

13 oplxity o vril liintion Suppos in on liintion stp w oput Tis rquirs x y1 yk ' x x y1 yk ' x x x k 1 yk i x y i i1 k Vl X Vl Y ultiplitions i y i For vlu o x y 1 y k w o k ultiplitions Vl X Vl Y i itions i For vlu o y 1 y k w o VlX itions oplxity is xponntil in nur o vrils in t intrit tor ri MU

14 liintion liqu Inu pnny uring rginliztion is ptur in liintion liqus Sution <-> liintion Intrit tr <-> liintion liqu B B D D F n tis l to n gnri inrn lgorit? G H F ri MU

15 Fro liintion to Mssg ssing liintion ssg pssing on liqu tr B B D p g G g D H F F Mssgs n rus ri MU

16 Fro liintion to Mssg ssing liintion ssg pssing on liqu tr notr qury... B B D D F G g H F Mssgs n r rus otrs n to roput ri MU

17 Fro liintion to ssg pssing Rll LIMINTION lgorit: oos n orring Z in wi qury no is t inl no l ll potntils on n tiv list liint no i y roving ll potntils ontining i tk su/prout ovr x i. l t rsultnt tor k on t list For TR grp: oos qury no s t root o t tr Viw tr s irt tr wit gs pointing towrs ro liintion orring s on pt-irst trvrsl liintion o no n onsir s ssg-pssing or Bli ropgtion irtly long tr rns rtr tn on so trnsor grps tus w n us t tr itsl s t-strutur to o gnrl inrn!! ri MU

18 Mssg pssing or trs Lt ij x i not t tor rsulting ro liinting vrils ro llow up to i wi is untion o x i : i Tis is rinisnt o ssg snt ro j to i. j k l ij x i rprsnts "li" o x i ro x j! ri MU

19 liintion on trs is quivlnt to ssg pssing long tr rns! i j k l ri MU

20 T ssg pssing protool: two-pss lgorit: X 1 21 X 1 12 X 2 32 X 2 X 2 42 X 2 X 3 X 4 24 X 4 23 X 3 ri MU

21 Bli ropgtion S-lgorit: Squntil iplnttion ri MU

22 Inrn on gnrl GM Now wt i t GM is not tr-lik grp? n w still irtly run ssg ssg-pssing protool long its gs? For non-trs w o not v t gurnt tt ssg-pssing will onsistnt! Tn wt? onstrut grp t-strutur ro tt s tr strutur n run ssg-pssing on it! Juntion tr lgorit ri MU

23 Sury: xt Inrn T sipl liint lgorit pturs t ky lgoriti Oprtion unrlying proilisti inrn: --- Tt o tking su ovr prout o potntil untions T oputtionl oplxity o t liint lgorit n ru to purly grp-torti onsirtions. Tis grp intrprttion will lso provi ints out ow to sign iprov inrn lgorits Wt n w sy out t ovrll oputtionl oplxity o t lgorit? In prtiulr ow n w ontrol t "siz" o t suns tt ppr in t squn o sution oprtion. ri MU

24 ppros to inrn xt inrn lgorits T liintion lgorit Bli propgtion T juntion tr lgorits ut will not ovr in til r pproxit inrn tniqus Vritionl lgorits Stosti siultion / spling tos Mrkov in Mont rlo tos ri MU

25 Mont rlo tos Drw rno spls ro t sir istriution Yil stosti rprsnttion o oplx istriution rginls n otr xptions n pproxit using spl-s vrgs N 1 t [ x ] x N t 1 syptotilly xt n sy to pply to ritrry ols llngs: ow to rw spls ro givn ist. not ll istriutions n trivilly spl? ow to k ttr us o t spls not ll spl r usul or qlly usul s n xpl ltr? ow to know w'v spl noug? ri MU

26 xpl: niv spling onstrut spls oring to proilitis givn in BN. lr xpl: oos t rigt spling squn 1 Spling:B=< > suppos it is ls B0. S or 0. B0 0=< > suppos it is ls... 2 Frquny ounting: In t spls rigt J0=J0/0=<1/9 8/9>. ri MU B0 0 M0 J0 0 B0 0 M0 J0 0 B0 0 M0 J1 0 B0 0 M0 J0 0 B0 0 M0 J0 0 B0 0 M0 J0 1 B0 1 M1 J1 0 B0 0 M0 J0 0 B0 0 M0 J0 0 B0 0 M0 J0

27 xpl: niv spling onstrut spls oring to proilitis givn in BN. lr xpl: oos t rigt spling squn 3 wt i w wnt to oput J1? w v only on spl... J1=J1/1=<0 1>. 4 wt i w wnt to oput JB1? No su spl vill! J1=JB1/B1 n not in. For ol wit unrs or or vrils rr vnts will vry r to grnr voug spls vn tr long ti or spling... 0 B0 0 M0 J0 0 B0 0 M0 J0 0 B0 0 M0 J1 0 B0 0 M0 J0 0 B0 0 M0 J0 0 B0 0 M0 J0 1 B0 1 M1 J1 0 B0 0 M0 J0 0 B0 0 M0 J0 0 B0 0 M0 J0 ri MU

28 Mrkov in Mont rlo MM onstrut Mrkov in wos sttionry istriution is t trgt nsity = X. Run or T spls urn-in ti until t in onvrgs/ixs/rs sttionry istriution. Tn ollt M orrlt spls x. Ky issus: Dsigning proposls so tt t in ixs rpily. Dignosing onvrgn. ri MU

29 Mrkov ins Dinition: Givn n n-insionl stt sp Rno vtor X = x 1 x n x t = x t ti-stp t x t trnsitions to x t+1 wit pro x t+1 x t x 1 = Tx t+1 x t = Tx t x t+1 Hoognous: in trin y stt x 0 ix trnsition krnl Q rows su to 1 quiliriu: x is sttionry quiliriu istriution i x' = x x Qxx'. i.. is lt ignvtor o t trnsition trix T = T Q ri MU X 1 X X 3

30 Gis spling T trnsition trix upts no on t ti using t ollowing proposl: Q x x x ' x p x ' x i i i i i i ' It is iint sin p x i x i only pns on t vlus in X i s Mrkov lnkt ri MU

31 Gis spling Gis spling is n MM lgorit tt is spilly pproprit or inrn in grpil ols. T prou w v vril st X={x 1 x 2 x 3... x N } or GM t stp on o t vrils X i is slt t rno or oring to so ix squns not t rining vrils s X -i n its urrnt vlu s x -i t-1 Using t "lr ntwork" s n xpl sy t ti t w oos X n w not t urrnt vlu ssignnts o t rining vrils X - otin ro prvious spls s t 1 t 1 t 1 t 1 t 1 x x x x x t onitonl istriution px i x -i t-1 is oput vlu x i t is spl ro tis istriution t spl x i t rpls t prvious spl vlu o X i in X. B J M i.. x x x t t 1 t ri MU

32 Mrkov Blnkt Mrkov Blnkt in BN vril is inpnnt ro otrs givn its prnts ilrn n ilrn s prnts sprtion. MB in MRF vril is inpnnt ll its non-nigors givn ll its irt nigors. px i X -i = px i MBX i Gis spling vry stp oos on vril n spl it y XMBX s on prvious spl. ri MU

33 Gis spling o t lr ntwork MB={B J M} MB={ B} To lult JB1M1 oos B101M1J1 s strt vins r B1 M1 vrils r J. oos nxt vril s Spl y MB=B1 0 M1 J1 suppos to ls. B1 0 0 M1 J1 oos nxt rno vril s spl ~B10... ri MU

34 xpl ri MU

35 xpl: ri MU

36 xpl ri MU

37 xpl J1 B1M1 = 0.90 J1 1M0 = J1 = M1 = M1J1 = 0.17 ri MU

38 T o siultion Run svrl ins Strt t ovr-isprs points Monitor t log lik. Monitor t sril orrltions Monitor ptn rtios R-prtriz to gt pprox. inp. R-lok Gis ollps int. ovr otr prs. Run wit troul prs. ix t rsonl vls. ri MU

39 Lrning Grpil Mols T gol: Givn st o inpnnt spls ssignnts o rno vrils in t st t ost likly? Bysin Ntwork ot DG n Ds B B R R BR=TFFTF BR=TFTTF.. BR=FTTTF ri MU B B

40 Lrning Grpil Mols ont. Snrios: opltly osrv GMs irt unirt prtilly osrv GMs irt unirt n opn rsr topi stition prinipls: Mxil liklioo stition ML Bysin stition Mxil onitionl liklioo Mxil "Mrgin" W us lrning s n or t pross o stiting t prtrs n in so ss t topology o t ntwork ro t. ri MU

41 ML or gnrl BN prtrs I w ssu t prtrs or D r glolly inpnnt n ll nos r ully osrv tn t logliklioo untion oposs into su o lol trs on pr no: l ; D log p D log i p xn i xn log p x i n i xn i i n i i n X 2 =1 X 5 =0 X 2 =0 X 5 =1 ri MU

42 xpl: oposl liklioo o irt ol onsir t istriution in y t irt yli GM: p x p x1 1 p x2 x1 2 p x3 x1 3 p x4 x2 x3 4 Tis is xtly lik lrning our sprt sll BNs o wi onsists o no n its prnts. X 1 X 1 X 1 X 1 X 2 X 3 X 2 X 2 X 3 X 3 X 4 X 4 ri MU

43 .g.: ML or BNs wit tulr Ds ssu D is rprsnt s tl ultinoil wr p X j X k Not tt in s o ultipl prnts will v oposit stt n t D will ig-insionl tl T suiint sttistis r ounts o ily onigurtions T log-liklioo is Using Lgrng ultiplir j to nor 1 w gt: ijk ijk n ijk i n x j n i i X i x k n i ijk l ; D log n log i j k ML ijk n ijk n i j' k i j k ijk n ij' k ijk ijk ri MU

44 Dinition o HMM Trnsition proilitis twn ny two stts y 1 y 2 y 3... y T j i p yt 1 yt 1 1 i j x 1 x 2 x 3... x T or i. i p yt yt 1 1 ~ Multinoil i 1 i 2 i M I Strt proilitis p ~ Multinoil. y M ission proilitis ssoit wit stt i. i p xt yt 1 ~ Multinoil i 1 i 2 i K I or in gnrl: i. i p x y 1 ~ I t t i ri MU

45 Suprvis ML stition Givn x = x 1 x N or wi t tru stt pt y = y 1 y N is known T T l θ; x y log p x y log p y p y y p x x Din: ij B ik = # tis stt trnsition ij ours in y = # tis stt i in y its k in x W n sow tt t xiu liklioo prtrs r: ML ij ML ik # i j # i # i k # i n T n t n T n t n i j 2 n t 1 n t T i y t 2 n t 1 1 T I y is ontinuous w n trt xn t yn t : t 1 : T n 1 : N s NT osrvtions o.g. Gussin n pply lrning ruls or Gussin y y t 1 i n t y x i n t y k n t k ' ij j ' Bik B ij ' ik ' n 1 n t n t1 n t n t t2 t1 ri MU

46 onsir t istriution in y t irt yli GM: N to oput px H x V inrn Wt i so nos r not osrv? x x x p x x p x x p x p x p X 1 X 4 X 2 X 3 46 ri MU

47 ML or BNs wit tulr Ds ssu D is rprsnt s tl ultinoil wr p X j X k Not tt in s o ultipl prnts will v oposit stt n t D will ig-insionl tl T suiint sttistis r ounts o ily onigurtions T log-liklioo is Using Lgrng ultiplir j to nor 1 w gt: ijk ijk n ijk i n x j n i i X i x k n i ijk l ; D log n log i j k ML ijk n ijk n i j' k i j k ijk n ij ' k ijk ijk ri MU

48 Sury GMM sris uniqu proility istriution Typil tsks: Tsk 1: How o w nswr quris out? W us inrn s n or t pross o oputing nswrs to su quris Tsk 2: How o w stit plusil ol M ro t D? i. W us lrning s n or t pross o otining point stit o M. ii. But or Bysin ty sk pm D wi is tully n inrn prol. iii. Wn not ll vrils r osrvl vn oputing point stit o M n to o inrn to iput t issing t. ri MU