st rt tt - - PowerPoint PPT Presentation

SLIDE 1

❋❛st ■♥❝r❡♠❡♥t❛❧ ❊①♣❡❝t❛t✐♦♥ ▼❛①✐♠✐③❛t✐♦♥✿ ❤♦✇ ♠❛♥② ✐t❡r❛t✐♦♥s ❢♦r ❛♥ ǫ✲st❛t✐♦♥❛r② ♣♦✐♥t ❄

• ❡rs❡♥❞❡ ❋♦rt

❈◆❘❙ ✫ ■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ❚♦✉❧♦✉s❡✱ ❋r❛♥❝❡

❈■❘▼ ✧❖♣t✐♠✐③❛t✐♦♥ ❢♦r ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣✧✱ ▼❛r❝❤ ✷✵✷✵✳

SLIDE 2

❇❛s❡❞ ♦♥ ❛ ❥♦✐♥t ✇♦r❦ ✇✐t❤

• P✐❡rr❡ ●❛❝❤ ✲ ✭■▼❚✱ ❯♥✐✈✳ P❛✉❧ ❙❛❜❛t✐❡r✱ ❋r❛♥❝❡✮
• ❊r✐❝ ▼♦✉❧✐♥❡s ✲ ✭❈▼❆P✱ ❊❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✱ ❋r❛♥❝❡✮

❆❝❦♥♦✇❧❡❞❣♠❡♥ts✿ ✲ ❋♦♥❞❛t✐♦♥ ❙✐♠♦♥❡ ❡t ❈✐♥♦ ❉❡❧ ❉✉❝❛ ✲ ❖♣t✐♠✐s❛t✐♦♥ ❡t ❙✐♠✉❧❛t✐♦♥ ▼♦♥t❡ ❈❛r❧♦ ✿ ❊♥tr❡❧❛❝❡♠❡♥ts ✲ ▲❛❜❡① ❈■▼■ ✲ ❙t♦❝❤❛st✐❝ ❉❡s❝❡♥t ❆❧❣♦r✐t❤♠s ✇✐t❤ ▼❛r❦♦✈✐❛♥ ■♥♣✉ts

SLIDE 3

✳

❖♣t✐♠✐③❛t✐♦♥ ♣❜♠ ❢♦r ❙t❛t✐st✐❝❛❧ ▲❡❛r♥✐♥❣

✳ ❆♥ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ♦❝❝✉rr✐♥❣✱ ❢♦r ❡①❛♠♣❧❡✱ ✐♥ ❧❛r❣❡ s❝❛❧❡ ❙t❛t✐st✐❝❛❧ ✐♥❢❡r❡♥❝❡

SLIDE 4

❚❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ argminθ∈ΘF(θ), F(θ) def = 1 n

n

• i=1

Li(θ) + R(θ) ✇✐t❤

• Θ ⊆ Rd
• n ✐s ❧❛r❣❡
• Li : Θ → R ✐s ♥♦t ❡①♣❧✐❝✐t ❛♥❞ ♦❢ t❤❡ ❢♦r♠

Li(θ) def = − log

• Z exp (si(z), φ(θ)) dµ(z)
• ◆♦ ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥s
• ✭❢♦r t❤❡ ❝✈❣ ❛♥❛❧②s✐s✮ ❘❡❣✉❧❛r✐t② ♣r♦♣❡rt✐❡s ♦❢ Li, R

SLIDE 5

▼♦t✐✈❛t✐♦♥✿ ❙t❛t✐st✐❝❛❧ ✐♥❢❡r❡♥❝❡ ✭✶✴✸✮

argminθ∈ΘF(θ), F(θ)

def

= 1 n

n

• i=1

Li(θ) + R(θ), Li(θ)

def

= − log

• Z

exp (si(z), φ(θ)) dµ(z)

• n ♦❜s❡r✈❛t✐♦♥s Y1, . . . , Yn✿ ❛ ♣❛r❛♠❡t❡r✐❝ st❛t✐st✐❝❛❧ ♠♦❞❡❧ ✐♥❞❡①❡❞ ❜② θ ∈ Θ
• R(θ)✿ ❛ ♣❡♥❛❧t② t❡r♠✱ ❛ r❡❣✉❧❛r✐③❛t✐♦♥ t❡r♠✱ ❛ ♣r✐♦r ✭✐♥ ❛ ❇❛②❡s✐❛♥ s❡tt✐♥❣ s♦❧✈❡❞

❜② ▼❆P✮

• ❆ ❧♦ss ❢✉♥❝t✐♦♥ ❛ss♦❝✐❛t❡❞ t♦ ❡❛❝❤ Yi✿ Li(θ)
• exp(−Li) ✐s ❛ ✧s✉♠✧ ♦✈❡r ❧❛t❡♥t ✈❛r✐❛❜❧❡s z ∈ Z✳ ❋♦r ❡①❛♠♣❧❡✱

Li(θ) = −

• ♥❡❣❛t✐✈❡

log

• ❧♦❣✲
• Z . . . dµ
• ❧✐❦❡❧✐❤♦♦❞ ♦❢ Yi

SLIDE 6

❊①❛♠♣❧❡✿ ♠✐①t✉r❡ ♠♦❞❡❧s ✭✷✴✸✮

• ❞❛t❛✿ y1, · · · , yn ♠♦❞❡❧❡❞ ❛s ✐✐❞ ❢r♦♠✿ L

ℓ=1 ωℓ N(µℓ, 1)

θ = (ω1:L, µ1:L)

• ❊q✉✐✈❛❧❡♥t❧②✿ zi ∈ {1, · · · , L} s✳t✳ L(Yi|Zi = ℓ) ∼ N(µℓ, 1)

L(Zi) ∼ (ωℓ)ℓ✳

• ❏♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ (Yi, Zi)

ωℓ/ωL = exp(αℓ)

L

• ℓ=1

1zi=ℓ ωℓ exp −(yi − µℓ)2/2 =

L

• ℓ=1

1zi=ℓ exp ln ωℓ − (yi − µℓ)2/2 = exp

• L
• ℓ=1

1zi=ℓ

• ln ωℓ − (yi − µℓ)2/2
• = exp

L−1

• ℓ=1

1zi=ℓ

• αℓ − (yi − µℓ)2/2 + (yi − µL)2/2

− ln(1 +

L−1

• ℓ=1

exp(αℓ)) − (yi − µL)2/2

• si(z)

def

= (1z=1, . . . , 1z=L−1, yi1z=1, . . . , yi1z=L−1, yi, −1) , φ(θ)

def

=

• α1 − µ2

1 − µ2 L

2 , . . . , αL−1 − µ2

L−1 − µ2 L

2 , µ1, . . . , µL−1, µL, ln(1 +

L−1

• ℓ=1

exp(αℓ)) + µ2

L/2

SLIDE 7

❊①❛♠♣❧❡✿ ▲♦❣✐st✐❝ r❡❣r❡ss✐♦♥ ✭✸✴✸✮

• ❞❛t❛✿ y1, · · · , yn ∈ {0, 1}n ♠♦❞❡❧❡❞ ❛s ✐♥❞❡♣ ❢r♦♠
• L(Yi|Zi) ∼ ❇❡r♥
• 1

1+exp(−ηi(zi))

• L(Zi) ∼ Np(θ, I)✳
• ❚❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ (Yi, Zi)
• exp(ηi(zi))

1 + exp(ηi(zi))

Yi

1 1 + exp(ηi(zi))

1−Yi

exp −zi − θ2/2 = exp(Yi ηi(zi)) 1 + exp(ηi(zi)) exp −zi − θ2/2 .

• ❚❤❡ ❧✐❦❡❧✐❤♦♦❞ ♦❢ Yi
• Rp

exp(Yi ηi(zi)) 1 + exp(ηi(zi)) exp −zi − θ2/2 dz =

• Z

exp(si(z), φ(θ)) dz

si(z) =

• yiηi(z) − ln(1 + exp(ηi(z))) − z2/2, 1, z′

φ(θ) =

• 1, −θ2/2, θ′

SLIDE 8

❲❤✐❝❤ ♥✉♠❡r✐❝❛❧ t♦♦❧ ❄ argminθ∈ΘF(θ), F(θ) def = 1 n

n

• i=1

Li(θ) + R(θ), Li(θ) def = − log

• Z exp (si(z), φ(θ)) µ(dz).

❆♥ ❛❧❣♦r✐t❤♠✐❝ s♦❧✉t✐♦♥ ❞❡s✐❣♥❡❞ ❢♦r ✲ ❧❛r❣❡ n✿ r❛r❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ ❛ s✉♠ ♦✈❡r n t❡r♠s ❛❧❧♦✇❡❞✱ ✲ ♥♦♥ ❝♦♥✈❡① s❡tt✐♥❣ ❙♦❧✉t✐♦♥✿ ❊①♣❡❝t❛t✐♦♥ ▼❛①✐♠✐③❛t✐♦♥✲❜❛s❡❞ ♠❡t❤♦❞s✿

❉❡♠♣st❡r ❡t ❛❧✳ ✭✶✾✼✼✮✱ ❲✉ ✭✶✾✽✸✮

SLIDE 9

✳

■■✲ ❊①♣❡❝t❛t✐♦♥ ▼❛①✐♠✐③❛t✐♦♥ ✭❊▼✮ ❛❧❣♦r✐t❤♠s

✳ ❊▼ ❛❧❣♦r✐t❤♠✿ ✐ts ❞❡r✐✈❛t✐♦♥ ❢♦r t❤✐s ♦♣t✐♠ ♣❜♠✱ ✐ts ✐♥tr❛❝t❛❜✐❧✐t②✱ ❛♥ ❛❧t❡r♥❛t✐✈❡✳

SLIDE 10

❊▼✿ ❆ ▼❛❥♦r✐③❡✲▼✐♥✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭✶✴✷✮

Argminθ F(θ) F(θ) = 1 n

n

• i=1

Li(θ) + R(θ) Li(θ) = − log

• Z

exp(si(z), φ(θ) dµ(z) .

• ❚❤❡ s✉rr♦❣❛t❡ ❢✉♥❝t✐♦♥ ❛t t❤❡ ❝✉rr❡♥t ♣♦✐♥t θk ∈ Θ✱ ✭❏❡♥s❡♥✬s ✐♥❡q✉❛❧✐t②✮

1 n

n

• i=1

Li(θ) ≤ 1 n

n

• i=1

Li(θk) + ¯ s(θk), φ(θk) − ¯ s(θk), φ(θ) ✇❤❡r❡ ¯ s(θk) def = 1 n

n

• i=1

¯ si(θk), ¯ si(θk) def =

• Z si(z) exp(si(z), φ(θk))

exp(−Li(θk)) µ(dz).

• ❬❊✲st❡♣❪ ❈♦♠♣✉t❡ ¯

s(θk)

• ❬▼✲st❡♣❪ ▼✐♥✐♠✐③❡ t❤❡ ♠❛❥♦r✐③✐♥❣ ❢✉♥❝t✐♦♥ ✭✉♥❞❡r ❤②♣✿ ✉♥✐q✉❡ ❛r❣♠✐♥✮

θk+1 = T ◦ ¯ s(θk) def = Argminθ∈Θ {− ¯ s(θk), φ(θ) + R(θ)} .

SLIDE 11

❊▼ ✐♥ t❤❡ ❙t❛t✐st✐❝✲s♣❛❝❡ ✭✷✴✷✮ ❊ ▼ ❊ ▼ θk ¯ s(θk) θk+1 = T ◦ ¯ s(θk) ¯ s ◦ T ◦ ¯ s(θk) θk+2 = T ◦ ¯ s ◦ T ◦ ¯ s(θk) F(θ) sk

T(sk)

sk+1 = ¯ s ◦ T(sk)

T ◦ ¯

s ◦ T(sk) V (s) ❲❡ ✇✐❧❧ s❡❡ ❊▼✲❜❛s❡❞ ❛❧❣♦r✐t❤♠s ❛s ❡✈♦❧✈✐♥❣ ✐♥ t❤❡ ✧s✲s♣❛❝❡✧❀ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✿ V = F ◦ T

• ■❢ ❝♦♥✈❡r❣❡♥❝❡✱ t♦ t❤❡ r♦♦ts ♦❢

h(s) def = ¯ s ◦ T(s) − s. ❯♥❞❡r ❛ss✉♠♣t✐♦♥s✱ t❤❡ r♦♦ts ♦❢ h ❛r❡ t❤❡ r♦♦ts ♦❢ ˙ V ✳

❉♦ ❙t♦❝❤❛st✐❝ ❊▼ ❛✈♦✐❞ tr❛♣s ❄ ♥♦t t❤❡ t♦♣✐❝ t♦❞❛②

❊▼ ✐s ❞❡s✐❣♥❡❞ t♦ ✜♥❞ t❤❡ r♦♦ts ♦❢ h

SLIDE 12

❋r♦♠ ❊▼ t♦ ❙t♦❝❤❛st✐❝ ❆♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤✐♥ ❊▼

• ❊❛❝❤ ✐t❡r❛t✐♦♥ ♦❢ ❊▼ r❡q✉✐r❡s

sk+1 = 1 n

n

• i=1

¯ si(T(sk)) ■t ❞♦❡s ♥♦t ❛♥s✇❡r t❤❡ s♣❡❝✐✜❝❛t✐♦♥s ❢♦r ❧❛r❣❡ s❝❛❧❡ ❧❡❛r♥✐♥❣✳

• ■t ✐s ❞❡s✐❣♥❡❞ t♦ ✜♥❞ t❤❡ ③❡r♦s ♦❢ h ❛♥❞ h ✐s ✐♥tr❛❝t❛❜❧❡✿

h(s) = 1 n

n

• i=1

¯ si ◦ T(s) − s = E [¯ sI ◦ T(s) − s] I ∼ U({1, · · · , n}) ✇❤❛t ❛❜♦✉t ❙t♦❝❤❛st✐❝ ❆♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤✐♥ ❊▼ ❛♣♣r♦❛❝❤❡s ❄ ❛♠♦♥❣ t❤❡ ♠❛♥② ✧❙t♦❝❤❛st✐❝ ❊▼✧ ❛❧❣♦r✐t❤♠s ❬❙❊▼❪ ❈❡❧❡✉① ✫ ❉✐❡❜♦❧t ✭✶✾✽✺✮❀ ❬▼❈❊▼❪ ❲❡✐

✫ ❚❛♥♥❡r ✭✶✾✾✵✮✱ ❋♦rt ✫ ▼♦✉❧✐♥❡s ✭✷✵✵✸✮❀ ❬❙❆❊▼❪ ❉❡❧②♦♥✱ ▲❛✈✐❡❧❧❡ ✫ ▼♦✉❧✐♥❡s ✭✶✾✾✾✮❀ ❑✉❤♥ ✫ ▲❛✈✐❡❧❧❡ ✭✷✵✵✹✮❀ ❬❖♥❧✐♥❡ ❊▼❪ ❈❛♣♣é ✫ ▼♦✉❧✐♥❡s ✭✷✵✵✾✮✱ ❬■♥❝r❡♠❡♥t❛❧ ❊▼❪ ◆❡❛❧ ✫ ❍✐♥t♦♥ ✭✶✾✾✾✮ . . .

SLIDE 13

❙t♦❝❤❛st✐❝ ❆♣♣r♦①✐♠❛t✐♦♥ ✭❙❆✮ ✇✐t❤✐♥ ❊▼✿

❊▼ → ❞❡s✐❣♥❡❞ t♦ ✜♥❞ t❤❡ r♦♦ts ♦❢ h(s)

def

= ¯ s ◦ T(s) − s

• ❲❤❛t ✐s ❙❆ ❄

❙♦❧✈❡ h(s) = 0 ✇❤❡♥ h(s) def = E [H(U, s)] ❜②✿

• Sk+1 =

Sk + γk+1 H(Uk+1, Sk)

• ❛♣♣r♦①✳ ♦❢ h(

Sk)

, Uk+1 ∼ U ♦r ✧❛❧♠♦st✧

• ❍❡r❡✱ ♣♦ss✐❜❧② t✇♦ s♦✉r❝❡s ♦❢ ✐♥tr❛❝t❛❜✐❧✐t②

h(s) = 1 n

n

• i=1

¯ si ◦ T(s) − s = 1 n

n

• i=1
• Z si(z) π(dz; T(s)) − s

❉❡❧②♦♥ ❡t ❛❧✳ ✭✶✾✾✾✮✱ t❤❡ s❡❝♦♥❞ ✐♥tr❛❝t❛❜✐❧✐t② ♦♥❧② ✇✐t❤ ✐✳✐✳❞✳ U′

k✳

❊①t❡♥s✐♦♥s t♦ ▼❈▼❈ s❛♠♣❧✐♥❣✱ ✜rst ❜② ▲❛✈✐❡❧❧❡ ✫ ❑✉❤♥ ✭✷✵✵✹✮

SLIDE 14

✶st ✐❞❡❛✿ ❙❆ h(s) = 1 n

n

• i=1

¯ si ◦ T(s) − s = E [¯ sJ ◦ T(s) − s] J ∼ U([[n]]) ■♥st❡❛❞ ♦❢ t❤❡ ❊▼ ✐t❡r❛t✐♦♥s sk+1 = h(sk) = ¯ s ◦ T(sk) ❞♦✿ ❉❛t❛✿

• S0 ∈ S✱ γk ∈ (0, ∞) ❢♦r k ≥ 1

❘❡s✉❧t✿ ❚❤❡ ❙❆ s❡q✉❡♥❝❡✿

• Sk, k = 0, . . . ,

✶ ❢♦r k ≥ 1 ❞♦ ✷

❙❛♠♣❧❡ Jk+1 ✉♥✐❢♦r♠❧② ♦♥ [[n]] ❀

✸

• Sk+1 =

Sk + γk+1

• ¯

sJk+1 ◦ T( Sk) − Sk

SLIDE 15

❋❛st ■♥❝r❡♠❡♥t❛❧ ❊▼ ✭❋■❊▼✮ ❬❋■❊▼❪ ❑❛r✐♠✐ ❡t ❛❧✳ ✭✷✵✶✾✮❀ ❬❙❆●❆❪ ❉❡❢❛③✐♦ ❡t ❛❧✳ ✭✷✵✶✹✮ h(s) = E

• ¯

sI ◦ T(s) − s + Vk+1

• I ∼ U([[n]]),

E[Vk+1] = 0 ❉❛t❛✿

• S0 ∈ S✱ γk ∈ (0, ∞) ❢♦r k ≥ 1

❘❡s✉❧t✿ ❚❤❡ ❋■❊▼ s❡q✉❡♥❝❡✿

• Sk, k = 0, . . . ,

✶ S0,i = ¯

si ◦ T( S0) ❢♦r ❛❧❧ i ∈ [[n]]❀

✷

• S0 = n−1 n

i=1 S0,i❀

✸ ❢♦r k ≥ 1 ❞♦ ✹

✯ ❙❛♠♣❧❡ Ik+1 ✉♥✐❢♦r♠❧② ♦♥ [[n]] ❀

✺

✯ Sk+1,i = Sk,i ❢♦r i = Ik+1 ❀

✻

✯ Sk+1,Ik+1 = ¯ sIk+1 ◦ T( Sk)❀

✼

✯ Sk+1 = Sk + n−1

Sk+1,Ik+1 − Sk,Ik+1

• ❀

✽

❙❛♠♣❧❡ Jk+1 ✉♥✐❢♦r♠❧② ♦♥ [[n]] ❀

✾

• Sk+1 =

Sk + γk+1

• ¯

sJk+1 ◦ T( Sk) − Sk −

• Sk+1,Jk+1 −

Sk+1 ✯ ✐t❡r❛t✐✈❡ ❝♦♠♣✉t✳ ♦❢ t❤❡ s✉♠✿

• Sk+1 = 1

n

n

• i=1

¯ si ◦ T( S❧❛st t✐♠❡ ✇❤❡♥ #i s❛♠♣❧❡❞)

SLIDE 16

✳

■■■✲ ◆♦♥ ❛s②♠♣t♦t✐❝ ❝♦♥✈❡r❣❡♥❝❡ ❜♦✉♥❞s

✳ ◆♦♥ ❛s②♠♣t♦t✐❝ → ❛ ♠❛①✐♠❛❧ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ❍♦✇ t♦ ❞❡✜♥❡ t❤❡ ❡st✐♠❛t❡ ♦❢ t❤❡ s♦❧✉t✐♦♥ ❄ ✏❈♦♥✈❡r❣❡♥❝❡ ❜♦✉♥❞s✑✿ ✐♥ ✇❤✐❝❤ s❡♥s❡ ❄ ❍♦✇ t♦ ❝❤♦♦s❡ t❤❡ st❡♣s✐③❡ s❡q✉❡♥❝❡ ❄

SLIDE 17

◆♦♥ ❛s②♠♣t♦t✐❝ ❝♦♥✈❡r❣❡♥❝❡ ❜♦✉♥❞s✿ ✇❤✐❝❤ ♦♥❡s ❄

• ❚❤❡ ✉s❡r ❝❤♦♦s❡s ❛ ♠❛①✐♠❛❧ ❧❡♥❣t❤ ♦❢ t❤❡ ♣❛t❤✿ Kmax✳

K ✐s ❛ r❛♥❞♦♠ st♦♣♣✐♥❣ t✐♠❡ ✐♥ t❤❡ r❛♥❣❡ {0, . . . , Kmax − 1}✳

• ❤❛❞✐♠✐ ✫ ▲❛♥ ✭✷✵✶✸✮
• ♠❡❛♥ ✧❞✐st❛♥❝❡✧ t♦ t❤❡ r♦♦ts ♦❢ t❤❡ ❣r❛❞✐❡♥t ♦❢ V def

= F ◦ T inf

k≤Kmax

E

• ˙

V ( Sk)2 ≤ Eg

def

= E

• ˙

V ( SK)2

• ♠❡❛♥ ✧❞✐st❛♥❝❡✧ t♦ t❤❡ r♦♦ts ♦❢ h

Eh

def

= E

• h(

SK)2

• ❯♥❞❡r t❤❡ st❛t❡❞ ❛ss✉♠♣t✐♦♥s

1 v2

max

Eg ≤ Eh

SLIDE 18

❆ss✉♠♣t✐♦♥s ❆✶ Θ ⊆ Rd ✐s ❛♥ ♦♣❡♥ s❡t✳ (Z, Z) ✐s ❛ ♠❡❛s✉r❛❜❧❡ s♣❛❝❡ ❛♥❞ µ ✐s ❛ σ✲✜♥✐t❡ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡ ♦♥

Z✳ ❚❤❡ ❢✉♥❝t✐♦♥s φ : Θ → Rq✱ si : Z → Rq ❢♦r ❛❧❧ i ∈ [[n]] ❛♥❞ R : Θ → R ❛r❡ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥s✳ ❋✐♥❛❧❧②✱ ❢♦r ❛♥② θ ∈ Θ ❛♥❞ i ∈ [[n]]✱ −∞ < Li(θ) < ∞✳

❆✷ ❋♦r ❛❧❧ θ ∈ Θ ❛♥❞ i ∈ [[n]]✱ t❤❡ ❡①♣❡❝t❛t✐♦♥ ¯

si(θ) ❡①✐sts✳ ❋♦r ❛♥② s ∈ Rq✱ Argminθ∈Θ (− s, φ(θ) + R ✐s ❛ ✭♥♦♥ ❡♠♣t②✮ s✐♥❣❧❡t♦♥ ❞❡♥♦t❡❞ ❜② {T(s)}✳

❆✸ φ, R ❛r❡ C1 ♦♥ Θ✳ T ✐s C1 ♦♥ Rq✳

❋♦r ❛♥② s ∈ Rq✱ B(s)

def

= ∇(φ ◦ T)(s) ✐s ❛ s②♠♠❡tr✐❝ q×q ♠❛tr✐① ❛♥❞ t❤❡r❡ ❡①✐st 0 < vmin ≤ vmax < ∞ s✉❝❤ t❤❛t ❢♦r ❛❧❧ s ∈ S✱ t❤❡ s♣❡❝tr✉♠ ♦❢ B(s) ✐s ✐♥ [vmin, vmax]✳ ❋♦r ❛♥② i ∈ [[n]]✱ ¯ si ◦ T ✐s ❣❧♦❜❛❧❧② ▲✐♣s❝❤✐t③ ♦♥ Rq ✇✐t❤ ❝♦♥st❛♥t Li✳ s → BT(s) (¯ s ◦ T(s) − s) ✐s ❣❧♦❜❛❧❧② ▲✐♣s❝❤✐t③ ♦♥ Rq ✇✐t❤ ❝♦♥st❛♥t L ˙

V ✳

❈♦r♦❧❧❛r②✿ R ✐s C1❀ Li ✐s C1❀ ˙ V ✐s ▲✐♣s❝❤✐t③❀ ˙ V (s) ≤ vmaxh(s)✳

SLIDE 19

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✱ V def = F ◦ T ❆♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ E

• h(

SK)2 ✇❤❡♥

• Sk+1 =

Sk + γk+1Hk+1

• ❆ ❚❛②❧♦r ❡①♣❛♥s✐♦♥✱ ✜rst ♦r❞❡r ✰ ●r❛❞✐❡♥t ▲✐♣s❝❤✐t③

V ( Sk+1) ≤ V ( Sk) + γk+1

• Hk+1, ˙

V ( Sk)

• + γ2

k+1

L ˙

V

2 Hk+12

• ❚❤❡ ❡①♣❡❝t❛t✐♦♥

E

• V (

Sk+1)

• ≤ E
• V (

Sk)

• + γk+1E
• h(

Sk), ˙ V ( Sk)

• + γ2

k+1

L ˙

V

2 E

• Hk+12
• ❚❤❡ ❛ss✉♠♣t✐♦♥ ✭▲②❛♣✉♥♦✈ ❝♦♥tr❛❝t✐♦♥✮

E

• V (

Sk+1)

• ≤ E
• V (

Sk)

• −γk+1vminE
• h(

Sk)2 + γ2

k+1

L ˙

V

2 E

• Hk+12
• ❆ s✉♠ ❢r♦♠ k = 0 t♦ k = Kmax − 1✱

vmin

Kmax−1

• k=0

γk+1E

• h(

Sk)2 ≤ E

• V (

S0)

• − E
• V (

SKmax)

• + L ˙

V

2

Kmax−1

• k=0

γ2

k+1E

Hk+12

• ❋❡✇ ♣❛❣❡s ❧❛t❡r✿

Kmax−1

• k=0

αk E

• h(

Sk)2 ≤ E

• V (

S0)

• − E
• V (

SKmax)

SLIDE 20

❘❡s✉❧t ✶ ✭●❛❝❤✱ ❋✳✱ ▼♦✉❧✐♥❡s✲✷✵✷✵✮ ❆ss✉♠❡ ❆✶ t♦ ❆✸ ❛♥❞ s❡t L2 def = n−1 n

i=1 L2 i ✳ ❈❤♦♦s❡ µ ∈ (0, 1)✳

▲❡t K ❜❡ ❛ {0, . . . , Kmax − 1}✲✈❛❧✉❡❞ ✉♥✐❢♦r♠ r✳✈✳ ❘✉♥ ❋■❊▼ ✇✐t❤ ❛ ❝♦♥st❛♥t st❡♣ s✐③❡ γℓ = √ C n2/3L ✇❤❡r❡ C ∈ (0, 1) ✐s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢ ✧❛♥ ❡q✉❛t✐♦♥ ❞❡♣❡♥❞✐♥❣ ✉♣♦♥ vmin, L, L ˙

V , µ✧

❚❤❡♥✱ ❢♦r ❛♥② n ≥ 2 ❛♥❞ Kmax ≥ 1

Eh ≤ n2/3

Kmax L √ C(1 − µ)vmin E

• V (

S0) − V ( SKmax)

• .

SLIDE 21

❈♦r♦❧❧❛r✐❡s ♦❢ ❘❡s✉❧t ✶

Eh ≤ n2/3

Kmax L √ C(1 − µ)vmin E

• V (

S0) − V ( SKmax)

• γk =

√ C n2/3L

• ❉❡♣❡♥❞❡♥❝❡ ✉♣♦♥ n ❛♥❞ Kmax✿

❛s ✐♥ ❑❛r✐♠✐ ❡t ❛❧✳ ✭✷✵✶✾✮

• ❈♦♥st❛♥t st❡♣s✐③❡ O(n−2/3) ❛s ✐♥ ❑❛r✐♠✐ ❡t ❛❧✳ ✭✷✵✶✾✮
• Pr❡❝✐s✐♦♥ ε✿ Kmax = M n2/3ε−1
• C ∈ (0, 1) ✐s ❡①♣❧✐❝✐t → ✧❞❡✜♥✐t✐♦♥✧ ♦❢ γk → ✐♠♣r♦✈❡s ♦♥ ♣r❡✈✐♦✉s r❡s✉❧ts

√ CL ˙

V

2L

1

n2/3 + C 1/2 − Cn−1/3

1

n + 2

• = µvmin

✇❤❡♥ n → ∞ C3/2 =

L 2L ˙

V µvmin

SLIDE 22

❘❡s✉❧t ✷ ✭●❛❝❤✱ ❋✳✱ ▼♦✉❧✐♥❡s✲✷✵✷✵✮ ❆ss✉♠❡ ❆✶ t♦ ❆✸ ❛♥❞ s❡t L2 def = n−1 n

i=1 L2 i ✳ ❈❤♦♦s❡ µ ∈ (0, 1)✳

▲❡t K ❜❡ ❛ {0, . . . , Kmax − 1}✲✈❛❧✉❡❞ ✉♥✐❢♦r♠ r✳✈✳ ❘✉♥ ❋■❊▼ ✇✐t❤ ❛ ❝♦♥st❛♥t st❡♣ s✐③❡ γℓ = √ C n1/3K1/3

maxL

✇❤❡r❡ C > 0 ✐s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢ ✧❛♥ ❡q✉❛t✐♦♥ ❞❡♣❡♥❞✐♥❣ ✉♣♦♥ vmin, L, L ˙

V , µ✧

❚❤❡♥✱ ❢♦r ❛♥② n ≥ 1 ❛♥❞ Kmax ≥ 1

Eh ≤ n1/3

K2/3

max

L √ C(1 − µ)vmin E

• V (

S0) − V ( SKmax)

• .

SLIDE 23

❈♦r♦❧❧❛r✐❡s ♦❢ ❘❡s✉❧t ✷

Eh ≤ n1/3

K2/3

max

L √ C(1 − µ)vmin E

• V (

S0) − V ( SKmax)

• γℓ =

√ C n1/3K1/3

maxL

.

• ❉❡♣❡♥❞❡♥❝❡ ✉♣♦♥ n ❛♥❞ Kmax✿ ♥❡✇ r❡s✉❧t✳
• ❈♦♥st❛♥t st❡♣s✐③❡ O(n−1/3K−1/3

max )

• Pr❡❝✐s✐♦♥ ε✿ Kmax = M √n ε−3/2

→ st❡♣s✐③❡ = O(√ǫ/√n)

• C ∈ (0, 1) ✐s ❡①♣❧✐❝✐t → ✧❞❡✜♥✐t✐♦♥✧ ♦❢ γk

SLIDE 24

❘❡s✉❧t ✸ ✭●❛❝❤✱ ❋✳✱ ▼♦✉❧✐♥❡s✲✷✵✷✵✮ ▲❡t K ❜❡ ❛ {0, . . . , Kmax −1}✲✈❛❧✉❡❞ r✳✈✳ ✇✐t❤ ✇❡✐❣❤ts p0, . . . , pKmax−1✱ infk pk > 0✳ ▲❡t C ∈ (0, 1) ❜❡ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢ ✧❛♥ ❡q✉❛t✐♦♥ ❞❡♣❡♥❞✐♥❣ ✉♣♦♥ vmin, L, L ˙

V , µ✧ ❘✉♥

❋■❊▼ ✇✐t❤ γk+1

def

= gk n2/3L , gk

def

= F −1

n,C

• pk

maxℓpℓ vmin √ C 2L 1 n2/3

• ;

Fn,C : x → 1 Ln2/3x (vmin − xfn(C)) , fn(C)

def

= L ˙

V

2L

• 1

n2/3 + C 1/2 − Cn−1/3

1

n + 2

• .

❋♦r ❛♥② n ≥ 2 ❛♥❞ Kmax ≥ 1✱ ✇❡ ❤❛✈❡

Eh ≤ n2/3 maxkpk

2L √ Cvmin E

• V (

S0) − V ( SKmax)

• .
• ❖♣t✐♠❛❧ s❛♠♣❧✐♥❣ str❛t❡❣② ✿ pk = 1/Kmax

→ s❛♠❡ ❛s ❘❡s✉❧t ✶✳

• ◆❡✇ r❡s✉❧t✳

SLIDE 25

❖♥ ❛ t♦② ❡①❛♠♣❧❡ ✭✶✴✹✮

• ❚♦② ❡①❛♠♣❧❡✿ Argminθ F(θ) = θ⋆ ✐s ❡①♣❧✐❝✐t✳
• ❈♦♠♣❛r❡ ❙❆ ❛♥❞ ❋■❊▼ ✭❧❡❢t✮ t❤r♦✉❣❤ ❜♦①♣❧♦ts ♦❢ θk − θ⋆ ❢♦r ❞✐✛❡r❡♥t k❀

s❛♠❡ t❤✐♥❣ ❢♦r ❋■❊▼ ❛♥❞ ❋■❊▼✲♦♣t ✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♣❤❛s❡ ✭r✐❣❤t✮

❋■❊▼✲♦♣t Sk+1 = Sk + γk+1 (Tk+1 + λk+1Vk+1)

3000 6000 8000 10000 12000 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 6000 8000 10000 12000 0.005 0.01 0.015 0.02

SLIDE 26

❖♥ ❛ t♦② ❡①❛♠♣❧❡ ✭✷✴✹✮

• ❋■❊▼ ❛♥❞ ❋■❊▼✲♦♣t

✭❧❡❢t✮ ❚❤❡ ❝♦❡✣❝✐❡♥t k → λk+1❀ ✭r✐❣❤t✮ ❚❤❡ L2✲♠♦♠❡♥t ♦❢ t❤❡ ✜❡❧❞ ❡st✐♠❛t❡❞ ❜② ▼♦♥t❡ ❈❛r❧♦✿ E

• Hk+12

✳

1000 2000 3000 4000 5000 6000 7000 8000 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

FIEM FIEM-coeff

1500 2000 2500 3000 3500 4000 4500 10 1 10 2 10 3

SA FIEM FIEM-Coeff

SLIDE 27

❖♥ ❛ t♦② ❡①❛♠♣❧❡ ✭✸✴✹✮

• ❋■❊▼ ❛♥❞ ❋■❊▼✲♦♣t ✲ ▼♦♥t❡ ❈❛r❧♦ ❡✈❛❧✉❛t✐♦♥ ♦❢ E [θk − θ⋆] ❛♥❞ std(θk − θ⋆)✳

✭❧❡❢t✮ r❛t✐♦ ♦❢ t❤❡ ❡①♣❡❝t❛t✐♦♥s ❋■❊▼ ✴ ❋■❊▼✲♦♣t ✭r✐❣❤t✮ r❛t✐♦ ♦❢ t❤❡ st❞ ❋■❊▼ ✴ ❋■❊▼✲♦♣t

1000 2000 3000 4000 5000 6000 0.999 0.9992 0.9994 0.9996 0.9998 1 1.0002 1000 2000 3000 4000 5000 6000 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45

SLIDE 28

❖♥ ❛ t♦② ❡①❛♠♣❧❡ ✭✹✴✹✮

• ❋♦r ❋■❊▼✱ ❝♦♥st❛♥t st❡♣s✐③❡ γ✱ ♦❜s❡r✈❡ k → θk − θ⋆

20 40 60 80 100 10 -3 10 -2 10 -1 10 0 10 1 10 2

1 1e-1 1e-2 GFM KM

2000 4000 6000 8000 10000 12000 10 -8 10 -6 10 -4 10 -2 10 0 10 2

1e-2 1e-4 1e-5 GFM KM

SLIDE 29

✳

■❱✲ ❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ¯ si ✐s ♥♦t ❡①♣❧✐❝✐t ❄

✳ ❚❤❡ ♣❡rt✉r❜❡❞ ❝❛s❡✿ ✇❤❡♥ t❤❡ ❡①♣❡❝t❛t✐♦♥ ❛r❡ ❛♣♣r♦①✐♠❛t❡❞✱ ✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ✐♥ ♦r❞❡r t♦ ❤❛✈❡ t❤❡ s❛♠❡ r❛t❡s ❛s ✐♥ t❤❡ ✧❡①❛❝t✧ ❝❛s❡❄

SLIDE 30

❚❤❡ P❡rt✉r❜❡❞ ❋■❊▼ ❛❧❣♦r✐t❤♠ ❉❛t❛✿

• S0 ∈ S✱ γk ∈ (0, ∞) ❢♦r k ≥ 1

❘❡s✉❧t✿ ❚❤❡ ❋■❊▼ s❡q✉❡♥❝❡✿

• Sk, k = 0, . . . ,

✶ S0,i = ˜

si✱ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ¯ si ◦ T( S0) ❢♦r ❛❧❧ i ∈ [[n]]❀

✷

• S0 = n−1 n

i=1 S0,i❀

✸ ❢♦r k ≥ 1 ❞♦ ✹

✯ ❙❛♠♣❧❡ Ik+1 ✉♥✐❢♦r♠❧② ♦♥ [[n]] ❀

✺

✯ Sk+1,i = Sk,i ❢♦r i = Ik+1 ❀

✻

✯ Sk+1,Ik+1 = ˇ

Sk+1 ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ¯

sIk+1 ◦ T( Sk)❀

✼

✯ Sk+1 = Sk + n−1

Sk+1,Ik+1 − Sk,Ik+1

• ❀

✽

❙❛♠♣❧❡ Jk+1 ✉♥✐❢♦r♠❧② ♦♥ [[n]] ❀

✾

❈♦♠♣✉t❡ ˜ sk+1✱ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ¯ sJk+1 ◦ T( Sk) ❀

✶✵

• Sk+1 =

Sk − γk+1

• ˜

sk+1 − Sk −

• Sk+1,Jk+1 −

Sk+1

SLIDE 31

❆ss✉♠♣t✐♦♥s ✭♦♥ t❤❡ ♣❡rt✉r❜❛t✐♦♥s✮ ❚❤❡r❡ ❡①✐st ♣♦s✐t✐✈❡ s❡q✉❡♥❝❡s {mk, k ≥ 0} ❛♥❞ {mk, k ≥ 0}✱ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs M(1) ❛♥❞ M(2) ❛♥❞ M(2)

ν

≥ 0 s✉❝❤ t❤❛t ❢♦r ❛❧❧ k ≥ 0✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ˜ sk+1 ❛♥❞ ˇ

Sk+1 s❛t✐s❢②

E[ˇ

Sk+1 − ¯

sIk+1 ◦ T( Sk)2] ≤ M(1) mk+1 , E[ E

• ˜

sk+1 − ¯ sJk+1 ◦ T( Sk)|Jk+1, Ik+1, Sk 2] ≤ M(2)

ν

m2

k+1

, E[˜ sk+1 − ¯ sJk+1 ◦ T( Sk)2] ≤ M(2) mk+1 . ❲❡❧❧ · · · ✐♥ ❛ ▼♦♥t❡ ❈❛r❧♦ ❛♣♣r♦①✐♠❛t✐♦♥ 1 n

n

• i=1

E[ 1 mk+1

mk+1

• ℓ=1

si(Zℓ,k) − ¯ si ◦ T( Sk)2] ≤ E

• C(

Sk)

• mk+1

≤ M(1) mk+1 ,

SLIDE 32

❈♦♥✈❡r❣❡♥❝❡ r❡s✉❧t ✭●❛❝❤✱ ❋✳✱ ▼♦✉❧✐♥❡s ✭✷✵✷✵✮✮

Eh ≤ n2/3

Kmax C0 + C1 n2/3{1 ∧ n Kmax }E

• ε(0)

❡rr♦r ✇❤❡♥ ✐♥✐t✐❛❧✐③✐♥❣ + C1 n5/3{1 ∧ n Kmax }

Kmax−1

• k=0

E

• η(1)

k+12

• ❊rr♦r✿ ♦♥ t❤❡ ❝♦♥tr♦❧ ✈❛r✐❛t❡

+ C2 1 Kmax

Kmax−1

• k=0

E

• E
• η(2)

k+1|Fk

• 2
• η(2)✿ s❡❝♦♥❞ ❡rr♦r

+ C1 2(1 + ν) 1 n2/3Kmax

Kmax−1

• k=0
• E
• η(2)

k+12

• + E
• E
• η(2)

k+1|Fk+3/4

• 2
• ;

SLIDE 33

■♥ t❤❡ ❝❛s❡ ♦❢ ❛ ▼♦♥t❡ ❈❛r❧♦ ❛♣♣r♦①✐♠❛t✐♦♥ ❋♦r ❛ ♣r❡❝✐s✐♦♥ ε✱

• ✇✐t❤ ❛♥ ✉♥❜✐❛s❡❞ ▼♦♥t❡ ❈❛r❧♦ ❛♣♣r♦①✐♠❛t✐♦♥ ✭❡①✳ ✐✳✐✳❞✳✮

+ Kmax = Mn2/3 ε−1 + m = n−2/3ε−1 m = (n−2/3ε−1) ∧ (n−1ε−2)

• ✇✐t❤ ❛ ❜✐❛s❡❞ ▼♦♥t❡ ❈❛r❧♦ ❛♣♣r♦①✐♠❛t✐♦♥ ✭❡①✳ ▼❛r❦♦✈ ❝❤❛✐♥ ▼♦♥t❡ ❈❛r❧♦✮

+ Kmax = Mn2/3 ε−1 + m = (n−2/3ε−1) ∨ ε−1/2 m = (n−2/3ε−1) ∧ (n−1ε−2)