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Jul 31, 2023 367 likes •789 views

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SLIDE 1

▲❡❛r♥✐♥❣ t♦ ❊①♣❧♦r❡

❏♦❤♥ ▲❛♥❣❢♦r❞ ▼✐❝r♦s♦❢t ❘❡s❡❛r❝❤ ❲♦r❦s❤♦♣ ♦♥ ❇✐❣ ❉❛t❛ ❛♥❞ ❙t❛t✐st✐❝❛❧ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣✱ ❏❛♥✉❛r② ✷✻✱ ✷✵✶✹ ❣✐t ❝❧♦♥❡ ❣✐t✿✴✴❣✐t❤✉❜✳❝♦♠✴❏♦❤♥▲❛♥❣❢♦r❞✴✈♦✇♣❛❧❴✇❛❜❜✐t✳❣✐t

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SLIDE 2

❊①❛♠♣❧❡s ♦❢ ■♥t❡r❛❝t✐✈❡ ▲❡❛r♥✐♥❣

❘❡♣❡❛t❡❞❧②✿ ✶✳ ❆ ✉s❡r ❝♦♠❡s t♦ ▼✐❝r♦s♦❢t ✭✇✐t❤ ❤✐st♦r② ♦❢ ♣r❡✈✐♦✉s ✈✐s✐ts✱ ■P ❛❞❞r❡ss✱ ❞❛t❛ r❡❧❛t❡❞ t♦ ❛♥ ❛❝❝♦✉♥t✮ ✷✳ ▼✐❝r♦s♦❢t ❝❤♦♦s❡s ✐♥❢♦r♠❛t✐♦♥ t♦ ♣r❡s❡♥t ✭✉r❧s✱ ❛❞s✱ ♥❡✇s st♦r✐❡s✮ ✸✳ ❚❤❡ ✉s❡r r❡❛❝ts t♦ t❤❡ ♣r❡s❡♥t❡❞ ✐♥❢♦r♠❛t✐♦♥ ✭❝❧✐❝❦s ♦♥ s♦♠❡t❤✐♥❣✱ ❝❧✐❝❦s✱ ❝♦♠❡s ❜❛❝❦ ❛♥❞ ❝❧✐❝❦s ❛❣❛✐♥✱✳✳✳✮ ❍♦✇ ❞♦ ②♦✉ ❝❤♦♦s❡ ❝♦♥t❡♥t❄

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SLIDE 3

❆♥♦t❤❡r ❊①❛♠♣❧❡✿ ❈❧✐♥✐❝❛❧ ❉❡❝✐s✐♦♥ ▼❛❦✐♥❣

❘❡♣❡❛t❡❞❧②✿ ✶✳ ❆ ♣❛t✐❡♥t ❝♦♠❡s t♦ ❛ ❞♦❝t♦r ✇✐t❤ s②♠♣t♦♠s✱ ♠❡❞✐❝❛❧ ❤✐st♦r②✱ t❡st r❡s✉❧ts ✷✳ ❚❤❡ ❞♦❝t♦r ❝❤♦♦s❡s ❛ tr❡❛t♠❡♥t ✸✳ ❚❤❡ ♣❛t✐❡♥t r❡s♣♦♥❞s t♦ ✐t ❚❤❡ ❞♦❝t♦r ✇❛♥ts ❛ ♣♦❧✐❝② ❢♦r ❝❤♦♦s✐♥❣ t❛r❣❡t❡❞ tr❡❛t♠❡♥ts ❢♦r ✐♥❞✐✈✐❞✉❛❧ ♣❛t✐❡♥ts✳

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SLIDE 4

❚❤❡ ❈♦♥t❡①t✉❛❧ ❇❛♥❞✐t ❙❡tt✐♥❣

❋♦r t = ✶, . . . , ❚✿ ✶✳ ❚❤❡ ✇♦r❧❞ ♣r♦❞✉❝❡s s♦♠❡ ❝♦♥t❡①t ① ∈ ❳ ✷✳ ❚❤❡ ❧❡❛r♥❡r ❝❤♦♦s❡s ❛♥ ❛❝t✐♦♥ ❛ ∈ ❆ ✸✳ ❚❤❡ ✇♦r❧❞ r❡❛❝ts ✇✐t❤ r❡✇❛r❞ r❛ ∈ [✵, ✶]

  • ♦❛❧✿

▲❡❛r♥ ❛ ❣♦♦❞ ♣♦❧✐❝② ❢♦r ❝❤♦♦s✐♥❣ ❛❝t✐♦♥s ❣✐✈❡♥ ❝♦♥t❡①t✳

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SLIDE 5

❚❤❡ ✏❉✐r❡❝t ♠❡t❤♦❞✑

❯s❡ ♣❛st ❞❛t❛ t♦ ❧❡❛r♥ ❛ r❡✇❛r❞ ♣r❡❞✐❝t♦r ˆ r(①, ❛)✱ ❛♥❞ ❛❝t ❛❝❝♦r❞✐♥❣ t♦ ❛r❣ ♠❛①❛ ˆ r(①, ❛)✳ ❊①❛♠♣❧❡✿ ❉❡♣❧♦②❡❞ ♣♦❧✐❝② ❛❧✇❛②s t❛❦❡s ❛✶ ♦♥ ①✶ ❛♥❞ ❛✷ ♦♥ ①✷✳ ❖❜s❡r✈❡❞✴❊st✐♠❛t❡❞✴❚r✉❡ ❛✶ ❛✷ ①✶ ✽✴ ✽✴ ✽ ❄✴✶ ①✷ ✴ ✸ ✷ ✴ ✷

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SLIDE 6

❚❤❡ ✏❉✐r❡❝t ♠❡t❤♦❞✑

❯s❡ ♣❛st ❞❛t❛ t♦ ❧❡❛r♥ ❛ r❡✇❛r❞ ♣r❡❞✐❝t♦r ˆ r(①, ❛)✱ ❛♥❞ ❛❝t ❛❝❝♦r❞✐♥❣ t♦ ❛r❣ ♠❛①❛ ˆ r(①, ❛)✳ ❊①❛♠♣❧❡✿ ❉❡♣❧♦②❡❞ ♣♦❧✐❝② ❛❧✇❛②s t❛❦❡s ❛✶ ♦♥ ①✶ ❛♥❞ ❛✷ ♦♥ ①✷✳ ❖❜s❡r✈❡❞✴❊st✐♠❛t❡❞✴❚r✉❡ ❛✶ ❛✷ ①✶ ✽✴ ✽✴ ✽ ❄✴✶ ①✷ ✴ ✸ ✷ ✴ ✷

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SLIDE 7

❚❤❡ ✏❉✐r❡❝t ♠❡t❤♦❞✑

❯s❡ ♣❛st ❞❛t❛ t♦ ❧❡❛r♥ ❛ r❡✇❛r❞ ♣r❡❞✐❝t♦r ˆ r(①, ❛)✱ ❛♥❞ ❛❝t ❛❝❝♦r❞✐♥❣ t♦ ❛r❣ ♠❛①❛ ˆ r(①, ❛)✳ ❊①❛♠♣❧❡✿ ❉❡♣❧♦②❡❞ ♣♦❧✐❝② ❛❧✇❛②s t❛❦❡s ❛✶ ♦♥ ①✶ ❛♥❞ ❛✷ ♦♥ ①✷✳ ❖❜s❡r✈❡❞ ✴❊st✐♠❛t❡❞✴❚r✉❡ ❛✶ ❛✷ ①✶ .✽ ✴ ✽✴ ✽ ❄ ✴✶ ①✷ ❄ ✴ ✸ .✷ ✴ ✷

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SLIDE 8

❚❤❡ ✏❉✐r❡❝t ♠❡t❤♦❞✑

❯s❡ ♣❛st ❞❛t❛ t♦ ❧❡❛r♥ ❛ r❡✇❛r❞ ♣r❡❞✐❝t♦r ˆ r(①, ❛)✱ ❛♥❞ ❛❝t ❛❝❝♦r❞✐♥❣ t♦ ❛r❣ ♠❛①❛ ˆ r(①, ❛)✳ ❊①❛♠♣❧❡✿ ❉❡♣❧♦②❡❞ ♣♦❧✐❝② ❛❧✇❛②s t❛❦❡s ❛✶ ♦♥ ①✶ ❛♥❞ ❛✷ ♦♥ ①✷✳ ❖❜s❡r✈❡❞✴❊st✐♠❛t❡❞ ✴❚r✉❡ ❛✶ ❛✷ ①✶ .✽✴.✽ ✴ ✽ ❄✴.✺ ✴✶ ①✷ ❄✴.✺ ✴ ✸ .✷ ✴.✷ ✴ ✷

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SLIDE 9

❚❤❡ ✏❉✐r❡❝t ♠❡t❤♦❞✑

❯s❡ ♣❛st ❞❛t❛ t♦ ❧❡❛r♥ ❛ r❡✇❛r❞ ♣r❡❞✐❝t♦r ˆ r(①, ❛)✱ ❛♥❞ ❛❝t ❛❝❝♦r❞✐♥❣ t♦ ❛r❣ ♠❛①❛ ˆ r(①, ❛)✳ ❊①❛♠♣❧❡✿ ❉❡♣❧♦②❡❞ ♣♦❧✐❝② ❛❧✇❛②s t❛❦❡s ❛✶ ♦♥ ①✶ ❛♥❞ ❛✷ ♦♥ ①✷✳ ❖❜s❡r✈❡❞✴❊st✐♠❛t❡❞ ✴❚r✉❡ ❛✶ ❛✷ ①✶ .✽✴.✽ ✴ ✽ ❄✴.✺ ✴✶ ①✷ ✳✸✴.✺ ✴ ✸ .✷ ✴.✷ ✴ ✷

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SLIDE 10

❚❤❡ ✏❉✐r❡❝t ♠❡t❤♦❞✑

❯s❡ ♣❛st ❞❛t❛ t♦ ❧❡❛r♥ ❛ r❡✇❛r❞ ♣r❡❞✐❝t♦r ˆ r(①, ❛)✱ ❛♥❞ ❛❝t ❛❝❝♦r❞✐♥❣ t♦ ❛r❣ ♠❛①❛ ˆ r(①, ❛)✳ ❊①❛♠♣❧❡✿ ❉❡♣❧♦②❡❞ ♣♦❧✐❝② ❛❧✇❛②s t❛❦❡s ❛✶ ♦♥ ①✶ ❛♥❞ ❛✷ ♦♥ ①✷✳ ❖❜s❡r✈❡❞✴❊st✐♠❛t❡❞ ✴❚r✉❡ ❛✶ ❛✷ ①✶ .✽✴.✽ ✴ ✽ ❄✴.✺ ✴✶ ①✷ ✳✸✴.✸ ✴ ✸ .✷ ✴.✷ ✴ ✷

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SLIDE 11

❚❤❡ ✏❉✐r❡❝t ♠❡t❤♦❞✑

❯s❡ ♣❛st ❞❛t❛ t♦ ❧❡❛r♥ ❛ r❡✇❛r❞ ♣r❡❞✐❝t♦r ˆ r(①, ❛)✱ ❛♥❞ ❛❝t ❛❝❝♦r❞✐♥❣ t♦ ❛r❣ ♠❛①❛ ˆ r(①, ❛)✳ ❊①❛♠♣❧❡✿ ❉❡♣❧♦②❡❞ ♣♦❧✐❝② ❛❧✇❛②s t❛❦❡s ❛✶ ♦♥ ①✶ ❛♥❞ ❛✷ ♦♥ ①✷✳ ❖❜s❡r✈❡❞✴❊st✐♠❛t❡❞✴❚r✉❡ ❛✶ ❛✷ ①✶ .✽✴.✽✴.✽ ❄✴.✺✴✶ ①✷ ✳✸✴.✸✴.✸ .✷ ✴.✷ ✴.✷

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SLIDE 12

❚❤❡ ✏❉✐r❡❝t ♠❡t❤♦❞✑

❯s❡ ♣❛st ❞❛t❛ t♦ ❧❡❛r♥ ❛ r❡✇❛r❞ ♣r❡❞✐❝t♦r ˆ r(①, ❛)✱ ❛♥❞ ❛❝t ❛❝❝♦r❞✐♥❣ t♦ ❛r❣ ♠❛①❛ ˆ r(①, ❛)✳ ❊①❛♠♣❧❡✿ ❉❡♣❧♦②❡❞ ♣♦❧✐❝② ❛❧✇❛②s t❛❦❡s ❛✶ ♦♥ ①✶ ❛♥❞ ❛✷ ♦♥ ①✷✳ ❖❜s❡r✈❡❞✴❊st✐♠❛t❡❞✴❚r✉❡ ❛✶ ❛✷ ①✶ .✽✴.✽✴.✽ ❄✴.✺✴✶ ①✷ ✳✸✴.✸✴.✸ .✷ ✴.✷ ✴.✷ ❇❛s✐❝ ♦❜s❡r✈❛t✐♦♥ ✶✿ ●❡♥❡r❛❧✐③❛t✐♦♥ ✐♥s✉✣❝✐❡♥t✳

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SLIDE 13

❚❤❡ ✏❉✐r❡❝t ♠❡t❤♦❞✑

❯s❡ ♣❛st ❞❛t❛ t♦ ❧❡❛r♥ ❛ r❡✇❛r❞ ♣r❡❞✐❝t♦r ˆ r(①, ❛)✱ ❛♥❞ ❛❝t ❛❝❝♦r❞✐♥❣ t♦ ❛r❣ ♠❛①❛ ˆ r(①, ❛)✳ ❊①❛♠♣❧❡✿ ❉❡♣❧♦②❡❞ ♣♦❧✐❝② ❛❧✇❛②s t❛❦❡s ❛✶ ♦♥ ①✶ ❛♥❞ ❛✷ ♦♥ ①✷✳ ❖❜s❡r✈❡❞✴❊st✐♠❛t❡❞✴❚r✉❡ ❛✶ ❛✷ ①✶ .✽✴.✽✴.✽ ❄✴.✺✴✶ ①✷ ✳✸✴.✸✴.✸ .✷ ✴.✷ ✴.✷ ❇❛s✐❝ ♦❜s❡r✈❛t✐♦♥ ✷✿ ❊①♣❧♦r❛t✐♦♥ r❡q✉✐r❡❞✳

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SLIDE 14

❚❤❡ ✏❉✐r❡❝t ♠❡t❤♦❞✑

❯s❡ ♣❛st ❞❛t❛ t♦ ❧❡❛r♥ ❛ r❡✇❛r❞ ♣r❡❞✐❝t♦r ˆ r(①, ❛)✱ ❛♥❞ ❛❝t ❛❝❝♦r❞✐♥❣ t♦ ❛r❣ ♠❛①❛ ˆ r(①, ❛)✳ ❊①❛♠♣❧❡✿ ❉❡♣❧♦②❡❞ ♣♦❧✐❝② ❛❧✇❛②s t❛❦❡s ❛✶ ♦♥ ①✶ ❛♥❞ ❛✷ ♦♥ ①✷✳ ❖❜s❡r✈❡❞✴❊st✐♠❛t❡❞✴❚r✉❡ ❛✶ ❛✷ ①✶ .✽✴.✽✴.✽ ❄✴.✺✴✶ ①✷ ✳✸✴.✸✴.✸ .✷ ✴.✷ ✴.✷ ❇❛s✐❝ ♦❜s❡r✈❛t✐♦♥ ✸✿ ❊rr♦rs = ❡①♣❧♦r❛t✐♦♥✳

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SLIDE 15

❚❤❡ ❊✈❛❧✉❛t✐♦♥ Pr♦❜❧❡♠

▲❡t π : ❳ → ❆ ❜❡ ❛ ♣♦❧✐❝② ♠❛♣♣✐♥❣ ❢❡❛t✉r❡s t♦ ❛❝t✐♦♥s✳ ❍♦✇ ❞♦ ✇❡ ❡✈❛❧✉❛t❡ ✐t❄ ▼❡t❤♦❞ ✶✿ ❉❡♣❧♦② ❛❧❣♦r✐t❤♠ ✐♥ t❤❡ ✇♦r❧❞✳ ❱❡r② ❊①♣❡♥s✐✈❡✦

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SLIDE 16

❚❤❡ ❊✈❛❧✉❛t✐♦♥ Pr♦❜❧❡♠

▲❡t π : ❳ → ❆ ❜❡ ❛ ♣♦❧✐❝② ♠❛♣♣✐♥❣ ❢❡❛t✉r❡s t♦ ❛❝t✐♦♥s✳ ❍♦✇ ❞♦ ✇❡ ❡✈❛❧✉❛t❡ ✐t❄ ▼❡t❤♦❞ ✶✿ ❉❡♣❧♦② ❛❧❣♦r✐t❤♠ ✐♥ t❤❡ ✇♦r❧❞✳ ❱❡r② ❊①♣❡♥s✐✈❡✦

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SLIDE 17

❚❤❡ ■♠♣♦rt❛♥❝❡ ❲❡✐❣❤t✐♥❣ ❚r✐❝❦

▲❡t π : ❳ → ❆ ❜❡ ❛ ♣♦❧✐❝② ♠❛♣♣✐♥❣ ❢❡❛t✉r❡s t♦ ❛❝t✐♦♥s✳ ❍♦✇ ❞♦ ✇❡ ❡✈❛❧✉❛t❡ ✐t❄ ❖♥❡ ❛♥s✇❡r✿ ❈♦❧❧❡❝t ❚ ❡①♣❧♦r❛t✐♦♥ s❛♠♣❧❡s ① ❛ r❛ ♣❛ ✇❤❡r❡ ① ❝♦♥t❡①t ❛ ❛❝t✐♦♥ r❛ r❡✇❛r❞ ❢♦r ❛❝t✐♦♥ ♣❛ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❝t✐♦♥ ❛ t❤❡♥ ❡✈❛❧✉❛t❡✿ ❱❛❧✉❡ ❆✈❡r❛❣❡ r❛ ✶ ① ❛ ♣❛

slide-18
SLIDE 18

❚❤❡ ■♠♣♦rt❛♥❝❡ ❲❡✐❣❤t✐♥❣ ❚r✐❝❦

▲❡t π : ❳ → ❆ ❜❡ ❛ ♣♦❧✐❝② ♠❛♣♣✐♥❣ ❢❡❛t✉r❡s t♦ ❛❝t✐♦♥s✳ ❍♦✇ ❞♦ ✇❡ ❡✈❛❧✉❛t❡ ✐t❄ ❖♥❡ ❛♥s✇❡r✿ ❈♦❧❧❡❝t ❚ ❡①♣❧♦r❛t✐♦♥ s❛♠♣❧❡s (①, ❛, r❛, ♣❛), ✇❤❡r❡ ① = ❝♦♥t❡①t ❛ = ❛❝t✐♦♥ r❛ = r❡✇❛r❞ ❢♦r ❛❝t✐♦♥ ♣❛ = ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❝t✐♦♥ ❛ t❤❡♥ ❡✈❛❧✉❛t❡✿ ❱❛❧✉❡(π) = ❆✈❡r❛❣❡ r❛ ✶(π(①) = ❛) ♣❛

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SLIDE 19

❚❤❡ ■♠♣♦rt❛♥❝❡ ❲❡✐❣❤t✐♥❣ ❚r✐❝❦

❚❤❡♦r❡♠

❋♦r ❛❧❧ ♣♦❧✐❝✐❡s π✱ ❢♦r ❛❧❧ ■■❉ ❞❛t❛ ❞✐str✐❜✉t✐♦♥s ❉✱ ❱❛❧✉❡(π) ✐s ❛♥ ✉♥❜✐❛s❡❞ ❡st✐♠❛t❡ ♦❢ t❤❡ ❡①♣❡❝t❡❞ r❡✇❛r❞ ♦❢ π✿ ❊(①,

r)∼❉

  • rπ(①)
  • = ❊[ ❱❛❧✉❡(π) ]

Pr♦♦❢✿ ❊❛ ♣

r❛✶ ① ❛ ♣❛ ❛ ♣❛ r❛✶ ① ❛ ♣❛

r

①

❊①❛♠♣❧❡✿ ❆❝t✐♦♥ ✶ ✷ ❘❡✇❛r❞ ✵ ✺ ✶ Pr♦❜❛❜✐❧✐t②

✶ ✹ ✸ ✹

❊st✐♠❛t❡ ✷ ⑤ ✵ ✵ ⑤ ✹

✸

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SLIDE 20

❚❤❡ ■♠♣♦rt❛♥❝❡ ❲❡✐❣❤t✐♥❣ ❚r✐❝❦

❚❤❡♦r❡♠

❋♦r ❛❧❧ ♣♦❧✐❝✐❡s π✱ ❢♦r ❛❧❧ ■■❉ ❞❛t❛ ❞✐str✐❜✉t✐♦♥s ❉✱ ❱❛❧✉❡(π) ✐s ❛♥ ✉♥❜✐❛s❡❞ ❡st✐♠❛t❡ ♦❢ t❤❡ ❡①♣❡❝t❡❞ r❡✇❛r❞ ♦❢ π✿ ❊(①,

r)∼❉

  • rπ(①)
  • = ❊[ ❱❛❧✉❡(π) ]

Pr♦♦❢✿ ❊❛∼♣

  • r❛✶(π(①)=❛)

♣❛

  • =

❛ ♣❛ r❛✶(π(①)=❛) ♣❛

= rπ(①) ❊①❛♠♣❧❡✿ ❆❝t✐♦♥ ✶ ✷ ❘❡✇❛r❞ ✵ ✺ ✶ Pr♦❜❛❜✐❧✐t②

✶ ✹ ✸ ✹

❊st✐♠❛t❡ ✷ ⑤ ✵ ✵ ⑤ ✹

✸

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SLIDE 21

❚❤❡ ■♠♣♦rt❛♥❝❡ ❲❡✐❣❤t✐♥❣ ❚r✐❝❦

❚❤❡♦r❡♠

❋♦r ❛❧❧ ♣♦❧✐❝✐❡s π✱ ❢♦r ❛❧❧ ■■❉ ❞❛t❛ ❞✐str✐❜✉t✐♦♥s ❉✱ ❱❛❧✉❡(π) ✐s ❛♥ ✉♥❜✐❛s❡❞ ❡st✐♠❛t❡ ♦❢ t❤❡ ❡①♣❡❝t❡❞ r❡✇❛r❞ ♦❢ π✿ ❊(①,

r)∼❉

  • rπ(①)
  • = ❊[ ❱❛❧✉❡(π) ]

Pr♦♦❢✿ ❊❛∼♣

  • r❛✶(π(①)=❛)

♣❛

  • =

❛ ♣❛ r❛✶(π(①)=❛) ♣❛

= rπ(①) ❊①❛♠♣❧❡✿ ❆❝t✐♦♥ ✶ ✷ ❘❡✇❛r❞ ✵.✺ ✶ Pr♦❜❛❜✐❧✐t②

✶ ✹ ✸ ✹

❊st✐♠❛t❡ ✷ ⑤ ✵ ✵ ⑤ ✹

✸

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SLIDE 22

❍♦✇ ❞♦ ②♦✉ t❡st t❤✐♥❣s❄

❯s❡ ❢♦r♠❛t✿ ❛❝t✐♦♥✿❝♦st✿♣r♦❜❛❜✐❧✐t② ⑤ ❢❡❛t✉r❡s ❊①❛♠♣❧❡✿ ✶✿✶✿✵✳✺ ⑤ t✉❡s❞❛② ②❡❛r ♠✐❧❧✐♦♥ s❤♦rt ❝♦♠♣❛♥ ✈❡❤✐❝❧ ❧✐♥❡ st❛t ✜♥❛♥❝ ❝♦♠♠✐t ❡①❝❤❛♥❣ ♣❧❛♥ ❝♦r♣ s✉❜s✐❞ ❝r❡❞✐t ✐ss✉ ❞❡❜t ♣❛② ❣♦❧❞ ❜✉r❡❛✉ ♣r❡❧✐♠✐♥ r❡✜♥ ❜✐❧❧✐♦♥ t❡❧❡♣❤♦♥ t✐♠❡ ❞r❛✇ ❜❛s✐❝ r❡❧❛t ✜❧❡ s♣♦❦❡s♠ r❡✉t s❡❝✉r ❛❝q✉✐r ❢♦r♠ ♣r♦s♣❡❝t ♣❡r✐♦❞ ✐♥t❡r✈✐❡✇ r❡❣✐st t♦r♦♥t r❡s♦✉r❝ ❜❛rr✐❝❦ ♦♥t❛r✐♦ q✉❛❧✐❢ ❜❧♥ ♣r♦s♣❡❝t✉s ❝♦♥✈❡rt✐❜❧ ✈✐♥❝ ❜♦r❣ ❛r❡q✉✐♣ ✳✳✳

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SLIDE 23

❍♦✇ ❞♦ ②♦✉ tr❛✐♥❄

❘❡❞✉❝❡ t♦ ❝♦st✲s❡♥s✐t✐✈❡ ❝❧❛ss✐✜❝❛t✐♦♥✿ ① ❝

✈✇ ✕❝❜ ✷ ✕❝❜❴t②♣❡ ❞r r❝✈✶✳tr❛✐♥✳t①t✳❣③ ✲❝ ✕♥❣r❛♠ ✷ ✕s❦✐♣s ✹ ✲❜ ✷✹ ✲❧ ✵✳✷✺ Pr♦❣r❡ss✐✈❡ ✵✴✶ ❧♦ss✿ ✵✳✵✹✺✽✷ ✈✇ ✕❝❜ ✷ ✕❝❜❴t②♣❡ ✐♣s r❝✈✶✳tr❛✐♥✳t①t✳❣③ ✲❝ ✕♥❣r❛♠ ✷ ✕s❦✐♣s ✹ ✲❜ ✷✹ ✲❧ ✵✳✶✷✺ Pr♦❣r❡ss✐✈❡ ✵✴✶ ❧♦ss✿ ✵✳✵✺✵✻✺ ✈✇ ✕❝❜ ✷ ✕❝❜❴t②♣❡ ❞♠ r❝✈✶✳tr❛✐♥✳t①t✳❣③ ✲❝ ✕♥❣r❛♠ ✷ ✕s❦✐♣s ✹ ✲❜ ✷✹ ✲❧ ✵✳✶✷✺ Pr♦❣r❡ss✐✈❡ ✵✴✶ ❧♦ss✿ ✵✳✵✹✻✼✾

slide-24
SLIDE 24

❍♦✇ ❞♦ ②♦✉ tr❛✐♥❄

❘❡❞✉❝❡ t♦ ❝♦st✲s❡♥s✐t✐✈❡ ❝❧❛ss✐✜❝❛t✐♦♥✿ (①, ❝)

✈✇ ✕❝❜ ✷ ✕❝❜❴t②♣❡ ❞r r❝✈✶✳tr❛✐♥✳t①t✳❣③ ✲❝ ✕♥❣r❛♠ ✷ ✕s❦✐♣s ✹ ✲❜ ✷✹ ✲❧ ✵✳✷✺ Pr♦❣r❡ss✐✈❡ ✵✴✶ ❧♦ss✿ ✵✳✵✹✺✽✷ ✈✇ ✕❝❜ ✷ ✕❝❜❴t②♣❡ ✐♣s r❝✈✶✳tr❛✐♥✳t①t✳❣③ ✲❝ ✕♥❣r❛♠ ✷ ✕s❦✐♣s ✹ ✲❜ ✷✹ ✲❧ ✵✳✶✷✺ Pr♦❣r❡ss✐✈❡ ✵✴✶ ❧♦ss✿ ✵✳✵✺✵✻✺ ✈✇ ✕❝❜ ✷ ✕❝❜❴t②♣❡ ❞♠ r❝✈✶✳tr❛✐♥✳t①t✳❣③ ✲❝ ✕♥❣r❛♠ ✷ ✕s❦✐♣s ✹ ✲❜ ✷✹ ✲❧ ✵✳✶✷✺ Pr♦❣r❡ss✐✈❡ ✵✴✶ ❧♦ss✿ ✵✳✵✹✻✼✾

slide-25
SLIDE 25

❍♦✇ ❞♦ ②♦✉ tr❛✐♥❄

❘❡❞✉❝❡ t♦ ❝♦st✲s❡♥s✐t✐✈❡ ❝❧❛ss✐✜❝❛t✐♦♥✿ (①, ❝)

✈✇ ✕❝❜ ✷ ✕❝❜❴t②♣❡ ❞r r❝✈✶✳tr❛✐♥✳t①t✳❣③ ✲❝ ✕♥❣r❛♠ ✷ ✕s❦✐♣s ✹ ✲❜ ✷✹ ✲❧ ✵✳✷✺ Pr♦❣r❡ss✐✈❡ ✵✴✶ ❧♦ss✿ ✵✳✵✹✺✽✷ ✈✇ ✕❝❜ ✷ ✕❝❜❴t②♣❡ ✐♣s r❝✈✶✳tr❛✐♥✳t①t✳❣③ ✲❝ ✕♥❣r❛♠ ✷ ✕s❦✐♣s ✹ ✲❜ ✷✹ ✲❧ ✵✳✶✷✺ Pr♦❣r❡ss✐✈❡ ✵✴✶ ❧♦ss✿ ✵✳✵✺✵✻✺ ✈✇ ✕❝❜ ✷ ✕❝❜❴t②♣❡ ❞♠ r❝✈✶✳tr❛✐♥✳t①t✳❣③ ✲❝ ✕♥❣r❛♠ ✷ ✕s❦✐♣s ✹ ✲❜ ✷✹ ✲❧ ✵✳✶✷✺ Pr♦❣r❡ss✐✈❡ ✵✴✶ ❧♦ss✿ ✵✳✵✹✻✼✾

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SLIDE 26

❘❡♠✐♥❞❡r✿ ❈♦♥t❡①t✉❛❧ ❇❛♥❞✐t ❙❡tt✐♥❣

❋♦r t = ✶, . . . , ❚✿ ✶✳ ❚❤❡ ✇♦r❧❞ ♣r♦❞✉❝❡s s♦♠❡ ❝♦♥t❡①t ① ∈ ❳ ✷✳ ❚❤❡ ❧❡❛r♥❡r ❝❤♦♦s❡s ❛♥ ❛❝t✐♦♥ ❛ ∈ ❆ ✸✳ ❚❤❡ ✇♦r❧❞ r❡❛❝ts ✇✐t❤ r❡✇❛r❞ r❛ ∈ [✵, ✶]

  • ♦❛❧✿

▲❡❛r♥ ❛ ❣♦♦❞ ♣♦❧✐❝② ❢♦r ❝❤♦♦s✐♥❣ ❛❝t✐♦♥s ❣✐✈❡♥ ❝♦♥t❡①t✳ ❲❤❛t ❞♦❡s ❧❡❛r♥✐♥❣ ♠❡❛♥❄ ❊✣❝✐❡♥t❧② ❝♦♠♣❡t✐♥❣ ✇✐t❤ s♦♠❡ ❧❛r❣❡ r❡❢❡r❡♥❝❡ ❝❧❛ss ♦❢ ♣♦❧✐❝✐❡s Π = {π : ❳ → ❆}✿ ❘❡❣r❡t = ♠❛①

π∈Π ❛✈❡r❛❣❡t(rπ(①) − r❛)

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SLIDE 27

❇✉✐❧❞✐♥❣ ❛♥ ❆❧❣♦r✐t❤♠

▲❡t ◗✶ ❂ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ❋♦r t = ✶, . . . , ❚✿ ✶✳ ❚❤❡ ✇♦r❧❞ ♣r♦❞✉❝❡s s♦♠❡ ❝♦♥t❡①t ① ∈ ❳ ✷✳ ❉r❛✇ ◗t ✸✳ ❚❤❡ ❧❡❛r♥❡r ❝❤♦♦s❡s ❛♥ ❛❝t✐♦♥ ❛ ∈ ❆ ✉s✐♥❣ ① ✳ ✹✳ ❚❤❡ ✇♦r❧❞ r❡❛❝ts ✇✐t❤ r❡✇❛r❞ r❛ ∈ [✵, ✶] ✺✳ ❯♣❞❛t❡ ◗t

✶

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SLIDE 28

❇✉✐❧❞✐♥❣ ❛♥ ❆❧❣♦r✐t❤♠

▲❡t ◗✶ ❂ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ❋♦r t = ✶, . . . , ❚✿ ✶✳ ❚❤❡ ✇♦r❧❞ ♣r♦❞✉❝❡s s♦♠❡ ❝♦♥t❡①t ① ∈ ❳ ✷✳ ❉r❛✇ π ∼ ◗t ✸✳ ❚❤❡ ❧❡❛r♥❡r ❝❤♦♦s❡s ❛♥ ❛❝t✐♦♥ ❛ ∈ ❆ ✉s✐♥❣ π(①)✳ ✹✳ ❚❤❡ ✇♦r❧❞ r❡❛❝ts ✇✐t❤ r❡✇❛r❞ r❛ ∈ [✵, ✶] ✺✳ ❯♣❞❛t❡ ◗t+✶

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SLIDE 29

❲❤❛t ✐s ❣♦♦❞ ◗t❄

◮ ❊①♣❧♦r❛t✐♦♥✿ ◗t ❛❧❧♦✇s ❞✐s❝♦✈❡r② ♦❢ ❣♦♦❞

♣♦❧✐❝✐❡s

◮ ❊①♣❧♦✐t❛t✐♦♥✿ ◗t ❧❛r❣❡ ♦♥ ❣♦♦❞ ♣♦❧✐❝✐❡s

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SLIDE 30

❍♦✇ ❞♦ ②♦✉ ✜♥❞ ◗t❄

❝r✉❞❡❧②✿ ❜② ❝r❡❛t✐♥❣ ❛ ❝♦✈❡r✳

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SLIDE 31

❍♦✇ ❞♦ ②♦✉ ✜♥❞ ◗t❄

❝r✉❞❡❧②✿ ❜② ❝r❡❛t✐♥❣ ❛ ❝♦✈❡r✳

All Policies

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SLIDE 32

❍♦✇ ❞♦ ②♦✉ ✜♥❞ ◗t❄

❝r✉❞❡❧②✿ ❜② ❝r❡❛t✐♥❣ ❛ ❝♦✈❡r✳

All Policies Good Policies

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SLIDE 33

❍♦✇ ❞♦ ②♦✉ ✜♥❞ ◗t❄

❝r✉❞❡❧②✿ ❜② ❝r❡❛t✐♥❣ ❛ ❝♦✈❡r✳

All Policies Good Policies Cover

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SLIDE 34

❍♦✇ ❞♦ ②♦✉ ❝r❡❛t❡ ❛ ❝♦✈❡r❄

❘❡❞✉❝❡ t♦ ❈♦st s❡♥s✐t✐✈❡ ❝❧❛ss✐✜❝❛t✐♦♥✿ ① ❝ ▲❡t

✶ ❚ ❆ ❂ ♠✐♥✐♠✉♠ ♣r♦❜❛❜✐❧✐t②✳

▲❡t ◗♥ ❛ ♠❛①

✶ ♥ ✐ ■ ✐ ①

❛ ❝❛ ✉♥❜✐❛s❡❞ ❝♦st ❡st✐♠❛t❡

◗♥ ❛

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SLIDE 35

❍♦✇ ❞♦ ②♦✉ ❝r❡❛t❡ ❛ ❝♦✈❡r❄

❘❡❞✉❝❡ t♦ ❈♦st s❡♥s✐t✐✈❡ ❝❧❛ss✐✜❝❛t✐♦♥✿ (①, ❝) ▲❡t

✶ ❚ ❆ ❂ ♠✐♥✐♠✉♠ ♣r♦❜❛❜✐❧✐t②✳

▲❡t ◗♥ ❛ ♠❛①

✶ ♥ ✐ ■ ✐ ①

❛ ❝❛ ✉♥❜✐❛s❡❞ ❝♦st ❡st✐♠❛t❡

◗♥ ❛

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SLIDE 36

❍♦✇ ❞♦ ②♦✉ ❝r❡❛t❡ ❛ ❝♦✈❡r❄

❘❡❞✉❝❡ t♦ ❈♦st s❡♥s✐t✐✈❡ ❝❧❛ss✐✜❝❛t✐♦♥✿ (①, ❝) ▲❡t µ =

✶

√

❚|❆| ❂ ♠✐♥✐♠✉♠ ♣r♦❜❛❜✐❧✐t②✳

▲❡t ◗♥ ❛ ♠❛①

✶ ♥ ✐ ■ ✐ ①

❛ ❝❛ ✉♥❜✐❛s❡❞ ❝♦st ❡st✐♠❛t❡

◗♥ ❛

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SLIDE 37

❍♦✇ ❞♦ ②♦✉ ❝r❡❛t❡ ❛ ❝♦✈❡r❄

❘❡❞✉❝❡ t♦ ❈♦st s❡♥s✐t✐✈❡ ❝❧❛ss✐✜❝❛t✐♦♥✿ (①, ❝) ▲❡t µ =

✶

√

❚|❆| ❂ ♠✐♥✐♠✉♠ ♣r♦❜❛❜✐❧✐t②✳

▲❡t ◗♥(❛) = ♠❛①{µ, ✶

♥

  • ✐ ■(π✐(①) = ❛)}

❝❛ ✉♥❜✐❛s❡❞ ❝♦st ❡st✐♠❛t❡

◗♥ ❛

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SLIDE 38

❍♦✇ ❞♦ ②♦✉ ❝r❡❛t❡ ❛ ❝♦✈❡r❄

❘❡❞✉❝❡ t♦ ❈♦st s❡♥s✐t✐✈❡ ❝❧❛ss✐✜❝❛t✐♦♥✿ (①, ❝) ▲❡t µ =

✶

√

❚|❆| ❂ ♠✐♥✐♠✉♠ ♣r♦❜❛❜✐❧✐t②✳

▲❡t ◗♥(❛) = ♠❛①{µ, ✶

♥

  • ✐ ■(π✐(①) = ❛)}

❝❛ = ✉♥❜✐❛s❡❞ ❝♦st ❡st✐♠❛t❡ − ǫ

µ ◗♥(❛)

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SLIDE 39

❍♦✇ ✇❡❧❧ ❞♦❡s t❤✐s ✇♦r❦❄

0.02 0.04 0.06 0.08 0.1 0.12 eps-greedy tau-first LinUCB* Cover loss losses on CCAT RCV1 problem

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SLIDE 40

❍♦✇ ❧♦♥❣ ❞♦❡s t❤✐s t❛❦❡❄

1 10 100 1000 10000 100000 1e+06 eps-greedy tau-first LinUCB* Cover seconds running times on CCAT RCV1 problem

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SLIDE 41

❋✉rt❤❡r r❡❛❞✐♥❣

❱❲ ✇✐❦✐✿ ❤tt♣s✿✴✴❣✐t❤✉❜✳❝♦♠✴❏♦❤♥▲❛♥❣❢♦r❞✴ ✈♦✇♣❛❧❴✇❛❜❜✐t✴✇✐❦✐ ◆■P❙ t✉t♦r✐❛❧✿ ❤tt♣✿✴✴❤✉♥❝❤✳♥❡t✴⑦❥❧✴✐♥t❡r❛❝t✳♣❞❢ ❈♦❞❡ ❱♦✇♣❛❧ ❲❛❜❜✐t ♦♣❡♥ s♦✉r❝❡ ♣r♦❥❡❝t✱ ❤tt♣✿✴✴❣✐t❤✉❜✳❝♦♠✴❏♦❤♥▲❛♥❣❢♦r❞✴✈♦✇♣❛❧❴ ✇❛❜❜✐t✴✇✐❦✐✱ ✷✵✵✼✳ ❊①♣❧♦r❡ ❆✳ ❆❣❛r✇❛❧✱ ❉✳ ❍s✉✱ ❙✳ ❑❛❧❡✱ ❏✳ ▲❛♥❣❢♦r❞✱ ▲✳ ▲✐✱ ❘✳ ❙❝❤❛♣✐r❡✱ ❚❛♠✐♥❣ t❤❡ ▼♦♥st❡r✿ ❆ ❋❛st ❛♥❞ ❙✐♠♣❧❡ ❆❧❣♦r✐t❤♠ ❢♦r ❈♦♥t❡①t✉❛❧ ❇❛♥❞✐ts✱ ❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✹✵✷✳✵✺✺✺