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▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞s ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥✿ ■♥t❡rt✇✐♥✐♥❣s

• ❡rs❡♥❞❡ ❋♦rt

❈◆❘❙ ■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ❚♦✉❧♦✉s❡✱ ❋r❛♥❝❡ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞s✱ ❙②❞♥❡②✱ ❏✉❧② ✷✵✶✾✳

✳

■♥t❡rt✇✐♥❡❞✱ ✇❤② ❄

✳

❚♦ ✐♠♣r♦✈❡ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞s t❛r❣❡tt✐♥❣✿ dπ = π dµ

• ❚❤❡ ✧♥❛✐✈❡✧ ▼❈ s❛♠♣❧❡r ❞❡♣❡♥❞s ♦♥ ❞❡s✐❣♥ ♣❛r❛♠❡t❡rs ✐♥ Rp ♦r ✐♥ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥ θ
• ❚❤❡♦r❡t✐❝❛❧ st✉❞✐❡s ❝❛r❛❝t❡r✐③❡ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ♦❢ t❤❡s❡s ♣❛r❛♠❡t❡rs θ⋆ ❜②

θ⋆ ∈ Θ s✳t✳

• H(θ, x) dπ(x) = 0

♦r θ⋆ ∈ argminθ∈Θ

• C(θ, x) dπ(x) = 0.
• ❙tr❛t❡❣✐❡s✿

✲ ❙tr❛t❡❣② ✶✿ ❛ ♣r❡❧✐♠✐♥❛r② ✧♠❛❝❤✐♥❡r②✧ ❢♦r t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ θ⋆❀ t❤❡♥ r✉♥ t❤❡ ▼❈ s❛♠♣❧❡r ✇✐t❤ θ ← θ⋆ ✲ ❙tr❛t❡❣② ✷✿ ❧❡❛r♥ θ ❛♥❞ s❛♠♣❧❡ ❝♦♥❝♦♠✐t❛♥t❧②

❚♦ ♠❛❦❡ ♦♣t✐♠✐③❛t✐♦♥ ♠❡t❤♦❞s tr❛❝t❛❜❧❡

• ■♥tr❛❝t❛❜❧❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥

θ s✳t✳ h(θ) = 0 ✇❤❡♥ h ✐s ♥♦t ❡①♣❧✐❝✐t h(θ) =

• X H(θ, x) dπθ(x)

♦r argminθ∈Θ

• X C(θ, x) dπθ(x)
• ■♥tr❛❝t❛❜❧❡ ❛✉①✐❧✐❛r② q✉❛♥t✐t✐❡s

❊①✲✶ ●r❛❞✐❡♥t✲❜❛s❡❞ ♠❡t❤♦❞s ∇f(θ) =

• X H(θ, x) dπθ(x)

❊①✲✷ ▼❛❥♦r✐③❡✲▼✐♥✐♠✐③❛t✐♦♥ ♠❡t❤♦❞s ❛t ✐t❡r❛t✐♦♥ t✱ f(θ) ≤ Ft(θ) =

• X Ht(θ, x) dπt,θ(x)
• ❙tr❛t❡❣✐❡s✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦ t❡❝❤♥✐q✉❡s t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ✉♥❦♥♦✇♥ q✉❛♥t✐t✐❡s

■♥ t❤✐s t❛❧❦✱ ▼❛r❦♦✈ ✦

• ❢r♦♠ t❤❡ ▼♦♥t❡ ❈❛r❧♦ ♣♦✐♥t ♦❢ ✈✐❡✇✿

✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ✉♣❞❛t✐♥❣ s❝❤❡♠❡ ❢♦r ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❛♠♣❧❡r ❄ ❈❛s❡✿ ▼❛r❦♦✈ ❝❤❛✐♥ ▼♦♥t❡ ❈❛r❧♦ s❛♠♣❧❡r

• ❢r♦♠ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣♦✐♥t ♦❢ ✈✐❡✇✿

✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ▼♦♥t❡ ❈❛r❧♦ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ st♦❝❤❛st✐❝ ♦♣t✐♠✐③❛t✐♦♥ ❄ ❈❛s❡✿ ❙t♦❝❤❛st✐❝ ❆♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s ✇✐t❤ ▼❛r❦♦✈✐❛♥ ✐♥♣✉ts

• ❆♣♣❧✐❝❛t✐♦♥ t♦ ❛ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ ♣❜♠✿ ♣❡♥❛❧✐③❡❞ ▼❛①✐♠✉♠

▲✐❦❡❧✐❤♦♦❞ t❤r♦✉❣❤ ❙t♦❝❤❛st✐❝ Pr♦①✐♠❛❧✲●r❛❞✐❡♥t ♠❡t❤♦❞s

✳

P❛rt ■✿ ❚❤❡♦r② ♦❢ ❝♦♥tr♦❧❧❡❞

✭♦r ❛❞❛♣t✐✈❡✮ ▼❛r❦♦✈ ❝❤❛✐♥s

✳

❊①❛♠♣❧❡ ✶✴ ❆❞❛♣t❡❞ ▼❛r❦♦✈ ❝❤❛✐♥ ▼♦♥t❡ ❈❛r❧♦ s❛♠♣❧❡rs

• ❍❛st✐♥❣s✲▼❡tr♦♣♦❧✐s ❛❧❣♦r✐t❤♠✱ ✇✐t❤ ●❛✉ss✐❛♥ ♣r♦♣♦s❛❧ ❛♥❞ t❛r❣❡t dπ ♦♥ X ⊆ Rd

Pr♦♣♦s❛❧✿ Yt+1 ∼ Nd(Xt, θ) ❆❝❝❡♣t✲❘❡❥❡❝t Xt+1 =

• Yt+1

✇✐t❤ ♣r♦❜❛❜✐❧✐t② α(Xt, Yt+1) Xt ♦t❤❡r✇✐s❡ s✉♠♠❛r✐③❡❞✿ Xt+1 ∼ Pθ(Xt, ·)

• ✧❖♣t✐♠❛❧✧ ❝❤♦✐❝❡ ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① θ

θopt = (2.38)2 d Covπ(X) = (2.38)2 d Γopt

❊①❛♠♣❧❡ ✶ ✭t♦ ❢♦❧❧♦✇✮✴ ❆❞❛♣t❡❞ ▼❛r❦♦✈ ❝❤❛✐♥ ▼♦♥t❡ ❈❛r❧♦ s❛♠♣❧❡rs

• ❚❤❡ ❛❧❣♦r✐t❤♠

❙❛♠♣❧❡ Xt+1 ∼ Pθt(Xt, ·) ❙❆ s❝❤❡♠❡✿ Γt+1 = ❡♠♣✐r✐❝❛❧ ❝♦✈ ♠❛tr✐① ♦❢ X1:t+1 ❝♦♠♣✉t❡❞ ❢r♦♠ Γt, Xt+1 θt+1 = (2.38)2d−1Γt+1

• ■♥ t❤✐s ❡①❛♠♣❧❡✱ ❛ ❢❛♠✐❧② ♦❢ tr❛♥s✐t✐♦♥ ❦❡r♥❡❧s {Pθ, θ ∈ Θ} ❛♥❞

∀θ, Pθ ✐♥✈❛r✐❛♥t ✇✳r✳t✳π

• ❈♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts✿

✭❙❛❦s♠❛♥✲❱✐❤♦❧❛✱ ✷✵✶✵❀ ❋✳✲▼♦✉❧✐♥❡s✲Pr✐♦✉r❡t✱ ✷✵✶✷✮

✲ limt θt = θopt ✲ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ (Xt)t ❝♦♥✈❡r❣❡s t♦ π ✭❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ t❛✐❧s ♦❢ π✮ ✲ str♦♥❣ ▲▲◆✱ ❈▲❚ ❢♦r t❤❡ s❛♠♣❧❡s {Xt}t

❊①❛♠♣❧❡ ✷✴ ❆❞❛♣t❡❞ ■♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❜② ❲❛♥❣✲▲❛♥❞❛✉ ❛♣♣r♦❛❝❤❡s

• ❆ ❤✐❣❤❧② ♠✉❧t✐♠♦❞❛❧ t❛r❣❡t ❞❡♥s✐t② dπ ♦♥ X ⊆ Rd✳
• ❆ ❢❛♠✐❧② ♦❢ ♣r♦♣♦s❛❧ ♠❡❝❛♥✐s♠s✿ ●✐✈❡♥ ❛ ♣❛rt✐t✐♦♥ X1, · · · , XI ♦❢ X✱

dπθ(x) ∝

I

• i=1

1Xi(x) dπ(x) θ(i) , θ = (θ(1), · · · , θ(I)) ❛ ✇❡✐❣❤t ✈❡❝t♦r

• ❖♣t✐♠❛❧ ♣r♦♣♦s❛❧✿ dπθ⋆ ✇✐t❤ θ⋆(i) =
• Xi dπ(u)✱
• θ⋆✱ ✉♥✐q✉❡ ❧✐♠✐t✐♥❣ ✈❛❧✉❡ ♦❢ ❛ ❙t♦❝❤❛st✐❝ ❆♣♣r♦①✐♠❛t✐♦♥ s❝❤❡♠❡

✇✐t❤ ♠❡❛♥ ✜❡❧❞

• X

H(θ, X) dπθ(x) ❛♥❞ Hi(θ, x) = θ(i)

 1Xi(x) −

I

• j=1

θ(j)1Xj(x)

  .

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 0.02 0.04 0.06 0.08 0.1 0.12

❊①❛♠♣❧❡ ✷ ✭t♦ ❢♦❧❧♦✇✮✴ ❆❞❛♣t❡❞ ■♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❜② ❲❛♥❣✲▲❛♥❞❛✉ ❛♣♣r♦❛❝❤❡s

• ❚❤❡ ❛❧❣♦r✐t❤♠

❙❛♠♣❧❡✿ Xt+1 ∼ Pθt(Xt, ·), ✇❤❡r❡ πθPθ = πθ ❙❆ s❝❤❡♠❡✿ θt+1 = θt + γt+1H(θt, Xt+1)

• ■♥ t❤✐s ❡①❛♠♣❧❡✱ ❛ ❢❛♠✐❧② ♦❢ tr❛♥s✐t✐♦♥ ❦❡r♥❡❧s {Pθ, θ ∈ Θ} s✉❝❤ t❤❛t

∀θ, Pθ ✐♥✈❛r✐❛♥t ✇✳r✳t✳ πθ

• ❈♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts✿

✭❋✳✲❏♦✉r❞❛✐♥✲▲❡❧✐❡✈r❡✲❙t♦❧t③✲✷✵✶✺✱✷✵✶✼✱✷✵✶✽✮

✲ θt ❝♦♥✈❡r❣❡s t♦ θ⋆ ❛✳s✳❀ ✲ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ Xt ❝♦♥✈❡r❣❡s t♦ dπθ⋆❀ ✲ θt ✐s ❛♥ ❡st✐♠❛t❡ ♦❢ t❤❡ ✐♠♣♦rt❛♥❝❡ r❛t✐♦ [dπ/dπθ⋆](x)✱ ❝♦♥st❛♥t ❛❧♦♥❣ ❡❛❝❤ Xi✳

■s ❛ ✏t❤❡♦r②✑ r❡q✉✐r❡❞ ❄ ❨❊❙ ✦ ❝♦♥✈❡r❣❡♥❝❡ ❝❛♥ ❜❡ ❧♦st ❜② t❤❡ ❛❞❛♣t✐♦♥ ♠❡❝❛♥✐s♠ ❊✈❡♥ ✐♥ ❛ s✐♠♣❧❡ ❝❛s❡ ✇❤❡♥ ∀θ ∈ Θ, Pθ ✐♥✈❛r✐❛♥t ✇rt dπ, ♦♥❡ ❝❛♥ ❞❡✜♥❡ ❛ s✐♠♣❧❡ ❛❞❛♣t✐♦♥ ♠❡❝❛♥✐s♠ Xt+1|♣❛st1:t ∼ Pθt(Xt, ·) θt ∈ σ(X1:t) s✉❝❤ t❤❛t lim

t E [f(Xt)] =

• f dπ.

❆ {0, 1}✲✈❛❧✉❡❞ ❝❤❛✐♥ {Xt}t ❞❡✜♥❡❞ ❜② Xt+1 ∼ PXt(Xt, ·) ✇❤❡r❡ t❤❡ tr❛♥s✐t✐♦♥ ♠❛tr✐❝❡s ❛r❡ P0 =

• t0

(1 − t0) (1 − t0) t0

• P1 =
• t1

(1 − t1) (1 − t1) t1

• ❚❤❡♥ P0 ❛♥❞ P1 ❛r❡ ✐♥✈❛r✐❛♥t ✇✳r✳t [1/2, 1/2] ❜✉t {Xt} ✐s ❛ ▼❛r❦♦✈ ❝❤❛✐♥ ✐♥✈❛r✐❛♥t ✇✳r✳t✳ [t1, t0]

❈♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts

• ❚❤❡ ❢r❛♠❡✇♦r❦✿

✲ ❛ ✜❧tr❛t✐♦♥ {Ft, t ≥ 0} ♦♥ (Ω, A, P) ✲ ❛ Ft✲❛❞❛♣t❡❞ X × Θ✲✈❛❧✉❡❞ ♣r♦❝❡ss {(Xt, θt), t ≥ 0} ❞❡✜♥❡❞ ♦♥ (Ω, A) ✲ ❛ ❢❛♠✐❧② ♦❢ tr❛♥s✐t✐♦♥ ❦❡r♥❡❧s {Pθ, θ ∈ Θ} ♦♥ ❛ ❣❡♥❡r❛❧ st❛t❡ s♣❛❝❡ (X, X) ✲ ❛ ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥ s❛t✐s❢②✐♥❣ E

• f(Xt+1)|Ft
• =
• Pθt(Xt, dx)f(x)

f ❜♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❛ ❝♦♥✈❡r❣❡♥❝❡ ✭✐♥ s♦♠❡ s❡♥s❡✮ ♦❢ t❤❡ ❦❡r♥❡❧s {Pθt, t ≥ 0}

• ◗✉❡st✐♦♥s✿

✲ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ ♦❢ Xt ❄ ✲ ❧✐♠✐t t❤❡♦r❡♠s

• ❍❡r❡❛❢t❡r✿

✲ ❢♦❝✉s ♦♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ ✲ θ ∈ Θ ⊆ Rp

❆ss✉♠♣t✐♦♥s ✭✶✴✸✮ ■♥✈❛r✐❛♥t ❞✐str✐❜✉t✐♦♥ ∀θ ∈ Θ, ∃πθ s✳t✳ t❤❡ ❦❡r♥❡❧ Pθ ✐♥✈❛r✐❛♥t ✇rt πθ

❆ss✉♠♣t✐♦♥s ✭✷✴✸✮ ✭●❡♥❡r❛❧✐③❡❞✮ ❈♦♥t❛✐♥♠❡♥t ❝♦♥❞✐t✐♦♥

• ❯♥✐❢♦r♠✲✐♥✲θ ❡r❣♦❞✐❝✐t② ❝♦♥❞✐t✐♦♥

sup

θ∈Θ

P r

θ (x; ·) − πθTV ≤ Cρr

■♥ ♣r❛❝t✐❝❡✿ ❛ ❞r✐❢t ❛♥❞ ❛ ♠✐♥♦r✐③❛t✐♦♥ ❝♦♥❞✐t✐♦♥ → ❡①♣❧✐❝✐t ❝♦♥tr♦❧ ♦❢ ❡r❣♦❞✐❝✐t② PθV ≤ λθV + bθ, Pθ(x, ·) ≥ δθνθ(·) ❢♦r x ∈ {V ≤ 2bθ(1 − λθ)−1 − 1}

• ❆ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥❞✐t✐♦♥✿ ❢♦r ❛♥② ǫ > 0✱ t❤❡r❡ ❡①✐sts ❛ ♥♦♥✲❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡

rǫ s✳t✳ limt rǫ(t)/t = 0 ❛♥❞ lim sup

t

E

• P rǫ(t)

θt−rǫ(t)(Xt−rǫ(t); ·) − πθt−rǫ(t)TV

• ≤ ǫ

✲ ❈♦♥tr♦❧❧❡❞ r❛t❡ ♦❢ ❣r♦✇t❤✲✐♥✲θ ❤❡r❡✱ rǫ(t) = t• P r

θ (x; ·) − πθTV ≤ Cθ ρr θ

t−τ θt < ∞ ❛✳s✳ lim sup

t

t−˜

τ

• Cθt ∨ (1 − ρθt)−1

< ∞ ❛✳s✳

❆ss✉♠♣t✐♦♥s ✭✸✴✸✮ ✭●❡♥❡r❛❧✐③❡❞✮ ❉✐♠✐♥✐s❤✐♥❣ ❛❞❛♣t❛t✐♦♥ ❝♦♥❞✐t✐♦♥

• ❲❤❡♥ ✉♥✐❢♦r♠✲✐♥✲θ ❡r❣♦❞✐❝ ❝♦♥❞✐t✐♦♥✱ ❝❤❡❝❦

lim

t E

D(θt, θt−1) = 0

✇❤❡r❡ D(θ, θ′) = supx Pθ(x, ·) − Pθ′(x, ·)TV✳

• ❖t❤❡r✇✐s❡✿ ❢♦r ❛♥② ǫ > 0✱

lim

t E

 

rǫ(t)−1

• j=0

D(θt−rǫ(t)+j, θt−rǫ(t))

  = 0

• ■♥ ♣r❛❝t✐❝❡

✲ Pr♦✈❡ ❛ ▲✐♣s❝❤✐t③ ♣r♦♣❡rt② D(θ, θ′) ≤ C θ − θ′ ✲ ❯s❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ θt ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ (Xℓ)ℓ≤t ❛♥❞ ♣♦ss✐❜❧② ♦t❤❡r ✧❡①t❡r♥❛❧✧ s❛♠♣❧❡❞ ♣♦✐♥ts ✲ ❘❡q✉✐r❡ ❝♦♥tr♦❧s ♦❢ t❤❡ ❢♦r♠ E [W(Xℓ)]✱ s♦❧✈❡❞ ❡✳❣✳ ❜② ❞r✐❢t ✐♥❡q✉❛❧✐t✐❡s E

W(Xℓ)|Fℓ−1 = Pθℓ−1W(Xℓ−1) ≤ λθℓ−1W(Xℓ−1) + bθℓ−1

❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❉✐str✐❜✉t✐♦♥ ✭✇❤❡♥ πθ = π ❢♦r ❛♥② θ✮ ❯♥❞❡r t❤❡s❡ ❝♦♥❞✐t✐♦♥s✱ ❢♦r ❛♥② ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥ f✱ lim

t E [f(Xt)] =

• f(x) dπ(x)

■♥ t❤❡ ❧✐t❡r❛t✉r❡

✭❘♦❜❡rts✲❘♦s❡♥t❤❛❧✱✷✵✵✼❀ ❋✳✲▼♦✉❧✐♥❡s✲Pr✐♦✉r❡t✱✷✵✶✷❀ ❋✳✲▼♦✉❧✐♥❡s✲Pr✐♦✉r❡t✲❱❛♥❞❡❦❡r❦❤♦✈❡✱✷✵✶✷✮

• ❇❛s❡❞ ♦♥ str❡♥❣❤t❡♥❡❞ ✧❝♦♥t❛✐♥♠❡♥t✧ ❛♥❞ ✧❞✐♠✐♥✐s❤✐♥❣ ❛❞❛♣t❛t✐♦♥✧ ❝♦♥❞✐t✐♦♥s✱

✲ str♦♥❣ ▲❛✇ ♦❢ ▲❛r❣❡ ◆✉♠❜❡rs ❢♦r {f(Xt)}t ❛♥❞ {f(θt, Xt)}t ✲ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ ❢♦r {f(Xt)}t

• ■♥ t❤❡ ❝❛s❡ θ ∈ Rp ❜✉t ❛❧s♦ ✐♥ ♠♦r❡ ❣❡♥❡r❛❧ s✐t✉❛t✐♦♥s✿ θ ♠❛② ❜❡ ❛ ❞✐str✐❜✉t✐♦♥

❝❛s❡ ♦❢ ✧✐♥t❡r❛❝t✐♥❣✧ ▼❈▼❈✳ ✭❉❡❧ ▼♦r❛❧✲❉♦✉❝❡t✱ ✷✵✶✵✮

• ❘❡s✉❧ts ✐♥ t❤❡ ❝❛s❡ ❡❛❝❤ ❦❡r♥❡❧ Pθ ❤❛s ✐ts ♦✇♥ ✐♥✈❛r✐❛♥t ❞✐str✐❜✉t✐♦♥ πθ✿

lim

t E [f(Xt)] = lim t

• f(x) dπθt(x)

✭❘❍❙✱ ❛ss✉♠❡❞ ❝♦♥st❛♥t ❛✳s✳✮

❆s ❛ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤✐s ♣❛rt ■

• ❆ ❢❛♠✐❧② ♦❢ ❡r❣♦❞✐❝ ❦❡r♥❡❧s❀ t♦ ❛❞❛♣t t❤❡ ♣❛r❛♠❡t❡rs θt✱ ❛ str❛t❡❣② ❜❛s❡❞ ♦♥

t❤❡ ♣❛st ♦❢ t❤❡ ❛❧❣♦r✐t❤♠

• ❚❤❡ ❡❛s✐❡st s✐t✉❛t✐♦♥✿

✲ ✉♥✐❢♦r♠✲✐♥✲θ ❡r❣♦❞✐❝✐t② ❝♦♥❞✐t✐♦♥s

• ❋❛r ♠♦r❡ ✢❡①✐❜❧❡ ❜✉t ❛❧s♦ ♠♦r❡ t❡❝❤♥✐❝❛❧✿

✲ ❛♥ ❡r❣♦❞✐❝ ❜❡❤❛✈✐♦r ❞❡♣❡♥❞✐♥❣ ♦♥ θ ✲ ❛♥❞ t❤❡ r❛t❡ ♦❢ ❣r♦✇t❤ ♦❢ t → |θt| ✐s ❝♦♥tr♦❧❧❡❞

• ■♥ ❜♦t❤ ❝❛s❡s✱

✲ t❤❡ ✉♣❞❛t✐♥❣ r✉❧❡ θt − → θt+1 ✐s s✳t✳ t❤❡ ❛❞❛♣t✐♦♥ ✐s ❞✐♠✐♥✐s❤✐♥❣ ❛❧♦♥❣ ✐t❡r❛t✐♦♥s✳

✳

P❛rt ■■✳ ❙t♦❝❤❛st✐❝ ❆♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤ ▼❛r❦♦✈✐❛♥ ❞②♥❛♠✐❝s

✳

❙t♦❝❤❛st✐❝ ❆♣♣r♦①✐♠❛t✐♦♥ ✭❙❆✮ ♠❡t❤♦❞s

• ❉❡s✐❣♥❡❞ t♦ s♦❧✈❡ ♦♥ Θ ⊆ Rp✿

h(θ) = 0 ✇❤❡♥ h ✐s ♥♦t ❡①♣❧✐❝✐t ❜✉t h(θ) =

• X H(θ, x) dπθ(x)
• ❆❧❣♦r✐t❤♠✿

✲ ❈❤♦♦s❡✿ ❛ ❞❡t❡r♠✐♥✐st✐❝ ♣♦s✐t✐✈❡ ✭❞❡❝r❡❛s✐♥❣✮ s❡q✉❡♥❝❡ {γt}t s✳t✳

t γt = +∞

✲ ■♥✐t✐❛❧✐s❛t✐♦♥✿ θ0 = θinit ∈ Θ, X0 = xinit ✲ ❯♥t✐❧ ❝♦♥✈❡r❣❡♥❝❡✿ Xt+1 ∼ Pθt(Xt, ·) θt+1 = θt + γt+1 H(θt, Xt+1) ✇❤❡r❡ Pθ ✐♥✈✳ ✇rt πθ✳ ❇❡✇❛r❡✦ ❛ ❜✐❛s❡❞ ❛♣♣r♦①✐♠❛t✐♦♥ E

• H(θt, Xt+1)|Ft
• − h(θt) =
• X
• Pθt(Xt, dx) − dπθt(x)
• H(θt, x)

❈♦♥✈❡r❣❡♥❝❡ ❛♥❛❧②s✐s ❢♦r ❙❆✿ t❤❡ s✉❝❝❡ss✐✈❡ st❡♣s ✶✲ ❚❤❡ s❡q✉❡♥❝❡ {θt}t ✐s st❛❜❧❡ ✐✳❡✳ ✭✇✳♣✳✶✮ t❤❡r❡ ❡①✐sts ❛ ❝♦♠♣❛❝t s✉❜s❡t K ♦❢ Θ s✉❝❤ t❤❛t θt ∈ K ❢♦r ❛♥② t✳ ✷✲ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ {θt}t t♦ L ✭♦r t♦ ❛ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ L❀ ♦r t♦ ❛ ♣♦✐♥t θ⋆ ∈ L✮✳

• ❘❡q✉✐r❡❞✿ t❤❡r❡ ❡①✐sts ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ▲②❛♣✉♥♦✈ ❢✉♥❝t✐♦♥ V ✿

V (θt+1) ≤ V (θt) − γt+1 φ2(θt) + γt+1 Wt+1

s✐❣♥❡❞

. ✇❤♦s❡ ❧❡✈❡❧ s❡ts ❛r❡ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ Θ✱ ❛♥❞ φ ✐s s✳t✳ t❤❛t inf

❝♦♠♣❛❝t⊂Θ\L φ2 > 0

✇✐t❤ L := {φ2 = 0} ⊂ {V ≤ M⋆}.

❈♦♥tr♦❧ ♦❢ t❤❡ ✧♥♦✐s❡✧✿ sup

t

|

t

• k=1

γk+1 (H(θk, Xk+1) − h(θk)) |

{θ: V(θ) ≤ M 1 } {θ: V(θ) ≤ M 0 } L

❙t❛❜✐❧✐t②✿ ❛ ❝r✉❝✐❛❧ ♣♦✐♥t ✲ ❉✐✛❡r❡♥t str❛t❡❣✐❡s

• ❙t❛❜❧❡ ❜② ❞❡✜♥✐t✐♦♥✿

θt+1 = θt + γt+1H(θt, Xt+1) q✉✐t❡ ✉♥❧✐❦❡❧②

• ❋♦r❝❡ t❤❡ st❛❜✐❧✐t② ❜② ❛ ♣r♦❥❡❝t✐♦♥ ♦♥ ❛ ❝♦♠♣❛❝t s✉❜s❡t K

θt+1 = ΠK

• θt + γt+1H(θt, Xt+1)
• ▲✐♠✐t✐♥❣ ♣♦✐♥ts✿ ✐♥ L ∩ K✳ ❍♦✇ t♦ ❝❤♦♦s❡ K ❄
• ❯s❡ t❤❡ ❈❤❡♥✬s t❡❝❤♥✐q✉❡✿ ♣r♦❥❡❝t✐♦♥ ♦♥ ❣r♦✇✐♥❣ ❝♦♠♣❛❝t s✉❜s❡ts✳

✭❈❤❡♥✲❩❤✉✱ ✶✾✽✻✮

❙❡❧❢✲st❛❜✐❧✐③❡❞ ❙t♦❝❤❛st✐❝ ❆♣♣r♦①✐♠❛t✐♦♥ ✭t❤❡ ❈❤❡♥✬s t❡❝❤♥✐q✉❡✮ ❈❤♦♦s❡ ❝♦♠♣❛❝t s✉❜s❡ts {Ki}i≥0 s✳t✳

i Ki = Θ ❛♥❞ Ki ⊂ Ki+1✳

• ✭❙t❛rt ✲ ❇❧♦❝❦ ✶✮✿

θ0 = θinit ∈ K0 ❛♥❞ X0 = xinit ❛♥❞ r❡♣❡❛t ❢♦r t ≥ 0 Xt+1 ∼ Pθt(Xt, ·) θt+1 = θt + γt+1H(θt, Xt+1) ✉♥t✐❧ θt+1 / ∈ K0✳ ❙❡t T1 = t + 1✳

• · · ·
• ✭❙t♦♣ ✫ r❡✲st❛rt✱ ❇❧♦❝❦ q + 1✮

θTq = θinit, XTq = xinit ❛♥❞ r❡♣❡❛t ❢♦r t ≥ 0 XTq+t+1 ∼ PθTq+t(XTq+t, ·) θTq+t+1 = θTq+t + γq+t+1H(θTq+t, XTq+t+1) ✉♥t✐❧ θTq+t+1 / ∈ Kq✳ ❙❡t Tq+1 = Tq + t + 1✳

• · · ·

❲❤❡♥ ❞♦❡s s❡❧❢✲st❛❜✐❧✐③❛t✐♦♥ ❙❆ ✧✇♦r❦✧ ❄ ✭✶✴✸✮

• ■❢ t❤❡ ♥✉♠❜❡r ♦❢ ✧st♦♣ ✫ r❡✲st❛rt✧ ✐s ✜♥✐t❡✱ ✐t ✇♦r❦s ✦

t❤❡♥ t❤❡r❡ ❡①✐sts L s✳t✳ ✭❛✮ {θt}t ✐s ✐♥ t❤❡ ❝♦♠♣❛❝t s❡t KL ✭❜✮ ❢♦r ❛♥② t ≥ 0 XTL+t+1 ∼ PθTL+t(XTL+t, ·) θTL+t+1 = θTL+t + γL+t+1 H(θTL+t, XTL+t+1)

• ■❢ ✐t ✐s ♥♦t✿ ❛s ✐❢ ✇✐t❤ ρt+1 ← γL+t+1 ❢♦r ❛r❜✐tr❛r✐❧② ❧❛r❣❡ L✿

θ0 = θinit, X0 = xinit, Xt+1 ∼ Pθt(Xt, ·), θt+1 = θt + ρt+1H(θt, Xt+1)

✐❢ ✐t ✐s ♥♦t ✜♥✐t❡ ✭✷✴✸✮

• ▲❡♠♠❛✳ ❆ss✉♠❡ t❤❛t h ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ t❤❡r❡ ❡①✐sts ❛ C1 ♥♦♥✲♥❡❣❛t✐✈❡ ❢✉♥❝✲

t✐♦♥ V s✳t✳ ✲ t❤❡ ❧❡✈❡❧ s❡ts {V ≤ M} ❛r❡ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ Θ❀ ✲ t❤❡ s❡t L = {∇V ; h = 0} ✐s ❝♦♠♣❛❝t❀ ✲ ❛♥❞ ♦♥ Lc✱ ∇V ; h < 0✳ ▲❡t θinit ∈ K′✳ ▲❡t M0 ❜❡ s✳t✳ K0 ∪ L ⊂ {V ≤ M0}✳ ❚❤❡r❡ ❡①✐st δ, λ > 0 s✉❝❤ t❤❛t

  sup

1≤k≤t

ρk ≤ λ, sup

1≤k≤t

|

k

• j=1

ρj

• H(θj, Xj+1) − h(θj)
• | ≤ δ

  =

⇒ θ1:t ∈ {V ≤ M0 + 1}.

✐❢ ✐t ✐s ♥♦t ✜♥✐t❡ ✭✸✴✸✮

• Pr♦✈❡ ❢♦r ❛♥② ❝♦♠♣❛❝t s✉❜s❡t K

lim

L→∞ P(xinit,θinit),γL+•

 sup

k≥1

1θ1:k∈K

• k
• j=1

γL+j

• H(θj, Xj+1) − h(θj)
• > δ

  = 0.

• ❆♣♣❧② t❤❡ ❇✲❚ ✐♥❡q✉❛❧✐t②

E(xinit,θinit)

 sup

k≥1

1θ1:k∈K

• k
• j=1

ρj

• H(θj, Xj+1) − h(θj)
• 



• ❯s❡ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❜❡❧♦✇ ❛♥❞ ✉s❡ ♣r♦♣❡rt✐❡s ♦♥ ❝♦♥tr♦❧❧❡❞ ▼❛r❦♦✈ ❝❤❛✐♥s

s✐♥❝❡ Xj+1 ∼ Pθj(Xj, ·)✳

❚❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥✿ ˆ Hθ s✳t✳ ˆ Hθ(x) − Pθ ˆ Hθ(x) = H(θ, x) − h(θ)✳

k

• j=1

ρj

• H(θj, Xj+1) − h(θj)

=

k

• j=1

ρj

ˆ

Hθj(Xj+1) − Pθj ˆ Hθj(Xj) +

k

• j=1

ρj

• Pθj ˆ

Hθj(Xj) − Pθj+1 ˆ Hθj+1(Xj+1) +

k

• j=1

ρj

• Pθj+1 ˆ

Hθj+1(Xj+1) − Pθj ˆ Hθj(Xj+1)

■♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ❙❆ ✇✐t❤ ▼❛r❦♦✈✐❛♥ ❞②♥❛♠✐❝s

✭❋✱✷✵✶✺❀ ❋✳✲▼♦✉❧✐♥❡s✲❙❝❤r❡❝❦✲❱✐❤♦❧❛✱✷✵✶✻❀ ▼♦rr❛❧✲❇✐❛♥❝❤✐✲❋✳✱✷✵✶✼❀ ❈r❡♣❡②✲❋✳✲●♦❜❡t✲❙t❛③❤✐♥s❦✐✱✷✵✶✽✮

• ■♥ t❤❡ ❝❛s❡ θ ∈ Rp✱

✲ ❙✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✲ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠s ✭❛❧♦♥❣ ❛ ❝♦♥✈❡r❣✐♥❣ ♣❛t❤✮ ❢♦r ❜♦t❤ t❤❡ s❡q✉❡♥❝❡ {θt}t ❛♥❞ t❤❡ ❛✈❡r❛❣❡❞ s❡q✉❡♥❝❡ ¯ θt = 1 t

t

• k=1

θk ✲ ❉✐str✐❜✉t❡❞ ❙❆

• ❙♦♠❡ r❡s✉❧ts ✐♥ t❤❡ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❢r❛♠❡✇♦r❦ ❢♦r θ❀ ✇✐t❤ ✐✳✐✳❞✳ ❞②♥❛♠✐❝s✳

✳

P❛rt ■■■✿ ❙t♦❝❤❛st✐❝ Pr♦①✐♠❛❧✲●r❛❞✐❡♥t ❛❧❣♦r✐t❤♠s

✳

P❡♥❛❧✐③❡❞ ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✐♥❢❡r❡♥❝❡

• ❆♥ ✐♥tr❛❝t❛❜❧❡ ❧♦❣✲❧✐❦❡❧✐❤♦♦❞ ♦❢ t❤❡ ♦❜s❡r✈❛t✐♦♥s Y1:n

✲ ❊①✿ ▲❛t❡♥t ✈❛r✐❛❜❧❡ ♠♦❞❡❧s ℓ(Y1:n; θ) = log

• p(Y1:n, x; θ) dν(x)
• ❆ s♣❛rs✐t② ❝♦♥❞✐t✐♦♥ ♦♥ θ t❤r♦✉❣❤ ❛ ♥♦♥ s♠♦♦t❤ ❛♥❞ ❝♦♥✈❡① ♣❡♥❛❧t②

✲ ❊①✲✶✿ g(θ) = λθ1

• ❙♦❧✈❡

argminθ

  

f(θ)

s♠♦♦t❤✱ ✐♥tr❛❝t❛❜❧❡

+ g(θ)

♥♦♥ s♠♦♦t❤✱ ❝♦♥✈❡①✱ tr❛❝t❛❜❧❡

  

▼♦♥t❡ ❈❛r❧♦ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r ❣r❛❞✐❡♥t✲❜❛s❡❞ ♦♣t✐♠✐③❛t✐♦♥ ♠❡t❤♦❞s

• ■♥ t❤✐s ✧❧❛t❡♥t ✈❛r✐❛❜❧❡ ♠♦❞❡❧✧ ❡①❛♠♣❧❡✱ ❛s ✐♥ ♠❛♥② ❡①❛♠♣❧❡s✿

∇f(θ) =

• H(θ, x) dπθ(x)

✇❤❡r❡ πθ✿ ✭t❤❡ ❛ ♣♦st❡r✐♦r✐✮ ❞✐str✐❜✉t✐♦♥ ❦♥♦✇♥ ✉♣ t♦ ❛ ♥♦r♠❛❧✐③❛t✐♦♥ ❝♦♥st❛♥t

✭❞❡♣❡♥❞❛♥❝❡ ✉♣♦♥ Y1:n ♦♠✐tt❡❞✮

֒ → ✐♥tr❛❝t❛❜❧❡ ✐♥t❡❣r❛❧✳

• ■❢ t❤❡ ❣r❛❞✐❡♥t ✇❡r❡ ❛✈❛✐❧❛❜❧❡✿ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠

ut+1 = Proxγt+1 g

• ut − γt+1∇f(ut)
• Proxγ g(τ) = argminu
• g(u) + 1

2γu − τ2

• ❙✐♥❝❡ ✐t ✐s ♥♦t✿ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠

θt+1 = Proxγt+1 g

 θt − γt+1

1 mt+1

mt+1

• k=1

H(θt, Xt+1,k)

 

Xt+1,k ∼ Pθt(Xt+1,k−1, ·)

◗✉❡st✐♦♥s ✲ ❉♦❡s t❤❡ st♦❝❤❛st✐❝ ✈❡rs✐♦♥ ✐♥❤❡r✐t t❤❡ s❛♠❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ❛s t❤❡ ✭❡①❛❝t✮

• r❛❞✐❡♥t✲Pr♦①✐♠❛❧ ❛❧❣♦r✐t❤♠ ❄ ✐✳❡✳ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ {θt}t

✲ ❍♦✇ t♦ ❝❤♦♦s❡ t❤❡ st❡♣s✐③❡ s❡q✉❡♥❝❡ {γt}t❄ ✲ ❍♦✇ t♦ ❝❤♦♦s❡ t❤❡ ♥✉♠❜❡r ♦❢ ▼♦♥t❡ ❈❛r❧♦ s❛♠♣❧❡s mt ❄ ■s t❤❡ ✧❙❆ r❡❣✐♠❡✧ ✭✐✳❡✳ mt = 1✮ ♣♦ss✐❜❧❡ ❄ ✲ ❲❤❛t ❛❜♦✉t t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❄ ✲ ■s t❤❡ r❛t❡ ✐♠♣r♦✈❡❞ ❜② ◆❡st❡r♦✈✲❜❛s❡❞ ❛❝❝❡❧❡r❛t✐♦♥ ❄ ✐s ✐t ✐♠♣r♦✈❡❞ ❜② ❆✈❡r✲ ❛❣✐♥❣ t❡❝❤♥✐q✉❡s ❄

❆ss✉♠♣t✐♦♥s

• ❖♥ t❤❡ ♥♦♥✲s♠♦♦t❤ ♣❛rt✿ g : Rp → [0, ∞]✱ ✐s ♥♦t ✐❞❡♥t✐❝❛❧❧② +∞✱ ❝♦♥✈❡① ❛♥❞

❧♦✇❡r s❡♠✐✲❝♦♥t✐♥✉♦✉s✳

• ❖♥ t❤❡ s♠♦♦t❤ ♣❛rt✿ f : Rp → R ✐s ❝♦♥✈❡①✱ C1 ♦♥ Rp ❛♥❞ t❤❡r❡ ❡①✐sts L s✉❝❤

t❤❛t ❢♦r ❛♥② θ, θ′ ∇f(θ) − ∇f(θ′) ≤ L θ − θ′

• ❖♥ t❤❡ s♦❧✉t✐♦♥ s❡t✿ L := argminθ(f + g) = {θ = Proxγ g(θ − γ∇f(θ))} ✐s ❛ ♥♦♥

❡♠♣t② s✉❜s❡t ♦❢ Θ = {g < ∞}✳

• ❖♥ t❤❡ st❡♣s✐③❡✿

t γt = ∞

• ❖♥ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ηt+1 := m−1

t+1

mt+1

j=1 H(θt, Xt+1,j) − h(θt)✿ t❤❡ s❡r✐❡s

• t

γtηt,

• t

γ2

t ηt2,

• t

γt Tγt(θt−1); ηt ❝♦♥✈❡r❣❡

❘❡s✉❧ts ✭❆t❝❤❛❞❡✲❋✲▼♦✉❧✐♥❡s✱ ✷✵✶✼✮ θt+1 = Proxγt+1 g

 θt − γt+1

1 mt+1

mt+1

• k=1

H(θt, Xt+1,k)

 

• ❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✐t❡r❛t❡s {θt}t✿ t❤❡r❡ ❡①✐sts θ⋆ ∈ L s✳t✳ limt θt = θ⋆✳
• ❋♦r ♥♦♥✲♥❡❣❛t✐✈❡ ✇❡✐❣❤ts {ak,t}k s✳t✳ t

k=1 ak,t = 1✱ ❛♥ ❡①♣❧✐❝✐t ✉♣♣❡r ❜♦✉♥❞ ♦❢

(f + g)

• ¯

θt

• − min(f + g) ≤

t

• k=1

ak,t (f + g)(θk) − min(f + g) ≤ · · · ✇❤❡r❡ ¯ θt =

t

• k=1

ak,t θk

❘❛t❡s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦♥ t❤❡ ❢✉♥❝t✐♦♥❛❧ (f + g)(θt) − min(f + g)

• ❘❛t❡ ♦❢ t❤❡ ❡①❛❝t ❛❧❣♦r✐t❤♠✿ O(1/t)
• ❙t♦❝❤❛st✐❝ ✈❡rs✐♦♥ ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ❜❛t❝❤ s✐③❡

✲ ❆❢t❡r t ✐t❡r❛t✐♦♥s✱ t❤❡ s❛♠❡ r❛t❡ ❜② ❝❤♦♦s✐♥❣ γt = γ mt = t θt = t−1

t

• k=1

θk ✲ ❇❯❚ t❤❡ t♦t❛❧ ▼♦♥t❡ ❈❛r❧♦ ❝♦st ✐s O(t2)✿ ❝♦♠♣❧❡①✐t② O(1/ √ t)✳

• ❙t♦❝❤❛st✐❝ ✈❡rs✐♦♥ ✇✐t❤ ✜①❡❞ ❜❛t❝❤ s✐③❡

✲ ❆❢t❡r t ✐t❡r❛t♦♥s✱ ❛ r❛t❡ O(1/ √ t) ❜② ❝❤♦♦s✐♥❣ γt = t−1/2 mt = m θt = t−1

t

• k=1

θk ✲ t❤❡ t♦t❛❧ ▼♦♥t❡ ❈❛r❧♦ ❝♦st ✐s O(t)✿ ❝♦♠♣❧❡①✐t② O(1/ √ t)✳

◆❡st❡r♦✈✬s ❛❝❝❡❧❡r❛t✐♦♥✱ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛❧ ut+1 = Proxγt+1 g

• ϑt − γt+1 ∇f(ϑt)
• ϑt = ut + µt−1 − 1

µt (ut − ut−1) ✇❤❡r❡ µt = O(t)✳

• ❘❛t❡ ♦❢ t❤❡ ❡①❛❝t ❛❧❣♦r✐t❤♠✿ O(1/t2)
• ❙t♦❝❤❛st✐❝ ✈❡rs✐♦♥ ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ❜❛t❝❤ s✐③❡

✲ ❆❢t❡r t ✐t❡r❛t✐♦♥s✱ t❤❡ s❛♠❡ r❛t❡ ❜② ❝❤♦♦s✐♥❣ γt = γ mt = t3 θt ✲ ❇❯❚ t❤❡ t♦t❛❧ ▼♦♥t❡ ❈❛r❧♦ ❝♦st ✐s O(t4)✿ ❝♦♠♣❧❡①✐t② O(1/ √ t)✳

❈♦♥❝❧✉s✐♦♥

✭❋✳✲❘✐ss❡r✲❆t❝❤❛❞❡✲▼♦✉❧✐♥❡s✱✷✵✶✽❀❋✲❖❧❧✐❡r✲❙❛♠s♦♥✱✷✵✶✾✮

• ✐✈❡♥ ❛ ▼♦♥t❡ ❈❛r❧♦ ❜✉❞❣❡t t✿
• ❚❤❡ ✭♣❡rt✉r❜❡❞✮ Pr♦①✐♠❛❧✲●r❛❞✐❡♥t ❝♦♠❜✐♥❡❞ ✇✐t❤ ❛✈❡r❛❣✐♥❣ ❤❛s t❤❡ s❛♠❡

❝♦♠♣❧❡①✐t② ❛s t❤❡ ✭♣❡rt✉r❜❡❞✮ ◆❡st❡r♦✈✲❛❝❝❡❧❡r❛t❡❞ Pr♦①✐♠❛❧✲●r❛❞✐❡♥t✿ O(1/ √ t)

• ◆❡st❡r♦✈✲❛❝❝❡❧❡r❛t❡❞ Pr♦①✐♠❛❧✲●r❛❞✐❡♥t ✰ ✇❡✐❣❤t❡❞ ❛✈❡r❛❣✐♥❣ str❛t❡❣✐❡s✿ ♥♦

✐♠♣r♦✈❡♠❡♥t

• ◆❡st❡r♦✈✲❛❝❝❡❧❡r❛t❡❞ Pr♦①✐♠❛❧✲●r❛❞✐❡♥t ✰ ♦t❤❡r r❡❧❛①❛t✐♦♥s µt = O(td) ❢♦r s♦♠❡

d ∈ (0, 1)✿ ♥♦ ✐♠♣r♦✈❡♠❡♥t

❏♦✐♥t ✇♦r❦s ✇✐t❤ ✲ ❨✈❡s ❆t❝❤❛❞❡✱ ❯♥✐✈✳ ▼✐❝❤✐❣❛♥✱ ❋r❛♥❝❡ ✲ ❏❡❛♥✲❋r❛♥ç♦✐s ❆✉❥♦❧✱ ❯♥✐✈✳ ❇♦r❞❡❛✉①✱ ❋r❛♥❝❡ ✲ ❙té♣❤❛♥❡ ❈r❡♣❡②✱ ❯♥✐✈✳ ❊✈r②✱ ❋r❛♥❝❡ ✲ ❈❤❛r❧❡s ❉♦ss❛❧✱ ❯♥✐✈✳ ❚♦✉❧♦✉s❡✱ ❋r❛♥❝❡ ✲ P✐❡rr❡ ●❛❝❤✱ ❯♥✐✈✳ ❚♦✉❧♦✉s❡✱ ❋r❛♥❝❡ ✲ ❊♠♠❛♥✉❡❧ ●♦❜❡t✱ ❊❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✱ ❋r❛♥❝❡ ✲ ❇❡♥❥❛♠✐♥ ❏♦✉r❞❛✐♥✱ ❊◆P❈✱ ❋r❛♥❝❡ ✲ ❚♦♥② ▲❡❧✐❡✈r❡✱ ❊◆P❈✱ ❋r❛♥❝❡ ✲ ❊r✐❝ ▼♦✉❧✐♥❡s✱ ❊❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✱ ❋r❛♥❝❡ ✲ P✐❡rr❡ Pr✐♦✉r❡t✱ ❯♥✐✈✳ P❛r✐s ✻✱ ❋r❛♥❝❡ ✲ ▲❛✉r❡♥t ❘✐ss❡r✱ ❯♥✐✈✳ ❚♦✉❧♦✉s❡✱ ❋r❛♥❝❡ ✲ ❆❞❡❧✐♥❡ ❙❛♠s♦♥✱ ❯♥✐✈✳ ●r❡♥♦❜❧❡✲❆❧♣❡s✱ ❋r❛♥❝❡ ✲ ❆♠❛♥❞✐♥❡ ❙❝❤r❡❝❦✱ ❚❡❧❡❝♦♠ P❛r✐s❚❡❝❤✱ ❋r❛♥❝❡ ✲ ●❛❜r✐❡❧ ❙t♦❧t③✱ ❊◆P❈✱ ❋r❛♥❝❡ ✲ P✐❡rr❡ ❱❛♥❞❡❦❡r❦❤♦✈❡✱ ❯♥✐✈✳ ▼❛r♥❡✲❧❛✲❱❛❧❧é❡✱ ❋r❛♥❝❡ ✲ ▼❛tt✐ ❱✐❤♦❧❛✱ ❯♥✐✈✳ ❏②✈❛s❦②❧❛✱ ❋✐♥❧❛♥❞