Using Predictions in Online Optimization: Looking Forward with an Eye on the Past

Transcription

1 Usng Predctons n Onlne Optmzton: Lookng Forwrd wth n Eye on the Pst Nngjun Chen Jont work wth Joshu Comden, Zhenhu Lu, Anshul Gndh, nd Adm Wermn 1

2 Predctons re crucl for decson mkng 2

3 Predctons re crucl for decson mkng The humn brn, t s beng ncresngly rgued n the scentfc lterture, s best vewed s n dvnced predcton mchne. 3

4 We know how to mke predctons 4

5 We know how to mke predctons But not how to desgn lgorthms to use predcton How should n lgorthm use predctons f errors re ndependent vs correlted 5

6 Ths pper: Onlne lgorthm desgn wth predctons n mnd 6

7 c " c " c ) c ( c * Predcton error c " (x " ) F x " Cost = c " x " 7

8 c ) c ( c * c ( c ( (x ( ) Predcton error F x " x ( β x ( x " : swtchng cost Cost = c " x " + β x ( x " + c ( x ( 8

9 c ) c * c ) Predcton error c ) (x ) ) F x " x ) x ( β x ) x ( Cost = c " x " + β x ( x " + c ( x ( + β x ) x ( + c ) x ) 9

10 Onlne convex optmzton usng predctons x ", y ", x (, y (, x ), y ) mn c x 8, y 8 8 convex onlne + β x 8 x 8<" swtchng cost e.g. onlne trckng cost c x 8, y 8 Gol: mnmze compettve dfference cost(alg) cost(opt) Gven predcton of y 8 t tme τ, y 8 A Tme Informton Avlble Decson 1 y " > y ( > y ) > x " 2 y " y ( " y ) " x ( 3 y " y ( y ) ( x ) 4 y " y ( y ) x * 10

11 Predcton nose model [Gn et l 2013] [Chen et l 2014] [Chen et l 2015] y 8 = y 8 A RSAT" f t s e(s) Relzton tht lgorthm s tryng to trck predcton error Predcton for tme t gven to lgorthm t tme τ 11

12 Predcton nose model [Gn et l 2013] [Chen et l 2014] [Chen et l 2015] y 8 = y 8 A RSAT" Per- step nose f t s e(s) 9

13 Predcton nose model [Gn et l 2013] [Chen et l 2014] [Chen et l 2015] y 8 = y 8 A Weghtng fctor RSAT" f t s e(s) How mportnt s the nose t tme t s for the predcton of t? σ f 0 ( + + f s ( ) t = s 13

14 Predcton nose model [Gn et l 2013] [Chen et l 2014] [Chen et l 2015] y 8 = y 8 A RSAT" f t s e(s) predcton error Predctons re refned s tme moves forwrd Predctons re more nosy s you look further hed Predcton errors cn be correlted Form of errors mtches mny clsscl models Predcton of wde- sense sttonry process usng Wener flter Predcton of lner dynmcl system usng Klmn flter 14

15 Lots of pplctons Dynmc cpcty mngement n dt centers [Gndh et l. 2012][Ln et l 2013] Power system generton/lod schedulng[lu et l. 2013] Portfolo mngement [Cover 1991][Boyd et l. 2012] Vdeo stremng [Sen et l. 2000][Lu et l. 2008] Network routng [Bnsl et l. 2003][Kodlm et l. 2003] Geogrphcl lod blncng [Hndmn et l. 2011] [Ln et l. 2012] Vsul speech generton [Km et l. 2015] 15

16 Lots of pplctons Dynmc cpcty mngement n dt centers [Gndh et l. 2012][Ln et l 2013] Power system generton/lod schedulng[lu et l. 2013] Portfolo mngement [Cover 1991][Boyd et l. 2012] Vdeo stremng [Sen et l. 2000][Lu et l. 2008] Network routng [Bnsl et l. 2003][Kodlm et l. 2003] Geogrphcl lod blncng [Hndmn et l. 2011] [Ln et l. 2012] Vsul speech generton [Km et l. 2015] 16

17 Lots of pplctons Dynmc cpcty mngement n dt centers [Gndh et l. 2012][Ln et l 2013] Power system generton/lod schedulng[lu et l. 2013] Portfolo mngement [Cover 1991][Boyd et l. 2012] Vdeo stremng [Sen et l. 2000][Lu et l. 2008] Network routng [Bnsl et l. 2003][Kodlm et l. 2003] Geogrphcl lod blncng [Hndmn et l. 2011] [Ln et l. 2012] Vsul speech generton [Km et l. 2015] 17

18 Lots of pplctons Dynmc cpcty mngement n dt centers [Gndh et l. 2012][Ln et l 2013] Power system generton/lod schedulng[lu et l. 2013] Portfolo mngement [Cover 1991][Boyd et l. 2012] Vdeo stremng [Sen et l. 2000][Lu et l. 2008] Network routng [Bnsl et l. 2003][Kodlm et l. 2003] Geogrphcl lod blncng [Hndmn et l. 2011] [Ln et l. 2012] Vsul speech generton [Km et l. 2015] 18

19 Lots of pplctons lots of lgorthms Most populr choce by fr: Recedng Horzon Control (RHC) [Morret l 1989][Myne1990][Rwlnget l 2000][Cmcho 2013] y 8T" 8, y 8T( 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, 8TX x 8T", x 8T(,, x 8TX = rgmn 7 c(x 8, y 8 R ) + β x 8 x 8<" " RS8T" 19

20 Lots of pplctons lots of lgorthms Most populr choce by fr: Recedng Horzon Control (RHC) [Morret l 1989][Myne1990][Rwlnget l 2000][Cmcho 2013] y 8T" 8, y 8T( 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, y 8T( 8T", y 8T) 8T",, y 8TXT" 8T", y 8TXT( 8T", y 8TXT) 8T", x 8T(, x 8T), x 8TXT" 20

21 Lots of pplctons lots of lgorthms Most populr choce by fr: Recedng Horzon Control (RHC) [Morret l 1989][Myne1990][Rwlnget l 2000][Cmcho 2013] y 8T" 8, y 8T( 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, y 8T( 8T", y 8T) 8T",, y 8TXT" 8T", y 8TXT( 8T", y 8TXT) 8T", y 8T) 8T(, y 8T* 8T(,, y 8TXT( 8T(, y 8TXT) 8T(, y 8TXT* 8T(, x 8T), x 8T*, x 8TXT( 21

22 Lots of pplctons lots of lgorthms Most populr choce by fr: Recedng Horzon Control (RHC) [Morret l 1989][Myne1990][Rwlnget l 2000][Cmcho 2013] Recent suggeston: Avergng Fxed Horzon Control (AFHC) [Ln et l 2012] [Chen et l 2015] [Km et l 2015] 22

23 Avergng Fxed Horzon Control Fxed Horzon Control (FHC) y 8T" 8, y 8T( 8,, y 8TX 8, y 8TXT" 8TX, y 8TXT( 8TX, 8TX x 8T", x 8T(,, x 8TX = rgmn 7 c(x 8, y 8 R ) + β x 8 x 8<" " RS8T" 23

24 Avergng Fxed Horzon Control Fxed Horzon Control (FHC) y 8T" 8, y 8T( 8,, y 8TX 8, y 8TXT" 8TX, y 8TXT( 8TX, x 8T", x 8T(,, x 8TX x 8TXT", x 8TXT(,, x 8T(X 24

25 Avergng Fxed Horzon Control Averge choces of FHC lgorthms X x \6]^ = " x _ S" 6]^ w FHC lgorthms x " 8<(, x " 8<", x " " 8, x 8TX<( x ( 8<", x ( ( ( 8, x 8T*, x 8TX<" ",, x ) 8, x ) ) ) 8T*, x 8Tc, x 8TX x 8TX<" ( x 8TX,, x 8TXT" ),, x 8 X 25

26 Algorthms Usng Nosy Predcton Most populr choce by fr: Recedng Horzon Control (RHC) [Morret l 1989][Myne1990][Rwlnget l 2000][Cmcho 2013] Recent suggeston: Avergng Fxed Horzon Control (AFHC) [Ln et l 2012] [Chen et l 2015] [Km et l 2015] Whch lgorthm s better? Uncler 26

27 AFHC nd RHC hve vstly dfferent behvor Emprclly AFHC s better n worst cse under perfect predctons RHC s better n stochstc cse when predcton errors re correlted 27

28 Ths pper: Onlne lgorthm desgn wth predctons n mnd How to desgn lgorthm optml for predcton nose? 28

29 Three key desgn choces 1. How fr to look- hed n mkng decsons? y 8T" 8, y 8T( 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, Lookhed w steps 29

30 Three key desgn choces 1. How fr to look- hed n mkng decsons? 2. How mny ctons to commt? y 8T" 8, y 8T( 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, 8TX x 8T", x 8T(,, x 8TX = rgmn 7 c(x 8, y 8 R ) + β x 8 x 8<" " RS8T" 30

31 Three key desgn choces 1. How fr to look- hed n mkng decsons? 2. How mny ctons to commt? y 8T" 8, y 8T( 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, 8TX x 8T", x 8T(,, x 8TX = rgmn 7 c(x 8, y 8 R ) + β x 8 x 8<" " commts v steps RS8T" 31

32 Three key desgn choces 1. How fr to look- hed n mkng decsons? 2. How mny ctons to commt? 3. How to ggregte cton plns? x " 8<(, x " 8<", x " " 8, x 8TX<( x ( 8<", x ( ( ( 8, x 8T*, x 8TX<" x ) 8, x ) ) ) 8T*, x 8Tc, x 8TX x 8 = g(x 8 ", x 8 (, x 8 ) ) Key: commtment blnces swtchng cost nd predcton errors Our focus: wht s the optml commtment level gven the structure of predcton nose? 32

33 Commtted Horzon Control FHC wth lmted commtment v, for t k mod w y 8T" 8, y 8T( 8, y 8T 8, y 8TT" 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, x 8T", x 8T(,, x 8T, x 8TT",, x 8TX = rgmn 7 c(x 8, y 8 R ) + β x 8 x 8<" " 8TX RS8T" x () = (, x 8T", x 8T(,, x 8T, ) 33

34 Commtted Horzon Control FHC wth lmted commtment v, for t k mod w y 8T" 8, y 8T( 8, y 8T 8, y 8TT" 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, y 8TT" 8T, y 8T( 8T, y 8T(T" 8T, y 8TTX 8T, y 8TTXT" 8T, x 8TT", x 8TT(,, x 8T(, x 8T(T",, x 8TXT x () = (, x 8T", x 8T(,, x 8T, x 8TT", x 8TT(,, x 8T( ) 34

35 Commtted Horzon Control FHC wth lmted commtment v, for t k mod v y 8T" 8, y 8T( 8, y 8T 8, y 8TT" 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, y 8TT" 8T, y 8T( 8T, y 8T(T" 8T, y 8TTX 8T, y 8TTXT" 8T, y 8T(T" 8T(, y 8T) 8T(,, y 8T(TX 8T, y 8T(TXT" 8T(, x 8T(T", x 8T(T(,, x 8T), x 8T)T",, x 8TXT( x () = (, x 8T", x 8T(,, x 8T, x 8TT", x 8TT(,, x 8T(, x 8T(T", x 8T(T(,, x 8T) ) 35

36 Commtted Horzon Control FHC wth lmted commtment v, for t k mod v y 8T" 8, y 8T( 8, y 8T 8, y 8TT" 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, y 8TT" 8T, y 8T( 8T, y 8T(T" 8T, y 8TTX 8T, y 8TTXT" 8T, y 8T(T" 8T(, y 8T) 8T(,, y 8T(TX 8T, y 8T(TXT" 8T(, x 8T(T", x 8T(T(,, x 8T), x 8T)T",, x 8TXT( x () = (, x 8T", x 8T(,, x 8T, x 8TT", x 8TT(,, x 8T(, x 8T(T", x 8T(T(,, x 8T), ) 36

37 Commtted Horzon Control FHC wth lmted commtment v, for t k mod v y 8T" 8, y 8T( 8, y 8T 8, y 8TT" 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, y 8TT" 8T, y 8T( 8T, y 8T(T" 8T, y 8TTX 8T, y 8TTXT" 8T, y 8T(T" 8T(, y 8T) 8T(,, y 8T(TX 8T, y 8T(TXT" 8T(, x 8T(T", x 8T(T(,, x 8T), x 8T)T",, x 8TXT( x (") " " " " " " " " " = (, x 8T", x 8T(,, x 8T, x 8TT", x 8TT(,, x 8T(, x 8T(T", x 8T(T(,, x 8T), ) x () = (, x 8T", x 8T( x () = (, x 8T", x 8T(,, x 8T,, x 8T, x 8TT", x 8TT", x 8TT(, x 8TT(,, x 8T(,, x 8T(, x 8T(T", x 8T(T", x 8T(T(,, x 8T), ), x 8T(T(,, x 8T), ) v FHC(v) lgorthms 37

38 Commtted Horzon Control FHC wth lmted commtment v, for t k mod v y 8T" 8, y 8T( 8, y 8T 8, y 8TT" 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, y 8TT" 8T, y 8T( 8T, y 8T(T" 8T, y 8TTX 8T, y 8TTXT" 8T, y 8T(T" 8T(, y 8T) 8T(,, y 8T(TX 8T, y 8T(TXT" 8T(, x 8T(T", x 8T(T(,, x 8T), x 8T)T",, x 8TXT( x (") " " " " " " " " " = (, x 8T", x 8T(,, x 8T, x 8TT", x 8TT(,, x 8T(, x 8T(T", x 8T(T(,, x 8T), ) x () = (, x 8T", x 8T( x () = (, x 8T", x 8T(,, x 8T,, x 8T, x 8TT" x^]^ t = 1 v 7 x 8 S", x 8TT", x 8TT(, x 8TT(,, x 8T(,, x 8T(, x 8T(T", x 8T(T", x 8T(T(,, x 8T), ), x 8T(T(,, x 8T), ) v FHC(v) lgorthms 38

39 Commtted Horzon Control FHC wth lmted commtment v, for t k mod v y 8T" 8, y 8T( 8, y 8T 8, y 8TT" 8,, y 8TX 8, y 8TXT" 8, y 8TXT( 8, y 8TT" 8T, y 8T( 8T, y 8T(T" 8T, y 8TTX 8T, y 8TTXT" 8T, y 8T(T" 8T(, y 8T) 8T(,, y 8T(TX 8T, y 8T(TXT" 8T(, x 8T(T", x 8T(T(,, x 8T), x 8T)T",, x 8TXT( x (") " " " " " " " " " = (, x 8T", x 8T(,, x 8T, x 8TT", x 8TT(,, x 8T(, x 8T(T", x 8T(T(,, x 8T), ) x () = (, x 8T", x 8T( x () = (, x 8T", x 8T(,, x 8T,, x 8T, x 8TT", x 8TT" x^]^ t = 1 v 7 x 8 S", x 8TT(, x 8TT( v = 1 v = w,, x 8T(,, x 8T( RHC, AFHC, x 8T(T", x 8T(T", x 8T(T(,, x 8T), ), x 8T(T(,, x 8T), ) v FHC(v) lgorthms 39

40 Mn Result Theorem For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s S" 40

41 Mn Result Theorem For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s Compettve dfference S" 41

42 Mn Result Theorem For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s S" Commtment level v? 42

43 Mn Result Theorem For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s Due to swtchng cost S" 43

44 Mn Result Theorem x " x ( D, x ", x ( F For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s Due to swtchng cost S" 44

45 Mn Result Theorem For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s Due to swtchng cost S" Due to predcton error 45

46 Mn Result Theorem c x, y " c x, y ( G y " y ( s, x, y", y ( For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s Due to swtchng cost S" Due to predcton error 46

47 Mn Result Theorem f ( E y8t y 8T 8 ( = σ ( 7 f s ( For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s Due to swtchng cost RS" Predcton error k- steps wy S" Due to predcton error 47

48 Mn Result k Theorem For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s Due to swtchng cost S" Due to predcton error 48

49 Mn Result Theorem For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s S" Due to Due to swtchng cost predcton error Commtment level v? 49

50 Mn Result Theorem For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s S" Due to Due to swtchng cost predcton error Commtment level v? Key: choose commtment level v to blnce these two terms 50

51 Mn Result Theorem For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s 1, s = 0 e.g...d. nose f s = v 0, s > 0 = 2TβD v S" + 2GTσ s Decresng functon of v AFHC s best when nose s..d 51

52 ..d. predcton nose 1, s = 0 f s = x 0, s > 0 52

53 ..d. predcton nose 1, s = 0 f s = x 0, s > 0 Long rnge correlted 1, s L f s = x, L > w 0, s > L 53

54 ..d. predcton nose 1, s = 0 f s = x 0, s > 0 Long rnge correlted 1, s L f s = x, L > w 0, s > L Short rnge correlted 1, s L f s = x, L w 0, s > L 54

55 ..d. predcton nose 1, s = 0 f s = x 0, s > 0 Long rnge correlted 1, s L f s = x, L > w 0, s > L Short rnge correlted 1, s L f s = x, L w 0, s > L Exponentlly decyng f s = R, < 1 55

56 Optml commtment level depends on predcton nose structure..d. predcton nose 1, s = 0 f s = x 0, s > 0 Long rnge correlted 1, s L f s = x, L > w 0, s > L Short rnge correlted 1, s L f s = x, L w 0, s > L Exponentlly decyng f s = R, < 1 56

57 More detl: long- rnge correlted nose Theorem If predcton nose s long- rnge 1, s L correlted, f s = x, L > w 0, s > L AFHC s optml f { > α 2w "T }~ RHC s optml f { < ( }~ st( Reltve mportnce of predcton error nd swtchng cost CHC s optml wth v (1, w) o/w Intermedte v 2TβD α + 2 4GTσs = rgmn RHC v α + 2 Predcton error domnnt Swtchng cost domnnt s + 2)T (GTσ s v s ( α + 2 AFHC 57

58 More detl: short- rnge correlted nose Theorem If predcton nose s short- rnge 1, s L correlted, f s = x, L w 0, s > L 1 AFHC s optml f { }~ > H(L) RHC s optml f { < ( }~ st( CHC s optml wth v (1, w) o/w Intermedte α + 2 L + 1 s ( αl v = rgmn 2TβD v RHC Predcton error domnnt Swtchng cost domnnt + 2GTσ s L + 1 s/( 2GTσs v AFHC H(L) 58

59 More detl: exponentlly decyng nose Theorem If predcton nose s exponentlly decyng, f s = R, 0 < < 1 AFHC s optml f { }~ > (("< ) RHC s optml f { }~ < (("T ) CHC s optml wth v (1, w) o/w Intermedte Predcton error domnnt Swtchng cost domnnt v = rgmn 2TβD RHC + 2GTσ v 1 ( ( 1 ( GTσ v 1 ( ( AFHC 59

60 Mn Result Theorem For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s We cn use predcton error structure to gude desgn of onlne lgorthm S" 60

61 Mn Result Theorem For c tht s α- Hölder contnuous n the second rgument nd fesble set F s bounded, Ecost CHC Ecost OPT 2TβD + 2GT v v 7 f s =: V Compettve dfference holds wth hgh probblty P(cost CHC cost OPT > V + u) > exp u( F(v) S" 61

62 Concluson Desgn of optml lgorthm depends on structure of predcton error Ths tlk: OCO wth predcton Commtment should be optml to predcton nose Future: cn we extend ths frmework to other onlne problems? 62