MMO / mmouchpos

50

gh\h]h gZijyfdm ihrmdm ih}^gm} gh\bc gZcdjml•rbc gZijyfhd kimkdm a ihi_j_^g•f gZijyfdhf ihrmdm pk+1 = -gk + βkpk-1.

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^_ \gmlj•rg•c ^h[mlhd ihi_j_^gvh€ af•gb \ ]jZ^•}gl• • ihlhqgh]h ]jZ^•}glZ jha^•e_gbc gZ d\Z^jZl ghjfb ihi_j_^gvh]h ]jZ^•}glZ

IZjlZg ± f_lh^

F•g•f•aZp•y d\Z^jZlbqgh€ p•evh\h€ nmgdp•€ F(w ) = − 1 (w, Aw )+ bw ,

2

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: IhqZldh\bc djhd

1)M ^h\•evg•c lhqp• Z0 agZoh^blvky ]jZ^•}gl J1 nmgdp•€ ) Z

2)gZ ijyf•c Z Z0 + λ g1 agZoh^blvky lhqdZ Z1 sh ^hklZ\ey} fZdkbfmf nmgdp•€ ) Z

51

; AZ]Zevgbc djhd G_oZc \`_ agZc^_g• lhqdb Z0, w1, ...,wk-1, wk.

1)m lhqp• Zk agZoh^blvky ]jZ^•}gl Jk+1;

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′ g 2+

w′ + = λ′ g + + w ^_ λ k = ( k 1 );

k 1 k k 1 k

g k+1 , Ag k+1

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F_lh^ AZml_g^ZcdZ

F_lh^ AZml_g^ZcdZ (f_lh^ ijh_dp•c j_Ze•am}lvky m

\•^ih\•^ghkl• a gZklmigbfb djhdZfb

Djhd IhqZlb \ Z0 • ihdeZklb ijh_dlmxqm fZljbpx J0=I.

nmgdp•x ) Z m gZijyfdm Vk lhqdZ f•g•fmfm Zk+1 Ih\lhjblb djhd I•key lh]h yd ijhc^_gh Q gZijyfd•\ ihrmdm ^_ Q d•evd•klv d_jm\Zevgbo af•ggbo ihqZlb agh\m a djhdm ijb [0 = xn.

Djhd Ydsh 3kgk • Jk aZd•gqblb ihrmd

52

Djhd Ydsh 3kgk • Jk ¹ agh\m ihqZlb a djhdm ihdeZ\rb x0 = xk.

;Z]ZlhiZjZf_ljbqgbc ihrmd

F_lh^ F•ey D_glj_eeZ ihey]Z} \ j_Ze•aZp•€ ihke•^h\ghkl•

djhd•\

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H[qbkexxlvky Zk, gk lZ Dw k−1 .

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k k k 1

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Ijhp_k aZd•gqm}lvky dheb | DF(w) |< e .

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D\ZaLgvxlhg•\kvd• f_lh^b

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53

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wk+1=wk - a k Akgk,

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\bdhjbklh\m} gZklmigm nhjfmem

Dw k = w k+1 - w k ,

Dg k = g k+1 - g k ,

A k = A k−1 + Dw k−1Dw k−1T - A k−1Dg k−1Dg k−1T A k−1 ,

Dg k−1T Dg k−1 Dg k−1T A k−1Dg k−1

ijbqhfm a\bqZcgh ajmqgh \b[bjZlb :0 , ^_ , ± h^bgbqgZ fZljbpy

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^_ 50 ± ^h\•evgZ ihablb\gh \bagZq_gZ kbf_ljbqgZ fZljbpy F_lh^ I•jkhgZ ‹ j_Ze•am}lvky m \•^ih\•^ghkl• a nhjfmehx

^_ 50 ± ^h\•evgZ ihablb\gh \bagZq_gZ kbf_ljbqgZ fZljbpy

Ijh_dlb\gbc f_lh^ GvxlhgZ JZnkhgZ j_Ze•am}lvky m \•^ih\•^ghkl• a nhjfmehx

^_ 50 ± ^h\•evgZ ihablb\gh \bagZq_gZ kbf_ljbqgZ fZljbpy

:e]hjblf E_\_g[_j]Z FZjd\Zj^lZ

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< Ze]hjblf• E_\_g[_j]Z FZjd\Zj^lZ \bdhjbklh\m}lvky f_lh^ Dhr• sh[ h[qbkeblb ydh[•Zg - p•evh\h€ nmgdp•€ sh^h d_jm\Zevgbo af•ggbo D_jm\ZevgL af•gg• af•gxxlvky \•^ih\•^gh ^h dhjb]m\Zevgh]h ijZ\beZ sh m fZljbqg•c nhjf• fZ} \b]ey^

H=JTJ, g=JTe, wk+1 = wk – [H k ,@-1gk,

55

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:e]hjblf E_\_g[_j]Z FZjd\Zj^lZ fZ} \b]ey^

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E•qbevgbd qbkeZ pbde•\ gZ\qZggy N

Djhd I_j_\•jdZ a[•`ghkl• lZ mfh\ aZd•gq_ggy jh[hlb Ydsh a[•`g•klv ^hky]gmlZ N!(SRFKV Z[h jh[hlZ ih\bggZ [mlb ijbibg_gZ

± aZd•gq_ggy jh[hlb Djhd Ydsh ≤

Djhd H[qbkeblb agZq_ggy ihfbedb • kdhj_]m\Zlb \•^ih\•^gbf qbghf d_jm\ZevgL af•gg•

H=JTJ, g=JTe,

wk+1 = wk – [H k ,@-1gk.

Djhd H[qbkeblb gh\_ agZq_ggy p•evh\h€ nmgdp•€ Ydsh \hgh

f_gr_ ihlhqgh]h lh ih\_jgmlb ihi_j_^g• agZq_ggy d_jm\Zevgbf af•ggbf af•gblb - • \bclb a pbdem •gZdr_ aZn•dkm\Zlb agZq_ggy d_jm\Zevgbo af•ggbo • af•gblb +.

Djhd I_j_o•^ ^h djhdm

Djhd N N

Djhd I_j_o•^ ^h djhdm

GZ\qZggy amibgy}lvky ydsh \bdhgm}lvky ohqZ [ h^gZ a mfh\

^hky]gmlh fZdkbfZevgh ijbimklbf_ qbkeh pbde•\ gZ\qZggy

Epochs;

p•evh\Z nmgdp•y f•g•f•ah\ZgZ p•ev ^hky]gmlZ

]jZ^•}gl p•evh\h€ nmgdp•€ f_gr_ f•g•fZevgh]h ijbimklbfh]h agZq_ggy

4) i_j_\bsm} fZdkbfZevg_ ijbimklbf_ agZq_ggy max;

p•evh\Z nmgdp•y a[•evrbeZkv [•evr_ g•` m 0$;B)$,/ jZa•\ ihqbgZxqb a hklZggvh]h jZam af_gr_ggy

56

MaZ]Zevg_gbc ]jZ^•}glgbc Ze]hjblf

Ih^•[g•klv jha]eygmlbo ]jZ^•}glgbo f_lh^•\ ^Z} i•^klZ\m ^ey jhajh[db maZ]Zevg_gh]h ]jZ^•}glgh]h Ze]hjblfm sh fZ} gZklmigbc \b]ey^

Djhd 1g•p•Ze•aZp•y AZ^Zlb ihqZldh\• agZq_ggy d_jm\Zevgbo af•ggbo Z fZdkbfZevgh ijbimklbf_ qbkeh pbde•\ gZ\qZggy (SRFKV

iZjZf_lj a[•`ghkl• Ze]hjblfm 01 iZjZf_lj a[•`ghkl• ma^h\` ijyfh€

LZd A[•`g•klv ^hky]gmlZ I_j_o•^ ^h djhdm

G• i_j_clb ^h djhdm

Djhd Qb \bdhgm}lvky g_j•\g•klv N!(SRFKV"

LZd >hky]gmlh fZdkbfZevg_ qbkeh pbde•\ gZ\qZggy a[•`g•klv g_ ^hky]gmlZ

I_j_o•^ ^h djhdm

G• i_j_clb ^h djhdm Djhd H[qbkeblb V Zk).

Djhd Qb \bdhgm}lvky g_j•\g•klv ∂Q(0(w k-1 , k))T s(w k )< 0 ?

∂w

LZd i_j_clb ^h djhdm

G• ihdeZklb s(w k )= − ∂Q(0(w k-1 , k)) I_j_clb ^h djhdm

∂w

Djhd AgZclb lZd_ agZq_ggy .k ijb ydhfm ) Zk .ks(wk)) → min,

\bdhjbklh\mxqb iZjZf_lj 02.

Djhd IhdeZklb Zk+1 = wk .ks(wk).

Djhd Qb \bdhgm}lvky g_j•\g•klv ) Zk+1) < F(wk) ? LZd i_j_clb ^h djhdm

G• AZd•gq_ggy ihrmdm g_fZ} af_gr_ggy nmgdp•€ I_j_o•^ ^h djhdm

57

LZd AZd•gq_ggy ihrmdm g_fZ} ijhkm\Zggy ^h j•r_ggy I_j_o•^ ^h djhdm

G• i_j_clb ^h djhdm

Djhd IhdeZklb N N I_j_clb ^h djhdm Djhd Amibg_ggy

G_h[o•^gh aZagZqblb sh \ ijhp_k• h^gh\bf•jgh]h ihrmdm g_h[o•^gh ih fh`eb\hkl• mgbdZlb lhqgbo h[qbke_gv lhfm sh gZ \bdhgZggy hi_jZp•c ihrmdm ma^h\` ijyfh€ \bljZqZ}lvky ^m`_ agZqgZ qZklbgZ aZ]Zevgh]h qZkm h[qbke_gv

Hdj_f• \biZ^db maZ]Zevg_gh]h ]jZ^•}glgh]h Ze]hjblfm

M \bs_hibkZghfm maZ]Zevg_ghfm ]jZ^•}glghfm Ze]hjblf• fh`gZ \bdhjbklh\m\Zlb j•ag• ]jZ^•}glg• f_lh^b reyohf \bagZq_ggy \•^ih\•^gbo gZijyfd•\ ihrmdm gZ djhp• <bagZqZxqb \•^ih\•^gbf

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>ey f_lh^m GvxlhgZ Z[h Ze]hjblfm a\hjhlgh]h ihrbj_ggy

ihfbedb ^jm]h]h ihjy^dm djhd maZ]Zevg_gh]h ]jZ^•}glgh]h Ze]hjblfm fZ} \b]ey^

>ey Ze]hjblf•\ kijy`_gbo ]jZ^•}gl•\ djhd maZ]Zevg_gh]h ]jZ^•}glgh]h Ze]hjblfm fZ} \b]ey^

s(wk)=-gk + βk s(wk-1).

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>ey Ze]hjblfm E_\_g[_j]Z FZjd\Zj^lZ djhd maZ]Zevg_gh]h ]jZ^•}glgh]h Ze]hjblfm fZ} \b]ey^

s(wk)= – [H k ,@-1gk.

59

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\Io ξ .

M pvhfm \biZ^dm ijb \bkhdhfm j•\g• \biZ^dh\bo i_j_rdh^ ξ

mkdeZ^g_g_ \bj•r_ggy a ^hklZlgvhx lhqg•klx aZ^Zq• ihrmdm fZdkbfmfm ih o p•evh\h€ nmgdp•€ Io lhfm sh agZq_ggy aZ ydbf kihkl_j•]Zxlv fh`_ \biZ^dh\h \by\blbky fZdkbfZevgbf m lhqp• o

\•^^Ze_g•c gZ agZqgm \•^klZgv \•^ lhqdb fZdkbfmfm [max p•evh\h€

nmgdp•€ Io Lhfm deZkbqgZ ihklZgh\dZ aZ^Zq• hilbf•aZp•€ agZcreZ rbjhd_ aZklhkm\Zggy ebr_ ijb ^_l_jf•g•klkvdhfm i•^oh^• ^h aZ^Zq hilbf•aZp•€ ijb \•^kmlghkl• \biZ^dh\bo rmf•\ gZijbdeZ^ ijb \bj•r_gg• fZl_fZlbqgbo aZ^Zq ijb fh^_ex\Zgg• h[ }dl•\ Z lZdh` m \biZ^dm g_\bkhdh]h j•\gy \biZ^dh\bo i_j_rdh^ yd• fh`gZ g_ \jZoh\m\Zlb sh fZxlv f•kp_ ijb ijh\_^_gg• eZ[hjZlhjgbo ^hke•^`_gv m ki_p•Zevgbo mfh\Zo < mfh\Zo `_ ijhfbkeh\h]h \bjh[gbpl\Z a \_ebdbf j•\g_f rmf•\ ihlj•[gh aZklhkh\m\Zlb ki_p•Zevg• f•jb agb`_ggy j•\gy \biZ^dh\bo ihfbehd kihkl_j_`_ggy sh ijb\h^blv ^h a[•evr_ggy \bljZl gZ j_Ze•aZp•x ihrmdh\h]h f_lh^m fZdkbf•aZp•€ yd• fh`gZ \jZom\Zlb \\_^_ggyf ^_ydh€ \Z]h\h€ nmgdp•€ k o Lh^• h^_j`m}fh sh nmgdp•y jbabdm

R(x* ) = òc(x)(1 − /(x* − x))f (x)dx = -* − c(x* )f (x* )

x

• agZoh^`_ggy €€ f•g•fmfm mkdeZ^gx}lvky ydsh \Z]h\Z nmgdp•y k o g_ ^hj•\gx} ^_yd•c dhgklZgl• F0.

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