Chapter 4. Neural Networks Based on Competition

Transcription

1 Chaper 4. Neural Neworks Based on Compeon Compeon s mporan for NN Compeon beween neurons has been observed n bologcal nerve sysems Compeon s mporan n solvng many problems To classfy an npu paern _1 no one of he m classes dea case: one class node has oupu 1, all oher 0 ; ofen more han one class _n nodes have non-zero oupu INPUT C_1 C_m CLASSIFICATION If hese class nodes compee wh each oher, maybe only one wll wn evenually wnner-akes-all. The wnner represens he compued classfcaon of he npu

2 Wnner-akes-all WTA: Among all compeng nodes, only one wll wn and all ohers wll lose We manly deal wh sngle wnner WTA, bu mulple wnners WTA are possble and useful n some applcaons Eases way o realze WTA: have an eernal, cenral arbraor a program o decde he wnner by comparng he curren oupus of he compeors break he e arbrarly Ths s bologcally unsound no such eernal arbraor ess n bologcal nerve sysem.

3 Ways o realze compeon n NN Laeral nhbon Mane, Mecan ha oupu of each node feeds y_ o ohers hrough nhbory connecons wh negave weghs Resource compeon < 0 oupu of _k s dsrbued o y_ y_ and y_ proporonal o w_k and w_k, as well as y_ and y_ w self decay k bologcally sound Learnng mehods n compeve neworks Compeve learnng Kohonen learnng self-organzng map, SOM Couner-propagaon ne Adapve resonance heory ART n Ch. 5 w < 0 _k y_ w < 0 w y_ w k

4 Fed-wegh Compeve Nes Mane Laeral nhbon beween compeors weghs : w 1 ε f oherwse acvaon funcon : f > 0 f 0 oherwse Noes: Compeon: erave process unl he ne sablzes a mos one node wh posve acvaon 0 < ε < 1 / m, oo small: akes oo long o converge oo bg: may suppress he enre nework no wnner where m s he # of compeors ε ε

5 Mecal Ha Archecure: For a gven node, close neghbors: cooperave muually ecaory, w > 0 farher away neghbors: compeve muually nhbory,w < 0 oo far away neghbors: rrelevan w 0 Need a defnon of dsance neghborhood: one dmensonal: orderng by nde 1,2, n wo dmensonal: lace

6 > < k k c k c w, dsance f 0, dsance f, dsance f weghs 2 1 ramp funcon : ma ma ma acvaon funcon > < f f f f

7 eample : 0 0.0, 0.5, 0.8, 1.0, 0.8, 0.5, , 0.38, 1.06, 1.16, 1.06, 0.38, , 0.39, 1.14, 1.66, 1.14, 0.39, 0.0 Equlbrum: negave npu posve npu for all nodes wnner has he hghes acvaon; s cooperave neghbors also have posve acvaon; s compeve neghbors have negave acvaons.

8 Hammng Nework Hammng dsance of wo vecors, dmenson n, Number of bs n dsagreemen. In bpolar: and y y a d where : a s number of bs n agreemen n and y d s number of bs dfferen n and y d n a hammng dsance y 2a n a 0.5 y n larger y larger a shorer Hammng dsance of

9 Suppose a space of paerns s dvded no k classes, each class has an eampler represenave vecor e. An npu belongs o class, f and only f s closer o e han o any oher e,.e., e e Hammng ne s such a classfer: Weghs: le represen class w Y 0.5e, b 0. 5n The oal npu o y _ n b w 1 2 e Y n a

10 Upper layer: MAX ne akes he y_n as s nal value, hen eraes oward sable sae one oupu node wh hghes y_n wll be he wnner because s wegh vecor s closes o he npu vecor As assocave memory: each corresponds o a sored paern; Y paern connecon/compleon; sorage capacy oal # of nodes: k oal # of paerns sored: k capacy: k or k/k 1

11 Implc laeral nhbon by compeng lmed resources: he acvaon of he npu nodes y_1 y_ y_m W y _ n y _ n larger y _ n y β w y decay y akes larger share of w

12 Compeve Learnng Unsupervsed learnng Goal: Learn o form classes/clusers of eamplers/sample paerns accordng o smlares of hese eampers. Paerns n a cluser would have smlar feaures No pror knowledge as wha feaures are mporan for classfcaon, and how many classes are here. Archecure: Oupu nodes: Y_1,.Y_m, represenng he m classes They are compeors WTA realzed eher by an eernal procedure or by laeral nhbon as n Mane

13 Tranng: Tran he nework such ha he wegh vecor w. assocaed wh Y_ becomes he represenave vecor of he class of npu paerns Y_ s o represen. Two phase unsupervsed learnng compeng phase: apply an npu vecor randomly chosen from sample se. compue oupu for all y: y deermne he wnner wnner s no gven n ranng samples so hs s unsupervsed rewardng phase: he wnner s reworded by updang s weghs weghs assocaed wh all oher oupu nodes are no updaed repea he wo phases many mes and gradually reduce he learnng rae unl all weghs are sablzed. w

14 Wegh updae: w Mehod 1: Mehod 2 w w w w w -w_ a-w_ w_ a w_ w_ a-w_ w_ w_ a In each mehod, s moved closer o Normalzng he wegh vecor o un lengh afer s updaed w w w w

15 w s movng o he cener of a cluser of sample vecors afer repeaed wegh updaes Three eamplers: S1, S2 and S3 w_0 w_3 Inal wegh vecor w_0 Afer successvely raned S3 w_1 by S1, S2, and S3, S1 he wegh vecor changes o w_1, S2 w_2, and w_3 w_2

16 Eamples A smple eample of compeve learnng pp vecors of dmenson 4 n 2 classes 4 npu nodes, 2 oupu nodes S1 1, 1, 0, 0 S2 0, 0, 0, 1 S3 1, 0, 0, 0 S4 0, 0, 1, 1 Inalzaon: 0.6, wegh mar: Tranng wh S1 D1 S1 w W D 1 S1 w , class 2 wns w hen W

17 W Smlarly, afer ranng wh S2 0, 0, 0, 1, n whch class 1 wns, wegh mar becomes A he end of he frs eraon each of he 4 vecors are used, wegh mar becomes Reduce Repea ranng. Afer 10 eraons, wegh mar becomes 6.7e e e e W W S1 and S3 belong o class 2 S2 and S4 belong o class 1 w_1 and w_2 are he cenrods of he wo classes

18 Commens 1. Ideally, when learnng sops, each s close o he cenrod of a group/cluser of sample npu vecors. 2. To sablze w, he learnng rae may be reduced slowly oward zero durng learnng. 3. # of oupu nodes: oo few: several clusers may be combned no one class oo many: over classfcaon ART model laer allows dynamc add/remove oupu nodes w 4. Inal : w ranng samples known o be n dsnc classes, provded such nfo s avalable random bad choces may cause anomaly

19 Eample w_2 w_1 w 1 w 2 wll always wn no maer he sample s from whch class s suck and wll no parcpae n learnng unsuck: le oupu nodes have some conscence emporarly sho off nodes whch have had very hgh wnnng rae hard o deermne wha rae should be consdered as very hgh

20 Kohonen Self-Organzng Maps SOM Compeve learnng Kohonen 1982 s a specal case of SOM Kohonen 1989 In compeve learnng, he nework s raned o organze npu vecor space no subspaces/classes/clusers each oupu node corresponds o one class he oupu nodes are no ordered: random map cluser_1 w_2 w_3 w_1 cluser_2 cluser_3 The opologcal order of he hree clusers s 1, 2, 3 The order of her maps a oupu nodes are 2, 3, 1 The map does no preserve he opologcal order of he ranng vecors

21 Topographc map a mappng ha preserves neghborhood relaons beween npu vecors, opology preservng or feaure preservng. f 1 and 2 are wo neghborng npu vecors by some dsance mercs, her correspondng wnnng oupu nodes classes, and mus also be close o each oher n some fashon one dmensonal: lne or rng, node has neghbors or ±1mod n wo dmensonal:grd. recangular: node, has neghbors:, ± 1, ± 1,, or addonal ± 1, ± 1 heagonal: 6 neghbors ±1

22 Bologcal movaon Mappng wo dmensonal connuous npus from sensory organ eyes, ears, skn, ec o wo dmensonal dscree oupus n he nerve sysem. Renoopc map: from eye rena o he vsual core. Tonoopc map: from he ear o he audory core These maps preserve opographc orders of npu. Bologcal evdence shows ha he connecons n hese maps are no enrely pre-programmed or pre-wred a brh. Learnng mus occur afer he brh o creae he necessary connecons for approprae opographc mappng.

23 Two layer nework: Oupu layer: SOM Archecure Each node represens a class of npus Node acvaon : y w w Neghborhood relaon s defned over hese nodes Each node cooperaes wh all s neghbors whn dsance R and compees wh all oher oupu nodes. Cooperaon and compeon of hese nodes can be realzed by Mecan Ha model R 0: all nodes are compeors no cooperave random map R > 0: opology preservng map

24 SOM Learnng 1. Inalze W for all oupu nodes, and o a small value 2. For a randomly seleced npu sample/eampler deermne he wnnng oupu node J eher W J s mamum or w J w s mnmum 3. For all oupu node wh J R, updae he wegh w w w 4. Perodcally reduce and R slowly. 5. Repea 2-4 unl he nework sablzed. 2

25 Noes 1. Inal weghs: small random value from -e, e 2. Reducon of : Lnear: Geomerc: β may be 1 or greaer han 1 3. Reducon of R: R R 1 whle R > 0 should be much slower han reducon. R can be a consan hrough ou he learnng. 4. Effec of learnng For each npu, no only he wegh vecor of wnner J s pulled closer o, bu also he weghs of J s close neghbors whn he radus of R. 5. Evenually, w becomes close smlar o w ± 1. The classes hey represen are also smlar. 6. May need large nal R

26 Eamples A smple eample of compeve learnng pp vecors of dmenson 4 n 2 classes 4 npu nodes, 2 oupu nodes S1 1, 1, 0, 0 S2 0, 0, 0, 1 S3 1, 0, 0, 0 S4 0, 0, 1, 1 Inalzaon: 0.6, wegh mar: Tranng wh S1 D1 S1 w W D 1 S1 w , class 2 wns w hen W

27 How o llusrae Kohonen map Inpu vecor: 2 dmensonal Oupu vecor: 1 dmensonal lne/rng or 2 dmensonal grd. Wegh vecor s also 2 dmenson Represen he opology of oupu nodes by pons on a 2 dmensonal plane. Plong each oupu node on he plane wh s wegh vecor as s coordnaes. Connecng neghborng oupu nodes by a lne oupu nodes: 1, 1 2, 1 1, 2 wegh vecors: 0.5, , , 0.9 C1, 2 C1, 1 C2, 1

28 Travelng Salesman Problem TSP by SOM Each cy s represened as a 2 dmensonal npu vecor s coordnaes, y, Oupu nodes C_ form a one dmensonal SOM, each node corresponds o a cy. Inally, C_1,..., C_n have random wegh vecors Durng learnng, a wnner C_ on an npu, y of cy I, no only moves s w_ oward, y, bu also ha of of s neghbors w_1, w_-1. As he resul, C_-1 and C_1 wll laer be more lkely o wn wh npu vecors smlar o, y,.e, hose ces closer o I A he end, f a node represens cy I, would end up o have s neghbors 1 or -1 o represen ces smlar o cy I,e., ces close o cy I. Ths can be vewed as a concurren greedy algorhm

29 Inal poson Two canddae soluons: ADFGHIJBC ADFGHIJCB

30 Addonal eamples

31

32 Couner propagaon nework CPN Basc dea of CPN Purpose: fas and coarse appromaon of vecor mappng y φ no o map any gven o s φ wh gven precson, npu vecors are dvded no clusers/classes. each cluser of has one oupu y, whch s hopefully he average of φ for all n ha class. Archecure: Smple case: FORWARD ONLY CPN, 1 z1 y1 v z w y k n z p y m from npu feaures o class from class o oupu feaures

33 Learnng n wo phases: ranng sample :y where y φ s he precse mappng Phase1: s raned by compeve learnng o become he represenave vecor of a cluser of npu vecors use sample only 1. For a chosen, feedforward o deermned he wnnng Reduce, hen repea seps 1 and 2 unl sop condon s me Phase 2: w s raned by dela rule o be an average oupu of φ where s an npu vecor ha causes z o wn use boh and y. 1. For a chosen, feedforward o deermned he wnnng 2. v new v old v old oponal 3. v w v new v old v old k new w old y w old k 4. Repea seps 1 3 unl sop condon s me k k z z

34 Noes A combnaon of boh unsupervsed learnng for n phase 1 and supervsed learnng for n phase 2. w Afer phase 1, clusers are formed among sample npu, each s a represenave of a cluser average. Afer phase 2, each cluser maps o an oupu vecor y, whch s he average of φ : cluser { } Vew phase 2 learnng as followng dela rule E w k w k yk w k where yk w k, because w k E 2 yk w k z 2 yk w k z when z wns w k w k I can be shown ha, when, v and w φ where s he mean of all ranng samples ha make wn v v

35 1 1 1 as rewreen be updaerule can ssmlar.wegh of proof Show only on v v w v k [ ] v v v v E E E E v E ] [1 1] [ ] [ 1] [ 1 hen If are drawn randomly from he ranng se,

36 Afer ranng, he nework works lke a look-up of mah able. For any npu, fnd a regon where falls represened by he wnng z node; use he regon as he nde o look-up he able for he funcon value. CPN works n mul-dmensonal npu space More cluser nodes z, more accurae mappng.

37 If boh y Full CPN φ and s nverse funcon φ we can esablsh b-dreconal appromaon Two pars of weghs marces: V o z and U z o y for appro. map o y φ W y o z and T z o for appro. map y o φ When :y s appled on X and y on Y, hey can only deermne he wnner J or separaely for z, pp for more deals 1 y es * J z Jy