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❉❡❡♣ ▲❡❛r♥✐♥❣ ❜♦♦❦✱ ❜② ■❛♥ ●♦♦❞❢❡❧❧♦✇✱ ❨♦s❤✉❛ ❇❡♥❣✐♦ ❛♥❞ ❆❛r♦♥ ❈♦✉r✈✐❧❧❡

❈❤❛♣t❡r ✻ ✿❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ❇❡♥♦✐t ▼❛ssé ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶ ✴ ✷✺

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ✭❛♥❞ ❝❧❛ss✐✜❝❛t✐♦♥✮

■♥♣✉t ✈❡❝t♦r ① ❖✉t♣✉t ✈❡❝t♦r ② P❛r❛♠❡t❡rs ❲❡✐❣❤t ❲ ❛♥❞ ❜✐❛s ❜ Pr❡❞✐❝t✐♦♥ ✿ ② = ❲⊤① + ❜

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✷ ✴ ✷✺

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ✭❛♥❞ ❝❧❛ss✐✜❝❛t✐♦♥✮

■♥♣✉t ✈❡❝t♦r ① ❖✉t♣✉t ✈❡❝t♦r ② P❛r❛♠❡t❡rs ❲❡✐❣❤t ❲ ❛♥❞ ❜✐❛s ❜ Pr❡❞✐❝t✐♦♥ ✿ ② = ❲⊤① + ❜

① ✉ ② ❲ ❜

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✷ ✴ ✷✺

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ✭❛♥❞ ❝❧❛ss✐✜❝❛t✐♦♥✮

■♥♣✉t ✈❡❝t♦r ① ❖✉t♣✉t ✈❡❝t♦r ② P❛r❛♠❡t❡rs ❲❡✐❣❤t ❲ ❛♥❞ ❜✐❛s ❜ Pr❡❞✐❝t✐♦♥ ✿ ② = ❲⊤① + ❜

①✶ ①✷ ①✸ ✉✶ ✉✷ ②✶ ②✷ ❲✶✶ . . . ❲✷✸ ❜✶ ❜✷

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✷ ✴ ✷✺

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ✭❛♥❞ ❝❧❛ss✐✜❝❛t✐♦♥✮

■♥♣✉t ✈❡❝t♦r ① ❖✉t♣✉t ✈❡❝t♦r ② P❛r❛♠❡t❡rs ❲❡✐❣❤t ❲ ❛♥❞ ❜✐❛s ❜ Pr❡❞✐❝t✐♦♥ ✿ ② = ❲⊤① + ❜

①✶ ①✷ ①✸ ②✶ ②✷ ❲, ❜

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✷ ✴ ✷✺

▲✐♥❡❛r r❡❣r❡ss✐♦♥ ✭❛♥❞ ❝❧❛ss✐✜❝❛t✐♦♥✮

❆❞✈❛♥t❛❣❡s ❊❛s② t♦ ✉s❡ ❊❛s② t♦ tr❛✐♥✱ ❧♦✇ r✐s❦ ♦❢ ♦✈❡r✜tt✐♥❣ ❉r❛✇❜❛❝❦s ❙♦♠❡ ♣r♦❜❧❡♠s ❛r❡ ✐♥❤❡r❡♥t❧② ♥♦♥✲❧✐♥❡❛r

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✸ ✴ ✷✺

❙♦❧✈✐♥❣ ❳❖❘

▲✐♥❡❛r r❡❣r❡ss♦r

①✶ ①✷ ② ❲ ❜

❚❤❡r❡ ✐s ♥♦ ✈❛❧✉❡ ❢♦r ❲ ❛♥❞ ❜ s✉❝❤ t❤❛t ①✶ ①✷ ✵ ✶ ✷ ❲ ①✶ ①✷ ❜ ①♦r ①✶ ①✷

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✹ ✴ ✷✺

❙♦❧✈✐♥❣ ❳❖❘

▲✐♥❡❛r r❡❣r❡ss♦r

①✶ ①✷ ② ❲, ❜

❚❤❡r❡ ✐s ♥♦ ✈❛❧✉❡ ❢♦r ❲ ❛♥❞ ❜ s✉❝❤ t❤❛t ∀(①✶, ①✷) ∈ {✵, ✶}✷ ❲⊤ ①✶ ①✷

• + ❜ = ①♦r(①✶, ①✷)

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✹ ✴ ✷✺

❙♦❧✈✐♥❣ ❳❖❘

❲❤❛t ❛❜♦✉t✳✳✳ ❄

①✶ ①✷ ✉✶ ✉✷ ② ❲, ❜ ❱, ❝

❙tr✐❝t❧② ❡q✉✐✈❛❧❡♥t ✿ ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❧✐♥❡❛r ♦♣❡r❛t✐♦♥ ✐s st✐❧❧ ❛ ❧✐♥❡❛r ♦♣❡r❛t✐♦♥

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✺ ✴ ✷✺

❙♦❧✈✐♥❣ ❳❖❘

❲❤❛t ❛❜♦✉t✳✳✳ ❄

①✶ ①✷ ✉✶ ✉✷ ② ❲, ❜ ❱, ❝

❙tr✐❝t❧② ❡q✉✐✈❛❧❡♥t ✿ ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ❧✐♥❡❛r ♦♣❡r❛t✐♦♥ ✐s st✐❧❧ ❛ ❧✐♥❡❛r ♦♣❡r❛t✐♦♥

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✺ ✴ ✷✺

❙♦❧✈✐♥❣ ❳❖❘

❆♥❞ ❛❜♦✉t✳✳✳ ❄

①✶ ①✷ ✉✶ ✉✷ ❤✶ ❤✷ ② ❲, ❜ φ ❱, ❝

■♥ ✇❤✐❝❤ φ(①) = ♠❛①{✵, ①}

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✻ ✴ ✷✺

❙♦❧✈✐♥❣ ❳❖❘

❆♥❞ ❛❜♦✉t✳✳✳ ❄

①✶ ①✷ ✉✶ ✉✷ ❤✶ ❤✷ ② ❲, ❜ φ ❱, ❝

■♥ ✇❤✐❝❤ φ(①) = ♠❛①{✵, ①} ■t ✐s ♣♦ss✐❜❧❡ ✦ ❲✐t❤ ❲ = ✶ ✶ ✶ ✶

• ✱ ❜ =

✵ −✶

• ✱ ❱ =

✶ −✷

• ❛♥❞ ❝ = ✵✱

❱φ(❲① + ❜) = ①♦r(①✶, ①✷)

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✻ ✴ ✷✺

◆❡✉r❛❧ ♥❡t✇♦r❦ ✇✐t❤ ♦♥❡ ❤✐❞❞❡♥ ❧❛②❡r

❈♦♠♣❛❝t r❡♣r❡s❡♥t❛t✐♦♥

① ❤ ② ❲, ❜, φ ❱, ❝

◆❡✉r❛❧ ♥❡t✇♦r❦ ❍✐❞❞❡♥ ❧❛②❡r ✇✐t❤ ♥♦♥✲❧✐♥❡❛r✐t② → ❝❛♥ r❡♣r❡s❡♥t ❜r♦❛❞❡r ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✼ ✴ ✷✺

❯♥✐✈❡rs❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ t❤❡♦r❡♠

❚❤❡♦r❡♠ ❆ ♥❡✉r❛❧ ♥❡t✇♦r❦ ✇✐t❤ ♦♥❡ ❤✐❞❞❡♥ ❧❛②❡r ❝❛♥ ❛♣♣r♦①✐♠❛t❡ ❛♥② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ▼♦r❡ ❢♦r♠❛❧❧②✱ ❣✐✈❡♥ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❢ : ❈♥ → R♠ ✇❤❡r❡ ❈♥ ✐s ❛ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ R♥✱ ∀ε, ∃❢ ε

◆◆ : ① → ❑

• ✐=✶

✈✐φ(✇✐

⊤① + ❜✐) + ❝

s✉❝❤ t❤❛t ∀① ∈ ❈♥, ||❢ (①) − ❢ ε

◆◆(①)|| < ε

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✽ ✴ ✷✺

Pr♦❜❧❡♠s

❖❜t❛✐♥✐♥❣ t❤❡ ♥❡t✇♦r❦ ❚❤❡ ✉♥✐✈❡rs❛❧ t❤❡♦r❡♠ ❣✐✈❡s ♥♦ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❍❖❲ t♦ ♦❜t❛✐♥ s✉❝❤ ❛ ♥❡t✇♦r❦ ❙✐③❡ ♦❢ t❤❡ ❤✐❞❞❡♥ ❧❛②❡r ❤ ❱❛❧✉❡s ♦❢ ❲ ❛♥❞ ❜ ❯s✐♥❣ t❤❡ ♥❡t✇♦r❦ ❊✈❡♥ ✐❢ ✇❡ ✜♥❞ ❛ ✇❛② t♦ ♦❜t❛✐♥ t❤❡ ♥❡t✇♦r❦✱ t❤❡ s✐③❡ ♦❢ t❤❡ ❤✐❞❞❡♥ ❧❛②❡r ♠❛② ❜❡ ♣r♦❤✐❜✐t✐✈❡❧② ❧❛r❣❡✳

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✾ ✴ ✷✺

Pr♦❜❧❡♠s

❖❜t❛✐♥✐♥❣ t❤❡ ♥❡t✇♦r❦ ❚❤❡ ✉♥✐✈❡rs❛❧ t❤❡♦r❡♠ ❣✐✈❡s ♥♦ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❍❖❲ t♦ ♦❜t❛✐♥ s✉❝❤ ❛ ♥❡t✇♦r❦ ❙✐③❡ ♦❢ t❤❡ ❤✐❞❞❡♥ ❧❛②❡r ❤ ❱❛❧✉❡s ♦❢ ❲ ❛♥❞ ❜ ❯s✐♥❣ t❤❡ ♥❡t✇♦r❦ ❊✈❡♥ ✐❢ ✇❡ ✜♥❞ ❛ ✇❛② t♦ ♦❜t❛✐♥ t❤❡ ♥❡t✇♦r❦✱ t❤❡ s✐③❡ ♦❢ t❤❡ ❤✐❞❞❡♥ ❧❛②❡r ♠❛② ❜❡ ♣r♦❤✐❜✐t✐✈❡❧② ❧❛r❣❡✳

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✾ ✴ ✷✺

❉❡❡♣ ♥❡✉r❛❧ ♥❡t✇♦r❦

❲❤② ❉❡❡♣ ❄ ▲❡t✬s st❛❝❦ ❧ ❤✐❞❞❡♥ ❧❛②❡rs ♦♥❡ ❛❢t❡r t❤❡ ♦t❤❡r❀ ❧ ✐s ❝❛❧❧❡❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♥❡t✇♦r❦✳

① ❤✶

✳ ✳ ✳

❤❧ ② ❲✶, ❜✶, φ ❲✷, ❜✷, φ ❲❧, ❜❧, φ ❱, ❝

Pr♦♣❡rt✐❡s ♦❢ ❉◆◆ ❚❤❡ ✉♥✐✈❡rs❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ t❤❡♦r❡♠ ❛❧s♦ ❛♣♣❧② ❙♦♠❡ ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛ ❉◆◆ ✇✐t❤ ◆ ❤✐❞❞❡♥ ✉♥✐t✱ ❛♥❞ ✇♦✉❧❞ r❡q✉✐r❡ O(❡◆) ❤✐❞❞❡♥ ✉♥✐ts t♦ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ s❤❛❧❧♦✇ ♥❡t✇♦r❦✳

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✵ ✴ ✷✺

❙✉♠♠❛r②

❈♦♠♣❛r✐s♦♥ ▲✐♥❡❛r ❝❧❛ss✐✜❡r

− ▲✐♠✐t❡❞ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ♣♦✇❡r + ❙✐♠♣❧❡

❙❤❛❧❧♦✇ ◆❡✉r❛❧ ♥❡t✇♦r❦ ✭❊①❛❝t❧② ♦♥❡ ❤✐❞❞❡♥ ❧❛②❡r✮

+ ❯♥❧✐♠✐t❡❞ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ♣♦✇❡r − ❙♦♠❡t✐♠❡s ♣r♦❤✐❜✐t✐✈❡❧② ✇✐❞❡

❉❡❡♣ ◆❡✉r❛❧ ♥❡t✇♦r❦

+ ❯♥❧✐♠✐t❡❞ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ♣♦✇❡r + ❘❡❧❛t✐✈❡❧② s♠❛❧❧ ♥✉♠❜❡r ♦❢ ❤✐❞❞❡♥ ✉♥✐ts ♥❡❡❞❡❞

❘❡♠❛✐♥✐♥❣ ♣r♦❜❧❡♠ ❍♦✇ t♦ ❣❡t t❤✐s ❉◆◆ ❄

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✶ ✴ ✷✺

❙✉♠♠❛r②

❈♦♠♣❛r✐s♦♥ ▲✐♥❡❛r ❝❧❛ss✐✜❡r

− ▲✐♠✐t❡❞ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ♣♦✇❡r + ❙✐♠♣❧❡

❙❤❛❧❧♦✇ ◆❡✉r❛❧ ♥❡t✇♦r❦ ✭❊①❛❝t❧② ♦♥❡ ❤✐❞❞❡♥ ❧❛②❡r✮

+ ❯♥❧✐♠✐t❡❞ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ♣♦✇❡r − ❙♦♠❡t✐♠❡s ♣r♦❤✐❜✐t✐✈❡❧② ✇✐❞❡

❉❡❡♣ ◆❡✉r❛❧ ♥❡t✇♦r❦

+ ❯♥❧✐♠✐t❡❞ r❡♣r❡s❡♥t❛t✐♦♥❛❧ ♣♦✇❡r + ❘❡❧❛t✐✈❡❧② s♠❛❧❧ ♥✉♠❜❡r ♦❢ ❤✐❞❞❡♥ ✉♥✐ts ♥❡❡❞❡❞

❘❡♠❛✐♥✐♥❣ ♣r♦❜❧❡♠ ❍♦✇ t♦ ❣❡t t❤✐s ❉◆◆ ❄

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✶ ✴ ✷✺

❖♥ t❤❡ ♣❛t❤ ♦❢ ❣❡tt✐♥❣ ♠② ♦✇♥ ❉◆◆

❍②♣❡r♣❛r❛♠❡t❡rs ❋✐rst✱ ✇❡ ♥❡❡❞ t♦ ❞❡✜♥❡ t❤❡ ❛r❝❤✐t❡❝t✉r❡ ♦❢ t❤❡ ❉◆◆ ❚❤❡ ❞❡♣t❤ ❧ ❚❤❡ s✐③❡ ♦❢ t❤❡ ❤✐❞❞❡♥ ❧❛②❡rs ♥✶✱ ✳ ✳ ✳ ✱ ♥❧ ❚❤❡ ❛❝t✐✈❛t✐♦♥ ❢✉♥❝t✐♦♥ φ ❚❤❡ ♦✉t♣✉t ✉♥✐t P❛r❛♠❡t❡rs ❲❤❡♥ t❤❡ ❛r❝❤✐t❡❝t✉r❡ ✐s ❞❡✜♥❡❞✱ ✇❡ ♥❡❡❞ t♦ tr❛✐♥ t❤❡ ❉◆◆ ❲✶✱ ❜✶✱ ✳ ✳ ✳ ✱ ❲❧✱ ❜❧

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✷ ✴ ✷✺

❖♥ t❤❡ ♣❛t❤ ♦❢ ❣❡tt✐♥❣ ♠② ♦✇♥ ❉◆◆

❍②♣❡r♣❛r❛♠❡t❡rs ❋✐rst✱ ✇❡ ♥❡❡❞ t♦ ❞❡✜♥❡ t❤❡ ❛r❝❤✐t❡❝t✉r❡ ♦❢ t❤❡ ❉◆◆ ❚❤❡ ❞❡♣t❤ ❧ ❚❤❡ s✐③❡ ♦❢ t❤❡ ❤✐❞❞❡♥ ❧❛②❡rs ♥✶✱ ✳ ✳ ✳ ✱ ♥❧ ❚❤❡ ❛❝t✐✈❛t✐♦♥ ❢✉♥❝t✐♦♥ φ ❚❤❡ ♦✉t♣✉t ✉♥✐t P❛r❛♠❡t❡rs ❲❤❡♥ t❤❡ ❛r❝❤✐t❡❝t✉r❡ ✐s ❞❡✜♥❡❞✱ ✇❡ ♥❡❡❞ t♦ tr❛✐♥ t❤❡ ❉◆◆ ❲✶✱ ❜✶✱ ✳ ✳ ✳ ✱ ❲❧✱ ❜❧

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✷ ✴ ✷✺

❍②♣❡r♣❛r❛♠❡t❡rs

❚❤❡ ❞❡♣t❤ ❧ ❚❤❡ s✐③❡ ♦❢ t❤❡ ❤✐❞❞❡♥ ❧❛②❡rs ♥✶✱ ✳ ✳ ✳ ✱ ♥❧

❙tr♦♥❣❧② ❞❡♣❡♥❞ ♦♥ t❤❡ ♣r♦❜❧❡♠ t♦ s♦❧✈❡

❚❤❡ ❛❝t✐✈❛t✐♦♥ ❢✉♥❝t✐♦♥

❘❡▲❯ ❣ ① ♠❛① ✵ ① ❙✐❣♠♦✐❞ ① ✶ ❡

① ✶

▼❛♥② ♦t❤❡rs ✿ t❛♥❤✱ ❘❇❋✱ s♦❢t♣❧✉s✳✳✳

❚❤❡ ♦✉t♣✉t ✉♥✐t

▲✐♥❡❛r ♦✉t♣✉t ② ❱ ❤❧ ❝

❋♦r r❡❣r❡ss✐♦♥ ✇✐t❤ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥ ② ② ■

❙✐❣♠♦✐❞ ♦✉t♣✉t ② ✇ ❤❧ ❜

❋♦r ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ ❇❡r♥♦✉✐❧❧✐ ❞✐str✐❜✉t✐♦♥ P ② ✶ ① ②

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✸ ✴ ✷✺

❍②♣❡r♣❛r❛♠❡t❡rs

❚❤❡ ❞❡♣t❤ ❧ ❚❤❡ s✐③❡ ♦❢ t❤❡ ❤✐❞❞❡♥ ❧❛②❡rs ♥✶✱ ✳ ✳ ✳ ✱ ♥❧

❙tr♦♥❣❧② ❞❡♣❡♥❞ ♦♥ t❤❡ ♣r♦❜❧❡♠ t♦ s♦❧✈❡

❚❤❡ ❛❝t✐✈❛t✐♦♥ ❢✉♥❝t✐♦♥ φ

❘❡▲❯ ❣ : ① → ♠❛①{✵, ①} ❙✐❣♠♦✐❞ σ : ① → (✶ + ❡−①)−✶ ▼❛♥② ♦t❤❡rs ✿ t❛♥❤✱ ❘❇❋✱ s♦❢t♣❧✉s✳✳✳

❚❤❡ ♦✉t♣✉t ✉♥✐t

▲✐♥❡❛r ♦✉t♣✉t ② ❱ ❤❧ ❝

❋♦r r❡❣r❡ss✐♦♥ ✇✐t❤ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥ ② ② ■

❙✐❣♠♦✐❞ ♦✉t♣✉t ② ✇ ❤❧ ❜

❋♦r ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ ❇❡r♥♦✉✐❧❧✐ ❞✐str✐❜✉t✐♦♥ P ② ✶ ① ②

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✸ ✴ ✷✺

❍②♣❡r♣❛r❛♠❡t❡rs

❚❤❡ ❞❡♣t❤ ❧ ❚❤❡ s✐③❡ ♦❢ t❤❡ ❤✐❞❞❡♥ ❧❛②❡rs ♥✶✱ ✳ ✳ ✳ ✱ ♥❧

❙tr♦♥❣❧② ❞❡♣❡♥❞ ♦♥ t❤❡ ♣r♦❜❧❡♠ t♦ s♦❧✈❡

❚❤❡ ❛❝t✐✈❛t✐♦♥ ❢✉♥❝t✐♦♥ φ

❘❡▲❯ ❣ : ① → ♠❛①{✵, ①} ❙✐❣♠♦✐❞ σ : ① → (✶ + ❡−①)−✶ ▼❛♥② ♦t❤❡rs ✿ t❛♥❤✱ ❘❇❋✱ s♦❢t♣❧✉s✳✳✳

❚❤❡ ♦✉t♣✉t ✉♥✐t

▲✐♥❡❛r ♦✉t♣✉t E[②] = ❱⊤❤❧ + ❝

❋♦r r❡❣r❡ss✐♦♥ ✇✐t❤ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥ ② ∼ N(E[②], ■)

❙✐❣♠♦✐❞ ♦✉t♣✉t ˆ ② = σ(✇⊤❤❧ + ❜)

❋♦r ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ ❇❡r♥♦✉✐❧❧✐ ❞✐str✐❜✉t✐♦♥ P(② = ✶|①) = ˆ ②

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✸ ✴ ✷✺

P❛r❛♠❡t❡rs ❚r❛✐♥✐♥❣

❖❜❥❡❝t✐✈❡ ▲❡t✬s ❞❡✜♥❡ θ = (❲✶, ❜✶, . . . , ❲❧, ❜❧)✳ ❲❡ s✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ s❡t ♦❢ ✐♥♣✉ts ❳ = (①✶, . . . , ①◆) ❛♥❞ ❛ s❡t ♦❢ ❡①♣❡❝t❡❞ ♦✉t♣✉ts ❨ = (②✶, . . . , ②◆)✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ✜♥❞ ❛ ♥❡✉r❛❧ ♥❡t✇♦r❦ ❢◆◆ s✉❝❤ t❤❛t ∀✐, ❢◆◆(①✐, θ) ≃ ②✐.

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✹ ✴ ✷✺

P❛r❛♠❡t❡rs ❚r❛✐♥✐♥❣

❈♦st ❢✉♥❝t✐♦♥ ❚♦ ❡✈❛❧✉❛t❡ t❤❡ ❡rr♦r t❤❛t ♦✉r ❝✉rr❡♥t ♥❡t✇♦r❦ ♠❛❦❡s✱ ❧❡t✬s ❞❡✜♥❡ ❛ ❝♦st ❢✉♥❝t✐♦♥ L(❳, ❨, θ)✳ ❚❤❡ ❣♦❛❧ ❜❡❝♦♠❡s t♦ ✜♥❞ ❛r❣♠✐♥

θ

L(❳, ❨, θ) ▲♦ss ❢✉♥❝t✐♦♥ ❙❤♦✉❧❞ r❡♣r❡s❡♥t ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ❡✈❡r② ②✐ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢◆◆(①✐, θ) ▼❡❛♥ sq✉❛r❡ ❡rr♦r ✭r❛r❡✮ ❈r♦ss✲❡♥tr♦♣②

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✺ ✴ ✷✺

P❛r❛♠❡t❡rs ❚r❛✐♥✐♥❣

❋✐♥❞ t❤❡ ♠✐♥✐♠✉♠ ❚❤❡ ❜❛s✐❝ ✐❞❡❛ ❝♦♥s✐sts ✐♥ ❝♦♠♣✉t✐♥❣ ˆ θ s✉❝❤ t❤❛t ∇θL(❳, ❨, ˆ θ) = ✵. ❚❤✐s ✐s ❞✐✣❝✉❧t t♦ s♦❧✈❡ ❛♥❛❧②t✐❝❛❧❧② ❡✳❣✳ ✇❤❡♥ θ ❤❛✈❡ ♠✐❧❧✐♦♥s ♦❢ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✳

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✻ ✴ ✷✺

P❛r❛♠❡t❡rs ❚r❛✐♥✐♥❣

• r❛❞✐❡♥t ❞❡s❝❡♥t

▲❡t✬s ✉s❡ ❛ ♥✉♠❡r✐❝❛❧ ✇❛② t♦ ♦♣t✐♠✐③❡ θ✱ ❝❛❧❧❡❞ t❤❡ ❣r❛❞✐❡♥t ❞❡s❝❡♥t ✭s❡❝t✐♦♥ ✹✳✸✮✳ ❚❤❡ ✐❞❡❛ ✐s t❤❛t ❢ (θ − ε✉) ≃ ❢ (θ) − ε✉⊤∇❢ (θ) ❙♦ ✐❢ ✇❡ t❛❦❡ ✉ = ∇❢ (θ)✱ ✇❡ ❤❛✈❡ ✉⊤✉ > ✵ ❛♥❞ t❤❡♥ ❢ (θ − ε✉) ≃ ❢ (θ) − ε✉⊤✉ < ❢ (θ). ■❢ ❢ ✐s ❛ ❢✉♥❝t✐♦♥ t♦ ♠✐♥✐♠✐③❡✱ ✇❡ ❤❛✈❡ ❛♥ ✉♣❞❛t❡ r✉❧❡ t❤❛t ✐♠♣r♦✈❡s ♦✉r ❡st✐♠❛t❡✳

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✼ ✴ ✷✺

P❛r❛♠❡t❡rs ❚r❛✐♥✐♥❣

• r❛❞✐❡♥t ❞❡s❝❡♥t ❛❧❣♦r✐t❤♠

✶ ❍❛✈❡ ❛♥ ❡st✐♠❛t❡ ˆ

θ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs

✷ ❈♦♠♣✉t❡ ∇θL(❳, ❨, ˆ

θ)

✸ ❯♣❞❛t❡ ˆ

θ ← − ˆ θ − ε∇θL

✹ ❘❡♣❡❛t st❡♣ ✷✲✸ ✉♥t✐❧ ∇θL < t❤r❡s❤♦❧❞

Pr♦❜❧❡♠ ❍♦✇ t♦ ❡st✐♠❛t❡ ❡✣❝✐❡♥t❧② ❳ ❨ ❄ ❇❛❝❦✲♣r♦♣❛❣❛t✐♦♥ ❛❧❣♦r✐t❤♠

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✽ ✴ ✷✺

P❛r❛♠❡t❡rs ❚r❛✐♥✐♥❣

• r❛❞✐❡♥t ❞❡s❝❡♥t ❛❧❣♦r✐t❤♠

✶ ❍❛✈❡ ❛♥ ❡st✐♠❛t❡ ˆ

θ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs

✷ ❈♦♠♣✉t❡ ∇θL(❳, ❨, ˆ

θ)

✸ ❯♣❞❛t❡ ˆ

θ ← − ˆ θ − ε∇θL

✹ ❘❡♣❡❛t st❡♣ ✷✲✸ ✉♥t✐❧ ∇θL < t❤r❡s❤♦❧❞

Pr♦❜❧❡♠ ❍♦✇ t♦ ❡st✐♠❛t❡ ❡✣❝✐❡♥t❧② ∇θL(❳, ❨, ˆ θ) ❄ ❇❛❝❦✲♣r♦♣❛❣❛t✐♦♥ ❛❧❣♦r✐t❤♠

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✽ ✴ ✷✺

❇❛❝❦✲♣r♦♣❛❣❛t✐♦♥ ❢♦r P❛r❛♠❡t❡r ▲❡❛r♥✐♥❣

❈♦♥s✐❞❡r t❤❡ ❛r❝❤✐t❡❝t✉r❡✿

① ✇✶ ✇✷ ② φ φ

✇✐t❤ ❢✉♥❝t✐♦♥✿ ② = φ

• ✇✷φ(✇✶①)
• ,

s♦♠❡ tr❛✐♥✐♥❣ ♣❛✐rs ❚ =

• ˆ

①♥, ˆ ②♥ ◆

♥=✶✱ ❛♥❞

❛♥ ❛❝t✐✈❛t✐♦♥✲❢✉♥❝t✐♦♥ φ()✳ ▲❡❛r♥ ✇✶, ✇✷ s♦ t❤❛t✿ ❋❡❡❞✐♥❣ ˆ ①♥ r❡s✉❧ts ˆ ②♥✳

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✶✾ ✴ ✷✺

Pr❡r❡q✉✐s✐t❡✿ ❞✐✛❡r❡♥t✐❛❜❧❡ ❛❝t✐✈❛t✐♦♥ ❢✉♥❝t✐♦♥

❋♦r ❧❡❛r♥✐♥❣ t♦ ❜❡ ♣♦ss✐❜❧❡ φ() ❤❛s t♦ ❜❡ ❞✐✛❡r❡♥t✐❛❜❧❡✳ ▲❡t φ′(①) = ∂φ(①)

∂①

❞❡♥♦t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ φ(①)✳ ❋♦r ❡①❛♠♣❧❡ ✇❤❡♥ φ(①) = ❘❡❧✉(①) ✇❡ ❤❛✈❡✿

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✷✵ ✴ ✷✺

• r❛❞✐❡♥t✲❜❛s❡❞ ▲❡❛r♥✐♥❣

▼✐♥✐♠✐③❡ t❤❡ ❧♦ss ❢✉♥❝t✐♦♥ L(✇✶, ✇✷, ❚)✳ ❲❡ ✇✐❧❧ ❧❡❛r♥ t❤❡ ✇❡✐❣❤ts ❜② ✐t❡r❛t✐♥❣✿ ✇✶ ✇✷ ✉♣❞❛t❡❞ = ✇✶ ✇✷

• − γ

  

∂L ∂✇✶ ∂L ∂✇✷

   , ✭✶✮ L ✐s t❤❡ ❧♦ss ❢✉♥❝t✐♦♥ ✭♠✉st ❜❡ ❞✐✛❡r❡♥t✐❛❜❧❡✮✿ ■♥ ❞❡t❛✐❧ ✐s L(✇✶, ✇✷, ❚) ❛♥❞ ✇❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥t✭s✮ ❛t ✇✶, ✇✷✳ γ ✐s t❤❡ ❧❡❛r♥✐♥❣ r❛t❡ ✭❛ s❝❛❧❛r t②♣✐❝❛❧❧② ❦♥♦✇♥✮✳

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✷✶ ✴ ✷✺

❇❛❝❦✲♣r♦♣❛❣❛t✐♦♥

❈❛❧❝✉❧❛t❡ ✐♥t❡r♠❡❞✐❛t❡ ✈❛❧✉❡s ♦♥ ❛❧❧ ✉♥✐ts✿

✶

❛ = ✇✶ˆ ①♥✳

✷

❜ = φ(✇✶ˆ ①♥)✳

✸

❝ = ✇✷φ(✇✶ˆ ①♥)✳

✹

❞ = φ

• ✇✷φ(✇✶ˆ

①♥)

• ✳

✺

L(❞) = L

• φ
• ✇✷φ(✇✶ˆ

①♥)

• ✳

❚❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❛r❡✿

✻

∂L(❞) ∂❞

= L′(❞)✳

✼

∂❞ ∂❝ = φ′(✇✷φ(✇✶ˆ

①♥))✳

✽

∂❝ ∂❜ = ✇✷✳

✾

∂❜ ∂❛ = φ′(✇✶ˆ

①♥)✳

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✷✷ ✴ ✷✺

❈❛❧❝✉❧❛t✐♥❣ t❤❡ ●r❛❞✐❡♥ts ■

❆♣♣❧② ❝❤❛✐♥ r✉❧❡✿ ∂L ∂✇✶ =∂L ∂❞ ∂❞ ∂❝ ∂❝ ∂❜ ∂❜ ∂❛ ∂❛ ✇✶ , ∂L(❞) ∂✇✷ =∂L(❞) ∂❞ ∂❞ ∂❝ ∂❝ ✇✷ . ❙t❛rt t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ❢r♦♠ ❧❡❢t✲t♦✲r✐❣❤t✳ ❲❡ ♣r♦♣❛❣❡ t❤❡ ❣r❛❞✐❡♥ts ✭♣❛rt✐❛❧ ♣r♦❞✉❝ts✮ ❢r♦♠ t❤❡ ❧❛st ❧❛②❡r t♦✇❛r❞s t❤❡ ✐♥♣✉t✳

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✷✸ ✴ ✷✺

❈❛❧❝✉❧❛t✐♥❣ t❤❡ ●r❛❞✐❡♥ts ■

❆♣♣❧② ❝❤❛✐♥ r✉❧❡✿ ∂L ∂✇✶ =∂L ∂❞ ∂❞ ∂❝ ∂❝ ∂❜ ∂❜ ∂❛ ∂❛ ✇✶ , ∂L(❞) ∂✇✷ =∂L(❞) ∂❞ ∂❞ ∂❝ ∂❝ ✇✷ . ❙t❛rt t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ❢r♦♠ ❧❡❢t✲t♦✲r✐❣❤t✳ ❲❡ ♣r♦♣❛❣❡ t❤❡ ❣r❛❞✐❡♥ts ✭♣❛rt✐❛❧ ♣r♦❞✉❝ts✮ ❢r♦♠ t❤❡ ❧❛st ❧❛②❡r t♦✇❛r❞s t❤❡ ✐♥♣✉t✳

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✷✸ ✴ ✷✺

❈❛❧❝✉❧❛t✐♥❣ t❤❡ ●r❛❞✐❡♥ts

❆♥❞ ❜❡❝❛✉s❡ ✇❡ ❤❛✈❡ ◆ tr❛✐♥✐♥❣ ♣❛✐rs✿ ∂L ∂✇✶ =

◆

• ♥=✶

∂L(❞♥) ∂❞♥ ∂❞♥ ∂❝♥ ∂❝♥ ∂❜♥ ∂❜♥ ∂❛♥ ∂❛♥ ✇✶ , ∂L ∂✇✷ =

◆

• ♥=✶

∂L(❞♥) ∂❞♥ ∂❞♥ ∂❝♥ ∂❝♥ ✇✷ .

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✷✹ ✴ ✷✺

❚❤❛♥❦ ②♦✉ ✦

❇❡♥♦✐t ▼❛ssé✱ ❉✐♦♥②ss♦s ❑♦✉♥❛❞❡s✲❇❛st✐❛♥ ❉❡❡♣ ❋❡❡❞❢♦r✇❛r❞ ◆❡t✇♦r❦s ✷✺ ✴ ✷✺