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▼❱❆ ❈♦✉rs❡

■♥tr♦❞✉❝t✐♦♥ t♦ ❙t❛t✐st✐❝❛❧ ▲❡❛r♥✐♥❣

◆✐❝♦❧❛s ❱❛②❛t✐s

❙❡ss✐♦♥ ✶ ✲ ■♥tr♦❞✉❝t✐♦♥✱ ❈❤❛♣t❡r ✶ ✭P❛rt ❆✮

❘❡s❡❛r❝❤ ♦♣♣♦rt✉♥✐t✐❡s ❛t

❤tt♣✿✴✴✇✇✇✳❝❡♥tr❡❜♦r❡❧❧✐✳❢r

• ■♥t❡r❞✐s❝♣❧✐♥❛r② r❡s❡❛r❝❤ ❝❡♥tr❡ ✿ ♠❛t❤❡♠❛t✐❝s✴ ♥❡✉r♦s❝✐❡♥❝❡s✴

❝❧✐♥✐❝❛❧ r❡s❡❛r❝❤

• ❚✇♦ ❧♦❝❛t✐♦♥s ✿ ❊◆❙ P❛r✐s✲❙❛❝❧❛② ✭●✐❢✲s✉r✲❨✈❡tt❡✮ ❛♥❞ ❈❡♥tr❡

❯♥✐✈❡rs✐t❛✐r❡ ❞❡s ❙❛✐♥ts✲Pèr❡s ✭P❛r✐s ✻è♠❡✮

• ❲✐❞❡ s♣❡❝tr✉♠ ♦❢ ✐♥t❡r♥s❤✐♣s✴P❤❉ ♣r♦❥❡❝ts r❛♥❣✐♥❣ ❢r♦♠ t❤❡♦r❡t✐❝❛❧

❢♦✉♥❞❛t✐♦♥s ♦❢ ♠❛❝❤✐♥❡ ❧❡❛r♥✐♥❣ t♦ r❡❛❧✲❧✐❢❡ ❛♣♣❧✐❝❛t✐♦♥s

• ▼❛✐♥ t♦♣✐❝s ✿ ❛❝t✐✈❡ ❧❡❛r♥✐♥❣✱ tr❛♥s❢❡r ❧❡❛r♥✐♥❣✱ ❣❧♦❜❛❧ ♦♣t✐♠✐③❛t✐♦♥✱

♥❡t✇♦r❦ s❝✐❡♥❝❡✱ ❜✐♦♠❡❞✐❝❛❧ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣

• ■♥t❡r❛❝t ✇✐t❤ ♦t❤❡r s❝✐❡♥❝❡s ✭♥❡✉r♦s❝✐❡♥❝❡s ❛♥❞ ❤✉♠❛♥ ❢❛❝t♦r✱ s♦❝✐❛❧

s❝✐❡♥❝❡s✱ ♣❤②s✐❝s✮✱ ❤❡❛❧t❤❝❛r❡ ✭❤♦s♣✐t❛❧s✱ s♦❝✐❛❧ s❡❝✉r✐t②✮✱ ✐♥❞✉str② ✭❡♥❡r❣②✱ tr❛♥s♣♦rt❛t✐♦♥✱ ✳✳✳✮

Pr❛❝t✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❝♦✉rs❡

• ❈♦✉rs❡ ✇❡❜s✐t❡ ✿

❤tt♣✿✴✴♥✈❛②❛t✐s✳♣❡rs♦✳♠❛t❤✳❝♥rs✳❢r✴■❙▲❝♦✉rs❡✲✷✵✷✵✳❤t♠❧

• ❙❝❤❡❞✉❧❡ ❛♥❞ ❧♦❝❛t✐♦♥
• ❚✉❡s❞❛② ♠♦r♥✐♥❣ ✶✶❛♠✲✶♣♠ ✴ ❊◆❙ P❛r✐s✲❙❛❝❧❛② ✴ ❆♠♣❤✐

▲❛❣r❛♥❣❡ ✭✶❩✶✹✮

• ✻ ❝♦✉rs❡s ✰ ✹ ❡①❡r❝✐s❡ s❡ss✐♦♥s ✰ ♣❡rs♦♥❛❧ r❡s❡❛r❝❤
• ❖✣❝❡ ❤♦✉rs ✿ ❚✉❡s❞❛② ✶♣♠✲✷♣♠
• ❊✈❛❧✉❛t✐♦♥ ✿
• ❚✇♦ ♠❛♥❞❛t♦r② ❡①❛♠s ✿ ▼✐❞✲t❡r♠ ❡①❛♠ ▼ ✰ ✜♥❛❧ ❡①❛♠ ❋
• ❋✐♥❛❧ ❣r❛❞❡ ● ❂ ♠❛① ✭❋ ❀ ✭❋✰▼✮✴✷✮

❈♦✉rs❡ ♦❜❥❡❝t✐✈❡s

• ❇❡ ✐♥tr♦❞✉❝❡❞ t♦ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦✉♥❞❛t✐♦♥s ♦❢ ♠❛❝❤✐♥❡

❧❡❛r♥✐♥❣

• ▲❡❛r♥ ❤♦✇ t♦ ❢♦r♠❛❧✐③❡ ❛ ❧❡❛r♥✐♥❣ ♣r♦❜❧❡♠ t♦ ❜❡tt❡r ❛❞❞r❡ss

✉s❡rs✬ ❡①♣❡❝t❛t✐♦♥s

• ●❡t ✐♥s✐❣❤ts t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❦❡② ♣r✐♥❝✐♣❧❡s ♦❢ s❤❛❧❧♦✇

♠❛❝❤✐♥❡ ❧❡❛r♥✐♥❣ ♠❡t❤♦❞s

• ■❞❡♥t✐❢② t❤❡ ❦❡② ❞r✐✈❡rs t♦ ♠♦♥✐t♦r t❤❡ t❤❡♦r❡t✐❝❛❧ ♣❡r❢♦r♠❛♥❝❡

♦❢ ♠❛❝❤✐♥❡ ❧❡❛r♥✐♥❣ ♠❡t❤♦❞s

❈♦✉rs❡ ♦✈❡r✈✐❡✇

• ■♥tr♦❞✉❝t✐♦♥ ✿

❉❡♠②st✐✜❝❛t✐♦♥ ✴ ▲❡❛r♥✐♥❣ ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✴ s❡t✉♣

• ❈❤❛♣t❡r ✶ ✿ ❖♣t✐♠❛❧✐t② ✐♥ st❛t✐st✐❝❛❧ ❧❡❛r♥✐♥❣

Pr♦❜❛❜✐❧✐st✐❝ ✈✐❡✇ ✴ P❡r❢♦r♠❛♥❝❡ ❝r✐t❡r✐❛ ✴ ❖♣t✐♠❛❧ ❡❧❡♠❡♥ts

• ❈❤❛♣t❡r ✷ ✿ ▼❛t❤❡♠❛t✐❝❛❧ ❢♦✉♥❞❛t✐♦♥s ♦❢ st❛t✐st✐❝❛❧ ❧❡❛r♥✐♥❣

❈♦♥❝❡♥tr❛t✐♦♥ ✐♥❡q✉❛❧✐t② ✴ ❈♦♠♣❧❡①✐t② ♠❡❛s✉r❡s ✴ ❘❡❣✉❧❛r✐③❛t✐♦♥

• ❈❤❛♣t❡r ✸ ✿ ❈♦♥s✐st❡♥❝② ♦❢ ♠❛✐♥str❡❛♠ ♠❛❝❤✐♥❡ ❧❡❛r♥✐♥❣

♠❡t❤♦❞s ❇♦♦st✐♥❣✱ ❙❱▼✱ ◆❡✉r❛❧ ♥❡t✇♦r❦s ✴ ❇❛❣❣✐♥❣✱ ❘❛♥❞♦♠ ❢♦r❡sts

■♥tr♦❞✉❝t✐♦♥

❉❡♠②st✐❢②✐♥❣ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣

❚❤❡ ❣♦❛❧ ♦❢ ✭s✉♣❡r✈✐s❡❞✮ ♠❛❝❤✐♥❡ ❧❡❛r♥✐♥❣

❋✐♥❞✐♥❣ ❛ ❢✉♥❝t✐♦♥

• ❊①❛♠♣❧❡ ✿ P❡❞❡str✐❛♥ ❞❡t❡❝t✐♦♥ ❢r♦♠ ✈✐❞❡♦ ❝❛♠❡r❛s
• ❲❤❛t ✐s t❤❡ s❡❛r❝❤ s♣❛❝❡ ❢♦r s✉❝❤ ❛ ❢✉♥❝t✐♦♥ ❄

▼♦❞❡❧✐♥❣ ✭s✉♣❡r✈✐s❡❞✮ ❝❧❛ss✐✜❝❛t✐♦♥ ❞❛t❛

• ❉❛t❛ s❛♠♣❧❡ ✿ {(Xi, Yi) : i = ✶, . . . , n } ✇❤❡r❡
• Xi✬s ❡♥❝♦❞❡ ✐♠❛❣❡s ✭♣✐①❡❧✲✇✐s❡ ♦r ❢❡❛t✉r❡✲✇✐s❡✮
• Yi✬s ❛r❡ ❜✐♥❛r② ✈❛❧✉❡s ✭♣r❡s❡♥❝❡✴❛❜s❡♥❝❡ ♦❢ ❛ ♣❡❞❡str✐❛♥✮
• Pr♦❜❛❜✐❧✐st✐❝ ✈✐❡✇ ♦♥ t❤❡ ♣❛✐rs (Xi, Yi)

✭❍✶✮ (X, Y ) r❛♥❞♦♠ ♣❛✐r ✇✐t❤ ❞✐str✐❜✉t✐♦♥ P ♦✈❡r Rd × {✵, +✶} ✭❝♦♥✈❡♥t✐♦♥✮ ✭❍✷✮ ❉❛t❛ {(Xi, Yi) : i = ✶, . . . , n } ❛r❡ ■■❉ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐t❤ ❞✐str✐❜✉t✐♦♥ P

❉❡❝✐s✐♦♥ r✉❧❡s ❛r❡ t❤♦s❡ ❢✉♥❝t✐♦♥s

• ❉❡❝✐s✐♦♥ r✉❧❡s ✐♥ t❤❡ ❝❛s❡ ♦❢ s✉♣❡r✈✐s❡❞ ❜✐♥❛r② ❝❧❛ss✐✜❝❛t✐♦♥

g : Rd → {✵, ✶} ❛❧s♦ ❝❛❧❧❡❞ ❝❧❛ss✐✜❡rs✳

• ❊①❛♠♣❧❡s ♦❢ s✉❝❤ ❝❧❛ss✐✜❡rs ✿
• ▲✐♥❡❛r ❝❧❛ss✐✜❡rs ✿

gθ,θ✵(x) = I(θTx + θ✵ ≥ ✵) , ✇❤❡r❡ θ ∈ Rd✱ θ✵ ∈ R✳

• ▼♦r❡ ❣❡♥❡r❛❧❧② ✿

gf (x) = I(f (x) ≥ ✵) ✇❤❡r❡ f : Rd → R ❛♥❞ f ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ❜② ❧♦❣✐st✐❝ r❡❣r❡ss✐♦♥✱ ❞❡❝✐s✐♦♥ tr❡❡s✱ ❜♦♦st✐♥❣✱ r❛♥❞♦♠ ❢♦r❡sts✱ ❙❱▼✱ ♥❡✉r❛❧ ♥❡t✇♦r❦s✱ ✳✳✳

▲❡❛r♥✐♥❣ ❞❛t❛ ✈s✳ ❧❡❛r♥✐♥❣ ♦❜❥❡❝t✐✈❡s

❙❛♠❡ ✭❝❧❛ss✐✜❝❛t✐♦♥ ✮ ❞❛t❛✱ ❞✐✛❡r❡♥t ♦❜❥❡❝t✐✈❡s✳✳✳

• ❈❧❛ss✐✜❝❛t✐♦♥ ♦❜❥❡❝t✐✈❡
• Pr❡❞✐❝t Y ❦♥♦✇✐♥❣ X
• ❙✉❝❝❡ss ✐❢ ♠✐s❝❧❛ss✐✜❝❛✐t✐♦♥ ❡rr♦r ✐s s♠❛❧❧
• ❙❝♦r✐♥❣ ♦❜❥❡❝t✐✈❡
• ❘❛♥❦ X✬s s♦ t❤❛t ❛s ♠❛♥② ❛s ♣♦ss✐❜❧❡ Y ✬s ❛r❡ ❛t t❤❡ t♦♣ ♦❢ t❤❡

❧✐st

• ❙✉❝❝❡ss ✐❢ ❆r❡❛ ❯♥❞❡r ❛♥ ❘❖❈ ❈✉r✈❡ ✐s ❧❛r❣❡

■♥tr♦❞✉❝t✐♦♥

▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ s❡t✉♣ ❢♦r t❤❡ ❝♦✉rs❡

❋r❛♠❡✇♦r❦ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s ❝♦✉rs❡

• ❱❡❝t♦r ❞❛t❛ ✐♥ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✭✈s✳ str✐♥❣s✱ ♠❛tr✐❝❡s✱ s✐❣♥❛❧s✱

✐♠❛❣❡s✱ ❡t❝✳✮

• ◆♦ ♣r❡♣r♦❝❡ss✐♥❣ ✭✈s✳ r❡♣r❡s❡♥t❛t✐♦♥ ❧❡❛r♥✐♥❣✱ ❞✐♠❡♥s✐♦♥

r❡❞✉❝t✐♦♥✮

• ❙✉♣❡r✈✐s❡❞ ❧❡❛r♥✐♥❣ ✭✈s✳ ✉♥s✉♣❡r✈✐s❡❞✮
• ❇❛t❝❤ ❧❡❛r♥✐♥❣ ✭✈s✳ ♦♥❧✐♥❡✴s❡q✉❡♥t✐❛❧ ❧❡❛r♥✐♥❣✮
• P❛ss✐✈❡ ❧❡❛r♥✐♥❣ ✭✈s✳ ❛❝t✐✈❡ ❧❡❛r♥✐♥❣✮
• ❇✐♥❛r② ❝❧❛ss✐✜❝❛t✐♦♥ ❞❛t❛ ✭✈s✳ ♠✉❧t✐❝❧❛ss ❝❧❛ss✐✜❝❛t✐♦♥✱

r❡❣r❡ss✐♦♥✱ ♠✉❧t✐t❛s❦✱ ✳✳✳✮

• ❙t❛t✐♦♥❛r✐t② ❛ss✉♠♣t✐♦♥ ✭✈s✳ tr❛♥s❢❡r ❧❡❛r♥✐♥❣✱ ❞♦♠❛✐♥

❛❞❛♣t❛t✐♦♥✱ ✳✳✳✮

• ◆♦ ❢❡❡❞❜❛❝❦ ❧♦♦♣ ✭✈s✳ r❡✐♥❢♦r❝❡♠❡♥t ❧❡❛r♥✐♥❣✮

❚r❡♥❞s ❛♥❞ ♣r♦♠✐s❡s ✐♥ ▼▲

■♥tr♦❞✉❝t✐♦♥

▲❡❛r♥✐♥❣ ❛♥❞ ✐♥❢♦r♠❛t✐♦♥

▲❡❛r♥✐♥❣ ❧✐❦❡ t❤❡ t✇❡♥t②✲q✉❡st✐♦♥ ❣❛♠❡

• ❆ss✉♠❡ ◆❛t✉r❡ ❤❛s ♣✐❝❦❡❞ ♦♥❡ ❢✉♥❝t✐♦♥ ❛♠♦♥❣ K ❛♥❞ ✇❡ ✇❛♥t

t♦ r❡✈❡❛❧ t❤✐s ❢✉♥❝t✐♦♥

• ❆ss✉♠❡ ✇❡ ❤❛✈❡ ❛♥ ♦r❛❝❧❡ ❛♥s✇❡r✐♥❣ ❨❊❙ ♦r ◆❖ ✇❤❡♥ ✇❡ ❛s❦

❛ q✉❡st✐♦♥ ❛❜♦✉t t❤✐s ❢✉♥❝t✐♦♥

• ❲❤❛t ✐s t❤❡ ♦♣t✐♠❛❧ ♥✉♠❜❡r n ♦❢ q✉❡st✐♦♥s t♦ ❛s❦ t♦ ✜♥❞ t❤❡

✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ ❄

❇r✉t❡ ❢♦r❝❡ ❧❡❛r♥✐♥❣ ❋✐♥✐t❡ ❝❛s❡

• ■❙❙❯❊ ✿ ❍♦✇ ♠❛♥② q✉❡st✐♦♥s ✇✐t❤ ❛♥s✇❡rs ❨❊❙ ♦r ◆❖ ♦♥❡ ❤❛s

t♦ ❛s❦ t❤❡ ♦r❛❝❧❡ t♦ ✜♥❞ ❚❍❊ ❢✉♥❝t✐♦♥ ❛♠♦♥❣ K ❢✉♥❝t✐♦♥s ❄

• ❙❚❘❆❚❊●❨ ✿ Pr♦❝❡❡❞ r❡❝✉rs✐✈❡❧② ❜② s♣❧✐tt✐♥❣ t❤❡ s❡t ♦❢

❢✉♥❝t✐♦♥s ✐♥ t✇♦ ❣r♦✉♣s ❛♥❞ ❛s❦✐♥❣ ✇❤❡t❤❡r ❚❍❊ ❢✉♥❝t✐♦♥ ✐s t❤❡ ✜rst ❣r♦✉♣ ❛♥❞ r❡♠♦✈✐♥❣ t❤❡ ❣r♦✉♣ ✇❤✐❝❤ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ t❤❡ ❢✉♥❝t✐♦♥✳ ❚❤✐s ❧❡❛❞s t♦ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞❡s✐r❡❞ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛❜♦✉t log K q✉❡st✐♦♥s✳

• ❆◆❙❲❊❘ ✿ ◆✉♠❜❡r ♦❢ q✉❡st✐♦♥s n = log K

log ✷

• ◆❇ ✿ t❤✐s q✉❛♥t✐t② r❡♣r❡s❡♥ts t❤❡ ♥✉♠❜❡r ♦❢ ❜✐ts ♦❢ ✐♥❢♦r♠❛t✐♦♥

❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ s❡t ♦❢ K ❢✉♥❝t✐♦♥s

❙❤❛♥♥♦♥✬s ■♥❢♦r♠❛t✐♦♥ t❤❡♦r② ❚❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ log K

• ❘❡❧❛t❡❞ t♦ t❤❡ ❡♥tr♦♣② ♦❢ ❛ ❞✐str✐❜✉t✐♦♥ P ✐♥ ✐♥❢♦r♠❛t✐♦♥

t❤❡♦r② ✿ H(P) = −

K

• k=✶

P(k) log P(k)

• ❚❤❡ ❡♥tr♦♣② ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❜✐ts t♦ ❡♥❝♦❞❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ K

s②♠❜♦❧s ✭❢✉♥❝t✐♦♥s✮

❋r♦♠ q✉❡st✐♦♥s t♦ ❞❛t❛ ❩❡r♦ ❡rr♦r ❝❛s❡

• ◆♦t❛t✐♦♥s ✿ ❉♦♠❛✐♥ s♣❛❝❡ X ❛♥❞ ❧❛❜❡❧ s♣❛❝❡ Y = {✵, ✶}
• ■❙❙❯❊ ✿ ❍♦✇ ♠❛♥② ❡①❛♠♣❧❡s (xi, yi) ∈ X × Y ❛r❡ r❡q✉✐r❡❞ t♦

✜♥❞ ❛♠♦♥❣ ❛ ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥ ✭s✐③❡ K✮ ♦❢ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥s f : X → {✵, ✶} t❤❡ ❞❡s✐r❡❞ ♦♥❡ ❄

• ❙❆▼❊ ❆◆❙❲❊❘ ✿ ◆✉♠❜❡r ♦❢ ❡①❛♠♣❧❡s n = log K

log ✷

• ❙❚❘❆❚❊●❨ ✿ ❖♥❡ ❤❛s t♦ ✜♥❞ ❛ ✈❡❝t♦r xi s✉❝❤ t❤❛t ❤❛❧❢ ♦❢ t❤❡

❢✉♥❝t✐♦♥s t❛❦❡ ✈❛❧✉❡ ✶ ❛♥❞ t❤❡ ♦t❤❡r ❤❛❧❢ t❛❦❡ ✈❛❧✉❡ ✵ ❛♥❞ ❛s❦ t❤❡ ♦r❛❝❧❡ ✇❤❡t❤❡r t❤❡ ❞❡s✐r❡❞ ❢✉♥❝t✐♦♥ t❛❦❡s ✈❛❧✉❡ ✶ ♦r ✵ ♦♥ t❤✐s ✈❡❝t♦r ❛♥❞ ❞✐s❝❛r❞ t❤♦s❡ ❢✉♥❝t✐♦♥s t❛❦✐♥❣ t❤❡ ♦♣♣♦s✐t❡ ✈❛❧✉❡✳ ❆♣♣❧② t❤✐s n t✐♠❡s✳

Pr♦❜❛❜❧② ❛♣♣r♦①✐♠❛t❡❧② ❝♦rr❡❝t ❧❡❛r♥✐♥❣ ❩❡r♦ ❡rr♦r ❝❛s❡

• ❘❊▼❆❘❑ ✿ ✐t ♠❛② ❜❡ ❤❛r❞ t♦ ✜♥❞ s✉❝❤ ❛♥ xi ✇❤✐❝❤ s♣❧✐ts t❤❡

❝♦❧❧❡❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ✐♥ t✇♦✳

• ◆❊❲ ▼❖❉❊▲ ✿ ❆ss✉♠❡ X✶, . . . , Xn ✐s ❛♥ ■■❉ s❛♠♣❧❡
• ◗❯❊❙❚■❖◆ ✿ ❍♦✇ ♠❛♥② ❡①❛♠♣❧❡s (Xi, Yi) ❛r❡ r❡q✉✐r❡❞ t♦ ✜♥❞

❛♠♦♥❣ ❛ ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥s f : X → {✵, ✶} t❤❡ ♦♥❡ t❤❛t ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶ − δ ✐s ε✲❝❧♦s❡ t♦ t❤❡ ❞❡s✐r❡❞ ♦♥❡ ❄

• ❆◆❙❲❊❘ ✿ ◆✉♠❜❡r ♦❢ ❡①❛♠♣❧❡s

n = log K − log δ ε ✭Pr♦♦❢ ❝♦♠✐♥❣ ❧❛t❡r✮

Pr♦❜❛❜❧② ❛♣♣r♦①✐♠❛t❡❧② ❝♦rr❡❝t ❧❡❛r♥✐♥❣

• ❡♥❡r❛❧ ❝❛s❡
• ❆❙❙❯▼❊ ✿ ❛♠♦♥❣ K ❢✉♥❝t✐♦♥s✱ ◆❖◆❊ ♦❢ t❤❡♠ ❝♦♠♠✐ts ③❡r♦

❡rr♦r ♦♥ t❤❡ s❛♠♣❧❡ (Xi, Yi)✳

• ❙❆▼❊ ■❙❙❯❊ ❆❙ ❇❊❋❖❘❊
• ❆◆❙❲❊❘ ✿ ◆✉♠❜❡r ♦❢ ❡①❛♠♣❧❡s ♦♥ ❛✈❡r❛❣❡

n = log K − log δ ε✷ ❙❛♠❡ ❞❡♣❡♥❞❡♥❝② ♦♥ K✱ t❤❡ ♦♥❧② ❝❤❛♥❣❡ ✐s ✐♥ t❤❡ ❝♦♥st❛♥t✳ ✭Pr♦♦❢ ❝♦♠✐♥❣ ❧❛t❡r✮

◗✉❡st✐♦♥s r❛✐s❡❞

P❆❈ ❂ Pr♦❜❛❜❧② ❆♣♣r♦①✐♠❛t❡❧② ❈♦rr❡❝t

• Pr♦♦❢ ❛r❣✉♠❡♥ts ❢♦r P❆❈ ❧❡❛r♥❛❜✐❧✐t② ✭✜♥✐t❡ ❝❛s❡✮ ❄
• P❆❈ ✿ ❋r♦♠ ✜♥✐t❡ t♦ ✐♥✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ❄
• ❋r♦♠ ✧str❛t❡❣✐❡s✧ t♦ ✧❧❡❛r♥✐♥❣ ❛❧❣♦r✐t❤♠s✧ ❄
• ❲❤❛t ✐s ❧♦st t❤r♦✉❣❤ s❛♠♣❧✐♥❣ ❄

❚❤❡ ✧❜✐❣ ♣✐❝t✉r❡✧ ♦❢ st❛t✐st✐❝❛❧ ❧❡❛r♥✐♥❣ t❤❡♦r②

❇✐❛s✲✈❛r✐❛♥❝❡ ❞✐❧❡♠♠❛

❚❤❡ ❦❡② tr❛❞❡✲♦✛ ✐♥ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣

• ❉❡♥♦t❡ ❜② L(h) t❤❡ ❡rr♦r ♠❡❛s✉r❡ ❢♦r ❛♥② ❞❡❝✐s✐♦♥ ❢✉♥❝t✐♦♥ h
• ❲❡ ❤❛✈❡ ✿ L(¯

h) = inf

H L , ❛♥❞ L(h∗) = inf L

• ❇✐❛s✲❱❛r✐❛♥❝❡ t②♣❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❡rr♦r ❢♦r ❛♥② ♦✉t♣✉t

h ✿ L( h) − L(h∗) = L( h) − L(¯ h)

• ❡st✐♠❛t✐♦♥ ✭st♦❝❤❛st✐❝✮

+ L(¯ h) − L(h∗)

• ❛♣♣r♦①✐♠❛t✐♦♥ ✭❞❡t❡r♠✐♥✐st✐❝✮

❖❜❥❡❝t✐✈❡s ❢♦r ❈❤❛♣t❡r ✶

• ❲❤❛t ✐s h∗ ❄
• ❖♥ ✇❤❛t ❞♦❡s ✐t ❞❡♣❡♥❞ ❄
• ❙❛♠❡ ❞❛t❛✱ ✈❛r✐♦✉s ♦❜❥❡❝t✐✈❡s ❄

❈❤❛♣t❡r ✶ ✲ ❖♣t✐♠❛❧✐t② ✐♥ st❛t✐st✐❝❛❧ ❧❡❛r♥✐♥❣

❖✈❡r✈✐❡✇ ♦❢ ❈❤❛♣t❡r ✶

• ▼♦❞❡❧✐♥❣ t❤❡ ❞❛t❛ ✿
• ❛ ♣r♦❜❛❜✐❧✐st✐❝ ✈✐❡✇
• ▼♦❞❡❧✐♥❣ t❤❡ ♦❜❥❡❝t✐✈❡ ✿
• ♣❡r❢♦r♠❛♥❝❡ ♠❡tr✐❝s ❛♥❞ r✐s❦ ❢✉♥❝t✐♦♥❛❧s ❢♦r ♣r❡❞✐❝t✐♦♥
• ❚❤❡ ❣♦❛❧ ♦❢ ❧❡❛r♥✐♥❣ ✿
• ♦♣t✐♠❛❧ ❡❧❡♠❡♥ts

❈❤❛♣t❡r ✶

❆✳ ▼♦❞❡❧✐♥❣ ❝❧❛ss✐✜❝❛t✐♦♥ ❞❛t❛ ❇✳ ❖♣t✐♠❛❧✐t② ✐♥ t❤❡ ❜✐♥❛r② ❝❧❛ss✐✜❝❛t✐♦♥ ♦❜❥❡❝t✐✈❡ ❈✳ ❈♦♥✈❡① r✐s❦ ♠✐♥✐♠✐③❛t✐♦♥

❈❤❛♣t❡r ✶

❆✳ ▼♦❞❡❧✐♥❣ ❝❧❛ss✐✜❝❛t✐♦♥ ❞❛t❛

• ❡♥❡r❛t✐✈❡ ✈s✳ ❞✐s❝r✐♠✐♥❛t✐✈❡
• (X, Y ) r❛♥❞♦♠ ♣❛✐r ✇✐t❤ ❞✐str✐❜✉t✐♦♥ P ♦✈❡r Rd × {−✶, +✶}

✶ ●❡♥❡r❛t✐✈❡ ✈✐❡✇ ✲ ❏♦✐♥t ❞✐str✐❜✉t✐♦♥ P ❛s ❛ ♠✐①t✉r❡

• ❈❧❛ss✲❝♦♥❞✐t✐♦♥❛❧ ❞❡♥s✐t✐❡s ✿ f+ ❛♥❞ f−
• ▼✐①t✉r❡ ♣❛r❛♠❡t❡r ✿ p = P{Y = +✶}

✷ ❉✐s❝r✐♠✐♥❛t✐✈❡ ✈✐❡✇ ✲ ❏♦✐♥t ❞✐str✐❜✉t✐♦♥ P ❞❡s❝r✐❜❡❞ ❜②

(PX, η)

• ▼❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥ ✿ X ∼ PX = dfX/dλd
• P♦st❡r✐♦r ♣r♦❜❛❜✐❧✐t② ❢✉♥❝t✐♦♥ ✿

η(x) = P{Y = ✶ | X = x} , ∀x ∈ Rd

❊①❡r❝✐s❡ ✶

❋✐♥❞ t❤❡ ❡①♣r❡ss✐♦♥s ♦❢ f+✱ f− ❛♥❞ η ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❛❜✐❧✐st✐❝ ♠♦❞❡❧s ✿

• ❉✐s❝r✐♠✐♥❛♥t ❆♥❛❧②s✐s ✿ ✜♥❞ η ❦♥♦✇✐♥❣

f+ = Nd(µ+, Σ+), f− = Nd(µ−, Σ−)

• ▲♦❣✐st✐❝ r❡❣r❡ss✐♦♥ ✿ ✜♥❞ f+✱ f− ❦♥♦✇✐♥❣

log

• ηθ(x)

✶ − ηθ(x)

• = h(x, θ) ,

t②♣✐❝❛❧❧② h(x, θ) = θTx

❈❤❛♣t❡r ✶

❇✳ ❖♣t✐♠❛❧✐t② ✐♥ t❤❡ ❜✐♥❛r② ❝❧❛ss✐✜❝❛t✐♦♥ ♦❜❥❡❝t✐✈❡

❈❧❛ss✐✜❡r✱ ❊rr♦r ♠❡❛s✉r❡✱ ❖♣t✐♠❛❧ ❊❧❡♠❡♥ts

• ❈❧❛ss✐✜❡r ✿ g : Rd → {−✶, +✶}
• ❈❧❛ss✐✜❝❛t✐♦♥ ❡rr♦r ✿

L(g) = P {g(X) = Y } L(g) = E

• η(X) · I{g(X) = −✶} + (✶ − η(X)) · I{g(X) = ✶}
• ❇❛②❡s r✉❧❡ ✿

g∗(x) = ✷I{η(x) > ✶/✷} − ✶ , ∀x ∈ Rd

• ❇❛②❡s ❡rr♦r ✿ L∗ = L(g∗) = E{min(η(X), ✶ − η(X))}
• ❊①❝❡ss r✐s❦ ✿

L(g) − L∗ = ✷E

• η(X) − ✶

✷

• · I{g(X) = g∗(X)}

▲✐♥❦ ✇✐t❤ ♣❛r❛♠❡tr✐❝s ✿ P❧✉❣✲✐♥ ♠❡t❤♦❞s ❞♦ t❤❡ ❥♦❜ ❜✉t✳✳✳

• ▲❡t

η ❛♥ ❡st✐♠❛t❡ ♦❢ t❤❡ ♣♦st❡r✐♦r η ❜❛s❡❞ ♦♥ ❛ s❛♠♣❧❡ Dn

• ❈♦♥s✐❞❡r

g ❛ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r ❜❛s❡❞ ♦♥ η

• g(x) = ✷I{

η(x) > ✶/✷} − ✶ , ∀x ∈ Rd

• ❲❡ ❤❛✈❡✱ ❝♦♥❞✐t✐♦♥❛❧❧② ♦♥ t❤❡ s❛♠♣❧❡ Dn ✿

L( g) − L∗ ≤ ✷EX

• |

η(X) − η(X)|

• ❇✉t ❡st✐♠❛t✐♦♥ ♦❢ η ❢♦r ❤✐❣❤ ❞✐♠❡♥s✐♦♥❛❧ ❞❛t❛ ✐s ❛ ❞✐✣❝✉❧t

♣r♦❜❧❡♠ ✦

• ◗ ✿ ❉♦ ✇❡ r❡❛❧❧② ♥❡❡❞ t♦ ❞♦ t❤✐s ❄

❱❛r✐❛t✐♦♥s ♦♥ ❜✐♥❛r② ❝❧❛ss✐✜❝❛t✐♦♥

• ❆s②♠♠❡tr✐❝ ❝♦st ✲ s❡t ω ∈ (✵, ✶)✱

Lω(g) = ✷E

• (✶ − ω)I{Y = +✶}I{g(X) = −✶}

+ ωI{Y = −✶}I{g(X) = +✶}

• ❈❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ ♠❛ss ❝♦♥str❛✐♥t ✲ s❡t u ∈ (✵, ✶)

min

g P(Y = g(X))

s✉❜❥❡❝t t♦ P(g(X) = ✶) = u ✭❘❡❢❡r t♦ ❈❧é♠❡♥ç♦♥ ❛♥❞ ❱❛②❛t✐s ✭✷✵✵✼✮✮

• ❈❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ r❡❥❡❝t ♦♣t✐♦♥ ✲ s❡t γ ∈ (✵, ✶/✷)

LR

d (g) = P(Y = g(X) , g(X) = ) + γP(g(X) = )

✭❘❡❢❡r t♦ ❍❡r❜❡✐ ❛♥❞ ❲❡❣❦❛♠♣ ✭✷✵✵✻✮✮

❊①❡r❝✐s❡ ✷

❋✐♥❞ g∗ ❛♥❞ L∗ ✐♥ t❤❡ t❤r❡❡ ♣r❡✈✐♦✉s s❝❡♥❛r✐♦s ✿

• ❆s②♠♠❡tr✐❝ ❝♦st
• ❈❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ ♠❛ss ❝♦♥str❛✐♥t
• ❈❧❛ss✐✜❝❛t✐♦♥ ✇✐t❤ r❡❥❡❝t ♦♣t✐♦♥

❈❤❛♣t❡r ✶

❈✳ ❈♦♥✈❡① r✐s❦ ♠✐♥✐♠✐③❛t✐♦♥

❈♦♥✈❡① ❘✐s❦ ▼✐♥✐♠✐③❛t✐♦♥ ✭❈❘▼✮

• ❇✐♥❛r② ❝❧❛ss✐✜❝❛t✐♦♥ ❞❛t❛ ✇✐t❤ Y ∈ {+✶, −✶}
• ❘❡❛❧✲✈❛❧✉❡❞ ❞❡❝✐s✐♦♥ r✉❧❡ f

: Rd → R

• ❈♦st ❢✉♥❝t✐♦♥ ϕ : R → R+ ❝♦♥✈❡①✱ ✐♥❝r❡❛s✐♥❣✱ ϕ(✵) = ✶
• ❊①♣❡❝t❡❞ ϕ✲r✐s❦ ✿

A(f ) = E (ϕ(−Y · f (X)))

• ▼❛✐♥ ❡①❛♠♣❧❡s ✿

ϕ(x) = ex, log✷(✶ + ex), (✶ + x)+

• ◆♦t❡ t❤❛t ✿ L(s❣♥(f )) ≤ A(f )

❊①❡r❝✐s❡ ✸

❋✐♥❞ t❤❡ ♦♣t✐♠❛❧ ❡❧❡♠❡♥ts ❢♦r ❝♦♥✈❡① r✐s❦ ♠✐♥✐♠✐③❛t✐♦♥ ✿ f ∗ = ❛r❣ ♠✐♥

f

A(f ) , A∗ = A(f ∗) ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡s ✿ ✭✐✮ ϕ(u) = exp(u) ✭✐✐✮ ϕ(u) = log✷(✶ + exp(u)) ✭✐✐✐✮ ϕ(u) = (✶ + u)+

❈♦♠✐♥❣ ♥❡①t

• ❈♦♥✈❡① r✐s❦ ♠✐♥✐♠✐③❛t✐♦♥ ✿ ❣❡tt✐♥❣ ❝❧♦s❡r t♦ ❛❧❣♦r✐t❤♠✐❝

str❛t❡❣✐❡s ❛♥❞ ✇❤② ✐t ♠❛❦❡s s❡♥s❡ ❢r♦♠ t❤❡ ♣❡rs♣❡❝t✐✈❡ ♦❢ ❜✐♥❛r② ❝❧❛ss✐✜❝❛t✐♦♥

• ▼♦❞❡❧✐♥❣ ❛ ❞❡t❡❝t✐♦♥ t❛s❦ ❛♥❞ ♦♣t✐♠❛❧✐t② ✐♥ ❞❡t❡❝t✐♦♥
• ❇r✐♥❣✐♥❣ t❤❡ ❞❛t❛ ✐♥ ✿ ❊♠♣✐r✐❝❛❧ ❘✐s❦ ▼✐♥✐♠✐③❛t✐♦♥