Machine Learning. Hopfield networks. Prof. Dr. Volker Sperschneider

Transcription

1 Mache Learg Hopfed eor Prof. Dr. Voer Spercheder AG Machee Lere ud Naürchprachche Seme Iu für Iforma Techche Fauä Aber-Ludg-Uverä Freburg

2 Hopfed eor I. Movao II. Bac defo III. Free eerg IV. Hebb earg V. Correao earg VI. Sovg combaora opmao probem b free eerg mmao

3 Hopfed eor I. Movao

4 Movao MLP ove a a mag a deco corog a devce recogg he face of gradmoher Compe formao provded o he e (vecor of feaure vaue, vecor of meaureme, o of pe) ad o a ma amou of formao devered b he e (e/o, coro vaue, recoged/o recoged)

5 Movao Wha abou he a of memorg he face of gradmoher - ad o of furher mage? o mp a a daa-bae h equea acce bu h pob of recoverage from a hgh dored mage or a para mage or from voce of gradmoher peae cogve paube maer Hebb earg rue appcao oud be ce

6 Movao Such or of memor caed aocave memor. Memor he core of hgher cogve fuco: cocoue, ef-cocoue, aguage, oca behavour, I bra aumed o be ocaed hgher corca rego of mave ercoeced euro, bu o aered rego Here a appcao eampe:

7 Ma-Pac Magdeburg

8 Eampe of a Hopfed eor (hrehod dcaed de he euro)

9 Reaed o Igmode of phc magec dpoe each dpoe eperece magec force from he oher

10 Hopfed eor II. Bac defo

11 bar euro compee ercoeced mmerc egh ero ef-acvao Threhod bar ae epu of euro ae a uua

12 mmerc egh ero ef acvao 0 hrehod θ ( ) { } ae epu 0, e ( ) θ

13 Sae rao Leg euro fre ae mea o repace b vaue. ( ) Leg euro ge mue ae mea o repace b vaue 0. ( 0)

14 Acve ad acve euro Neuro acve ae f ( e ( ) 0 0) ( e ( ) < 0 > )

15 Acve ad acve euro Neuro acve ae f ( e ( ) 0 0) ( e ( ) < 0 > ) Neuro acve ae f ( e ( ) < 0 0) e ( ) 0 ( e ( ) > 0 )

16 Acve ad acve euro Neuro acve ae f ( e ( ) 0 0) ( e ( ) < 0 > ) Neuro acve ae f ( e ( ) < 0 0) e ( ) 0 ( e ( ) > 0 ) Sae abe f ever euro acve

17 Damc of a Hopfed eor

18 Damc of a Hopfed eor Achroou damc: WHILE acua ae o abe DO eec a acve euro ad e fre/go mue END

19 Damc of a Hopfed eor Schroou damc: WHILE acua ae o abe DO e a acve euro fre/go mue parae END

20 Schroou damc more reguar ha achroou oe, bu ee ha ma happe: - ½ - ½ - euro havg hrehod -½, egh - ae (0,0) che parae o ae (,) ae (,) che parae o ae (0,0) Iead of covergg o a abe ae he e che forever beee o ae: -cce

21 Schroou damc o cogve adequae: o bra chroaor o Schroou damc eher coverge o a abe ae or ru o a -cce: proof o foo Achroou damc aa coverge o a abe ae: proof o foo, ue cocep of free eerg

22 Sabe ae or -cce chroou damc Ne h chroou damc muaed b a e ogeher h a cop of he e ad egh a ho he eampe:

23 Le ae rao he chroou e be () () (3) The fr chroou ep muaed b 3 ( he eampe) achroou ep o he cop e: forh The ecod chroou ep muaed b 3 ( he eampe) achroou ep o he orga e: bac ad o o

24 The he achroou e ae rao ca be geeraed a foo: ()() bac (3)() forh (3)(4)... Sabe ae be arrved eher oe of he foog a: Thu () (). ()() ()() ()() ()()

25 If furhermore () () () e oba a abe ae. Ohere e oba a -cce

26 Hopfed eor III. Free eerg

27 Free eerg of a ae ae ( ) E( ) θ vecor oao : E( ) T T θ

28 Ver ma eampe (a hrehod 0)

29 Sae, free eerge, e pu, acve euro eo

30 ae rao

31 eerg dece - euro fre ae acve euro 0 e () > 0 che o ae for a

32 E E ) ( ) ( θ θ

33 E E ) ( ) ( θ θ θ θ

34 E E ) ( ) ( θ θ θ θ θ

35 E E ) ( ) ( θ θ θ θ θ θ

36 ) ( ) ( ) ( ) ( > e e E E θ θ θ θ θ θ θ

37 eerg dece - euro ge mue ae acve euro e () < 0 che o ae 0 for a Eerce: Sho ha E() - E() - e () >

38 Theorem: Uder achroou damc a Hopfed e aa ru o abe ae. Proof: I ever ep eerg of he acua ae rc decreae. A eerg fuco adop o fe ma vaue (gve a emao of ho ma) eerg dece mu fa come o a ed

39 Eerce: Adap a oo for bar Hopfed e o he cae of bpoar euro. Aume ae raformed o ae b he aco of euro. Sho ha E( ) E( ) e ( ) e ( ) f euro fre, ad f euro ge mue. E( ) E( ) e ( ) e ( )

40 Coruc a eampe of a bpoar Hopfed eor hch ru o a -cce uder chroou damc

41 Hopfed eor IV. Hebb earg

42 Hebb earg We ue a Hopfed eor h euro. Neuro are bpoar (, -) euro. Threhod are e o ero. We coder a rag e of mage. To mage (ha, bpoar ae) ad are caed orhogoa f her er produc ero: T

43 Cae pe 00 radom e pe 00 radom e pe

44 Hebb earg Trag e {,, } {, } euro,,, Threhod 0 Hebb egh

45 Sab of orhogoa vecor Aume orhogoa of rag vecor: ( p ) T q 0 for p q The for a,, ad,, : e ( ) Proof:

46 e ) (

47 e ) ( ) (

48 e ) ( ) (

49 e ) ( ) (

50 T e ) ) (( ) ( ) (

51 T e ) (0 ) ) (( ) ( ) (

52 T e ) (0 ) ) (( ) ( ) (

53 Sorage capac uder Hebb earg Hopfed e h euro ca ore ad afe reca a mo 3,8% ae uder Hebb rue. Thu he cae e houd be abe o afe ore o more ha 80 mage. Ug a dffere earg rue (correao earg) he auhor a abe o ore more ha 000 mage afe eve h 50% oe

54 Coder euro huma core. Though hee euro are b o mea compee coeced e u u for fu emae ho og oe coud ore mage per ecod, 0 hour a da, aumg he 0,38 emao: , ear Loer coecv (0.000 per euro) mgh be compeaed b more cever earg rue

55 Hebb earg dffer omeho for bar euro ad bpoar euro: euro euro egh updae O muaou acve euro ead o roger egh: orga Hebb rue

56 Hopfed eor V. Correao earg

57 Correao-baed earg Correao.r.. a ae rag e Meaure of ho e ha amog a ae rag e euro have he ame acvao κ

58 Correao earg Correao.r.. o a abe ae of acua e h egh vecor Meaure of ho e ha amog he abe ae euro have he ame acvao: κ ( ) abe. r.. S umber of a abe ae S

59 κ epree ho rog euro acvao are correaed h rag ae: ac, dered correao κ - epree ho rog euro acvao are correaed h abe ae of he acua e: damc, de faco correao

60 If for euro ad de faco correao maer (greaer) ha dered correao h ca be correced b creag (decreag) he egh beee ad. I a mpemeao egave correao are emaed b ampg a uffce umber of abe ae (ead of compug a)

61 Correao earg rue η( )( κ κ ( )) Learg rae η() decreae (a uua) h me. Paub of he earg rue be eabhed he e chaper o Boma mache

62 Hopfed eor VI. Sovg combaora opmao probem b free eerg mmao

63 Ug Hopfed eor for he ouo of combaora opmao probem A Hopfed e mme free eerg. Sabe ae are oca mma of free eerg fuco uder Hammg dace of ae (Eerce). Free eerg a quadrac fuco. Quadrac fuco are que epreve

64 Traveg aema probem (TSP) Le ode be gve:,,., Le a dace mar h dace d beee a o ode ad be gve. Aume d d > 0 d 0 Fd a our hrough a ode, ha a permu- ao π of a ode, h mma our co d π ( ) π ( ) dπ ( ) π ()

65 TSP repreeed b a Hopfed e Ue a Hopfed e h ma bar euro ha are arraged a quare mar. Neuro ro ad coum refered b N. Tour π repreeed b ae a foo: π ( )

66 A eampe h 6 ode ma hep carfg ho our π are repreeed b ae: π( ) I he repreeg ae euro h vaue are ho bac

67 Eracg dace co from ae Le our π be repreeed b ae. The for < d a b d π ( ) π ( ) ab ab a b a, b Wh? Sce o a π() gve a corbuo o he ouer um, ad o b π() gve a corbuo o he er um. d ab

68 The ame hod for he fa ep bac o he ar ode: d a b d π ( ) π () ab ab a b a, b Thu TSP am o mme ab d ab ab a, b a, b over a ae ha repree our. d ab d ab

69 No ever ae repree a our O ae ha coa eac oe vaue ever ro ad eac oe vaue ever coum repree our. Th ca be repreeed b a e of cora: a b b a ab ab

70 Th ca be epreed b he mmao of he foog quadrac fuco: ( a b ( a b (( ab ab ) ) a b ab ) ( a b b b a ( ( ba ab ba ) ) ) )

71 Boh quadrac fuco (our co fuco ad our admb cora fuco) are combed a commo fuco: a, b C a b d (( ab a, b b ab ) ( a b b a d ab ba ) ) Coa C > 0 coro heher mma of our co or admb of our cou more

72 Noe ha a oca mmum of h commo quadrac fuco he acheved ae mu o ecear be a ae repreeg a our. Eve f ae repreeed a our mu o be a oca or eve goba mmum

73 Commo quadrac fuco erpreed a free eerg of a Hopfed e A euro of he ued Hopfed e are ed b a doube de N ab e have o dea h egh havg 4 dce ad hrehod havg dce. ( ab)( cd ) ( ab) Thu free eerg formua read a foo: E( ) θ ad ab ( ab)( cd ) cd a, b, c, d a, b θ ab ( ab)

74 We raform commo quadrac fuco o free eerg form: a, b C a, b C a b (( b a, b ab ) ( a b b a d d ab ab a, b a a b b d d ) ( abac ab baca a b, c b b, c b ab ab ba ) ba )

75 a, b a b d ab a, b b C aac Cab 4 a,, c, a, b a, b a d ab C ab C For mmao coa erm C doe o pa a roe, hu e deee. a, b a b d ab a, b b C aac Cab 4 a,, c, a, b a, b a d ab C ab

76 a, b ab a b C aac Cab 4 a,, c, a, b a, b a b d d ab ab a, b ab a a b b d d ab ab C C ab a a,, a C a C aac Cab 4 a,, c, a, b a a, C a

77 No mach obaed quadrac form o free eerg: ab a b d ab ab a b d ab C a a,, a C a C aac Cab 4 a,, c, a, b a a, C a E( ) θ ab ( ab)( cd ) cd a, b, c, d a, b ab ( ab)

78 ( a)( b ) ( b )( a) d ab < a b a ( a)( b) ( b)( a) d ab a b a ( a)( ac) ( ac)( a) C a c ( a)( b) ( b)( a) C a b a A oher egh are e o ero. θ ( ab) 6C a b