Xác định gradient của một hàm bằng phương pháp Monte-Carlo. pot

VJiduu:'c xac' djnh b5.ngmot mo hinh ng&unhien.. Mo hinh phfii hop giira phuong ph ap do tirn ngiu nhien vo i pluro'ng phap bien phan dia phtro'ng la mot hurmg dan diro'c nghien ciru tro

Tii-p chi Tin h9C va Di'eu khdn hoc, T 17, S.2 (2001), 4 -5

TRAN CANH

Abstract. In the work the gradient: grad f(x) = (iJ£t1, , iJ£tl) of a differentiable function f(x)

isdetermined b random model The construction of an unbiassed estimator d x) = ( ~ l(x) , , ~n(x) of grad f (x) is established successfully

T'om tat> . Trong cong t" rm'h' nay gradilent: grad f() - iJ/(x x 1 - iJx,' , D i /( J x 1)., cua mo"th' am.khaa VJiduu:'c xac' djnh b5.ngmot mo hinh ng&unhien Vi~c thiet l~pmot u'o'c hro'n kho g chech dx) = (~ x), '~n(x))

du'o'cxac l~p than h congo

1 M( ) " D AU

Liroc do do tlm ngiu nhien da d uo'c s11:dung mi?t each hiru hieu doi vo i mot loai bai to an di'eu khign co' Ian d~ cho lo'i giai toi uu toan cue (xem [1] 0 'day su' hi?i tu cua lo'i giai gan d ung ve IO'i giai dung (theo quan digm xac suilt) va vi~c danh gia "sai so" thee so phep l~p N o ciing dtro'c chi ra

Tuy nhie ,nhieu bai toan di"eu khi~n loai nay, nhat la cac bai toan C1rC tr] toan cue (xem [ 3] ) d oi hoi mot di? chin h xac cao hon, bui?c chung ta phai cai tien mo hlnh da neu de' lam tang toc di? hi?i

tu Mo hinh phfii hop giira phuong ph ap do tirn ngiu nhien vo i pluro'ng phap bien phan dia phtro'ng

la mot hurmg dan diro'c nghien ciru trong vi~c cai tien mo hirih Cling vo ihu'cng nay cluing toi se

de ng hi mfit huong di tien khac do la mo hlnh phdi ho-p giira phirong p ap do tlm ngiu nhien voi phiro'ng phap gradient ng5:u nhien

Nham muc dich k~ tren, trong bai nay mot loai iro'c hrong k ong chech cua vec to' gradient

d uo'c thiet l~p tren CO' so' cac ket qua cua mo hlnh ng5:u n hien tinh t&ng cil a chu6i va gio'i h an cua day so

2 M O H INH N G AU NHIEN TiNH T O N G CD - A C HU G I v A G I O " I H4 - N

, - "

2.1. Xet mdt chu6i so hi?i tu co to'ng la s:

00

L Si = S.

i=O

( 1) C·ia, su, to tai.d-ay so"{} qi h

i>O , sac c 0 :

00

Lqi = 1, qi >0 (Vi ~ 0),

i=O

(2 ) ( 3 )

VO'i nh iing dieu kieri nay ro rang chuoi (1) la hi?i tu tuy~t doi

• Cor i g t rlnh d tr oc su h t r o c ua a U ,i KT 0 4- 1 15 thu o c c hu a ng trlnh N g ie n c i ru Co' ba n Nh a r n ro'c

G9i v E {O,1,2, }la d ai hro'n ngh nhien ro'i rac vo'i phan b5 xac suat:

G9i ~E [0,2ela d ai hro.n ngiu nhien phan b5 deu voi m~t de?xa suat

1

trang do X10,2cl ( x ) la ham d~c trung (chi thi] cu a t~p [0;2el

{ e khi ~< ' ! J +e

-e khi C '! JL +e

( 6 )

B 5 d e 1 VO ' i g i r l thi et ( 2 ) , (3) d i lu oviq ngdu nhi e n 17 c o kif v qng va ph ua iiq s ai hii : u han :

00

C hU' n g mi nh Tu: (4), (6) v a corig th irc tinh ky v9ng co di'eu ki~nta co:

00

E{17} = E{E(rJ / v)} = LP{v = i}E{17 / V = i}

i= O

00

= Lq d e P{ ~ < S i + e } - e P{ ~ ;::: Si + e} ]

S·

Tu: ( 3 ) ta suy ra 0: : ::: - 2 +e :::::2e, do do dua vao tinh ph an b5 deu ciia ~ta thu dtro'c:

S· j 2 C 1 1 ( s , )

p{ ~; :: : 2 + e}= , - dx= - c- - 2

'

(10)

(11)

Thay (10) , (11) vao (9) ta thu d u'o'c(7):

E {17} =f qi[e21e C ' +c) - e21e(e - .S , ) ]=f S i =:S<00

D~ cmrng minh (E8) ta tinh:

00

E{172} = E{E(172/V)} = L qiE{rJ2 /v = i}

i=o

= ~ qi e P { ~ < q i +e} +e P { ~ ; ::: q i +c}

= f e2 q i [p{ ~< s, + e } + P{ ~ ; ::: s, + e }]

0

2 '" 2

X.ACD~NH GRADIENT CUA MQT HAM BANG PHU'O"NG PH.AP MONTE-CARLO 47

D{1]} = E {1] 2 } - ( E {1]}) 2= c 2 - s2

Nghiern lai t5ng cda chu6i sau:

S = "(_1)"_ = e-1/ 2 :::: : °606

n O

o

Ta c oh n v 1a,o3ai·1rrang nga"u nh·" ien ro.irac co'h'p a b" P 0 Olsson:q. , = e"AAn- va c o, h n c= eA> -e-A

vo'i A :::: :0,5 Bay gio·ta ph ai so sanh ~E [0, 2eAIvo i d ai hrong Sv +eA Sau khi rut gon bie'u thirc

ta c 1p at so san <" 1 V 0 c i rrq g + ( 2A ) V' tro g uO ci p an 0 ueu tren 0, 1 et qua tfnh

tren may v6i A = 0,8 (Xem trong b n 1, C9t"t5ng cu a chu6i")

Bdng 1 Ket qui tfnh tren may

So ran l~p T5 g ciia chu6i Gi6i h n c aday T5ng chu6i Fourier

Vidu 2 Tfnh t5ng cu a chu6i

S - "( - I) n 2 (0 <x< 2)

Chu6i nay chinh la khai trie'n Fourier ciiaham so:

{X khi 0< x < 1

-2- x khi 1< x ::;2

qua t n tren mriy irng v i x = 1,1 (Xem trong ban 1,C9t "t5ng cu a chu6i Fourier")

2.2 Xet mdt day so h i tu {fn}n ~ O

lim I n = f

f I, "OQ

Gii\.thit1t di.ng to tai m9t h g so c>0 va mot day so {q i } i>O, sao cho:

0

i O

(14)

Giin v i cac dai hro g ngh nhie ~, l eLin u, ta Hip d ai lu'o g ngh nhien:

khi ~ < Jv - J -1 +C

' I v

{ c

c;=

~ - ( } v

(1 )

B5 e 2. Gid s d ' cac g id thiet (3) , (14) d ' c(c th o a man Khi do gio ' h.an (12) t o ic: h 1i: u h.at i v a dq.i

I U ' q' ng n ga u nhi e n c; co ky uotiq va ph u oiu; s ai huu luui :

E { ; } = lim In = I ,

n ~o o D{ c ; } =c 2 - 1 2.

(1 6) (17)

C hu'ng min h x a chu6i 2: ;:" = 0 S n, trong do:

Sn := In - I n- 1 (n ;:: 1); So:= 10. (18)

T'ir cac gii thiet (1 3 ) , (14) ta suy ra cac di'eu kien dang (2), (3) doi v 'i chu6i 2: ; " =0 S n diro'c thoa

man, do do chuoi nay h9i tu (tuy~t doi) Dong thai 'tir (18) ta c6:

00

L S = lim ( s o + + s n) = lim In f

(19)

M~t khac, dua vao (18), (15) ta co:

c; = { C

-c

khi ~ <! 'J +c

< I v

~ - ( I v

n hia Ia d ai hro'n n 5:u nhien c ; co d ang 17 trong (6) con cac dieu kien (13), (14) c6 dang cua

dieu kien (3), (2) to g B5 de L1 S11· dung b de nay doi vo i d ai lu'o'ng ngh nhien c;va (19)

Vi du 3 N gh iern lai gi&i h an cua day

1+22+32+"'+n2 1

(n 00) •

Ta d~t 1 - 1 = 0, 10 = a, a la rndt so tiiy y cho tru'o'c, thl gioi han tre chuye'n thanh t&ng cua chu6i

XAC DINH GRADIENT CUA MOT HAM BANG PHU'UNG PHAP MONTE-CARLO 4

Ta ch n v Ii d ai hro'n nga:u nhien roi r~c co phan bo: qn = ( n + l ) l(n +2)' So c can tlm Ii

max ciia cac so trong t~p ho sau :

{l:cl = 2 1al; l:: U ~ 6 1 - al; 1 - 3 2 + n + 11 ( n + l )( n + 2), (n 2' : 2)}

Ket qua tinh tren may irng vo'i a = 1, c= 4,5, (Xem trong bang 1, C9t "gi6'i h an c a day"),

Xet ham f : G ( x)b > RI, G (x) c Rm I Ian c~n Ioi vi mo' cua di~m x Gilt 513: tr en G( x )

ham f kha vi len tuc theo Lipschitz cap a ( x )

l a f (xl ) _ af ( x2 ) I ::: c ( x ) ll xl - x2 11 "( x ) (20)

Chon hai day so d on di~u giarn {qn}n>O,- { 8 n}n >- thoa man cac dieu kien:

00

O«s < 1 a/x ) (Vn > 0)',

i = O

tru'ong hop con lai:

f}n) ( x ) = ; [f(x + 6 ne ; ) f(x )] (Vn 2 ': 1),

n

(2 3 )

Cr day ei I v cto: chi p urmg thu i trong Rm, L U ll y rhg do tinh mo cua G( x) nen ta co th~ chon

80 du be sao cho:

(i = 1 ;-.m) cu a vec to' nga:u nhien dx ) = ( ~ I ( x ) , ~ m ( x ) theo cong th trc sau:

tong do v, ~Ii h ai d ai hro ng ngh nhien d9C lap vo'i ph an bo xac sat nhu da noi C:)' (4) vi (5),

Dirh l~ 1 Ham f(x) vO'i gid thiet ( 20) cung vO ' i cdc thiet ke (21) , (22) , (23) , (24) va (25) ta co :

E{ ~; ( x )} = a~~~ ) , D{ ~ ; x ) } = c2(x) - (a~~~)r ,

( 26 ) (27)

C h u'ng minh , Ap dung cong thu'c so gia gio'i noi VaG(23 ) ta co:

f} n) (x) = _ [ f (x + ' 5 n i) _ f ( x) ]= a f (x + eJ" ) (x) 6 n i) ,

trong do: 0 < ~ n ) (x) < 1 Tu' tinh khOng tang cu a day { on } n2:0 v a tinh lOi cu a G(x) ta co th~ dira

VaG (24) suy ra:

TRAN C ANH

x + 5 ei E [ X - 5 0e i, X + 5 0ei] c G( x ).

Do tinh lOi cu a G ( x ) ta con c6: X + B~n ) 5 ei EG( x ).

Tr e n w so nay, tir ( 2 ) , (2 0 ) ta suy ra:

I f ( n ) (x ) - f - 1 ) x ) l::; e ( x ) II B~ n ) x ) 5 nei - B ~n-1) ( x ) 5 _ ei ll a(x)

I ( n ) (n - 1 ) I '(X)

=e ( x ) B i ( x ) 5n + B i ( x ) 5 - 1 ( ( n ) ( n - 1 ) ) ' (X )

Khi d6 ti ( 2 1) ta suy ra:

I f } n ) x ) - f i(n - 1) ( x ) l : e ( x )( 5 + 5 _) " ' (x) < e ( x )(2 5 _ o( x ) ::;e ( x ) qn

Khi ket hop dieu kien nay vo i(22) va (2) ta nhan thfiy gi<l.thiet cu a B5 de 2 d iro'c tho a man d5i

v6i bai toan gi&ihan cua day so: {f i(n) (x)} n ;::o ( i =1-7 m).

Ap dun B5 de 2 ta thu dtro:c:

E {\; ( x )} = lim f} n ) ( x ) ,

n - + oo

D{ \ ; (x)} = e 2 - ( lim f i(n) ( x )) 2

n - + oo

( 29 ) (30) M~t kh ac tir (21) d~ dang nhan thay rhg

n - C X) 2 n- X)

nghia la:

lim 5n = lim 5n = o

n - oo n - + -C X)

(31)

Do su' ton tai cua cac d ao ham rien a~!~) ( \I x E G(x) , 1: :; i ::;m) nen tir ( 31 ) , ( 23) ta suy ra

hm f (x) = hm =- f x + 5 nei - f ( x ) =

T'ir (2 9 ) va ( 3 2) ta thu d u'o'c (26) con (27) cling thu duoc ti ( 3 0) va (32 ).

(32)

o

[ Tran Canh, Pluro'ng phap do tlm ngiu nhien giai mdt IO,!-ibai toan di'eu khie'n, Tuy f n t4p

Ccc co ng t r i nli khoa ho c [nganh To an], HNKH Trtro ng DH Khoa hoc tu' nhien, Ha n9i, 1998,

tr 2 - 40.

[ 2 ] Sob l 1.M Cdc phu·o · ng pluip iinh to an Monte - Car l o, FML Moskva, 1 973 (tieng Nga)

[3] Zielinski R., Neumann P., Stoeha s ty s czne Metody Po s zukiwania Minimum Funkeij, WNT

Warsza-w a, 1 8

N l uin bdi ngay 10 th ri ng 4 ndm 2 000 Nh4n bdi s au khi sd:a ngay 21 iluinq {] niim 2 001

Tru o ru; Dei ho c Xay d1 [ng Hd Nqi