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✶✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❚❤❡ ❍❡❞❣❡ ❆❧❣♦r✐t❤♠ ♦♥ ❛ ❈♦♥t✐♥✉✉♠

■❈▼▲ ✷✵✶✺ ✲ ▲✐❧❧❡✱ ❋r❛♥❝❡ ❲❛❧✐❞ ❑r✐❝❤❡♥❡ ▼❛①✐♠✐❧✐❛♥ ❇❛❧❛♥❞❛t ❈❧❛✐r❡ ❚♦♠❧✐♥ ❆❧❡①❛♥❞r❡ ❇❛②❡♥

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✶✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❖✉t❧✐♥❡

✶ ❚❤❡ Pr♦❜❧❡♠ ✷ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ✸ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s

✶✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❖✉t❧✐♥❡

✶ ❚❤❡ Pr♦❜❧❡♠ ✷ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ✸ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s

✷✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❖♥❧✐♥❡ ▲❡❛r♥✐♥❣ ♦✈❡r ❛ ✜♥✐t❡ s❡t

❆ ❞❡❝✐s✐♦♥ ♠❛❦❡r ❢❛❝❡s ❛ s❡q✉❡♥t✐❛❧ ♣r♦❜❧❡♠✿ ❖♥❧✐♥❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ♦✈❡r ❛ ✜♥✐t❡ s❡t {✶, . . . , N}✳

✶✿ ❢♦r t ∈ N ❞♦ ✷✿

❉❡❝✐s✐♦♥ ♠❛❦❡r ❝❤♦♦s❡s ❞✐str✐❜✉t✐♦♥ x(t) ♦✈❡r {✶, . . . N}✳

✸✿

❆ ❧♦ss ✈❡❝t♦r ℓ(t) ∈ [✵, M]N ✐s r❡✈❡❛❧❡❞✳

✹✿

❚❤❡ ❞❡❝✐s✐♦♥ ♠❛❦❡r ✐♥❝✉rs ❡①♣❡❝t❡❞ ❧♦ss N

n=✶ ℓ(t) n x(t) n

=

• x(t), ℓ(t)

✺✿ ❡♥❞ ❢♦r

✸✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❆♣♣❧✐❝❛t✐♦♥s

❆♣♣❧✐❝❛t✐♦♥s

• ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♣❧❛②❡r ❞②♥❛♠✐❝s ✐♥ ❣❛♠❡s ✭❇❧❛❝❦✇❡❧❧ ❬✶❪✱ ❍❛♥♥❛♥❬✺❪✮

{✶, . . . , N} ✐s t❤❡ s❡t ♦❢ ❛❝t✐♦♥s✳ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ ✶ ✐s t❤❡ tr❛✐♥✐♥❣ s❡t✳ ✏▼♦❞❡❧✲❢r❡❡✑ ♣♦rt❢♦❧✐♦ ♦♣t✐♠✐③❛t✐♦♥ ✭❈♦✈❡r ❬✹❪✱ ❇❧✉♠ ❬✷❪✮ ✶ ✐s t❤❡ s❡t ♦❢ st♦❝❦s✳ ▼❛♥② ♦t❤❡rs

❬✶❪❉❛✈✐❞ ❇❧❛❝❦✇❡❧❧✳ ❆♥ ❛♥❛❧♦❣ ♦❢ t❤❡ ♠✐♥✐♠❛① t❤❡♦r❡♠ ❢♦r ✈❡❝t♦r ♣❛②♦✛s✳ P❛❝✐✜❝ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✻✭✶✮✿✶✕✽✱ ✶✾✺✻ ❬✺❪❏❛♠❡s ❍❛♥♥❛♥✳ ❆♣♣r♦①✐♠❛t✐♦♥ t♦ ❇❛②❡s r✐s❦ ✐♥ r❡♣❡❛t❡❞ ♣❧❛②s✳ ❈♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ ❚❤❡♦r② ♦❢ ●❛♠❡s✱ ✸✿✾✼✕✶✸✾✱ ✶✾✺✼ ❬✹❪❚❤♦♠❛s ▼✳ ❈♦✈❡r✳ ❯♥✐✈❡rs❛❧ ♣♦rt❢♦❧✐♦s✳ ▼❛t❤❡♠❛t✐❝❛❧ ❋✐♥❛♥❝❡✱ ✶✭✶✮✿✶✕✷✾✱ ✶✾✾✶ ❬✷❪❆✈r✐♠ ❇❧✉♠ ❛♥❞ ❆❞❛♠ ❑❛❧❛✐✳ ❯♥✐✈❡rs❛❧ ♣♦rt❢♦❧✐♦s ✇✐t❤ ❛♥❞ ✇✐t❤♦✉t tr❛♥s❛❝t✐♦♥ ❝♦sts✳ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣✱ ✸✺✭✸✮✿✶✾✸✕✷✵✺✱ ✶✾✾✾

✸✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❆♣♣❧✐❝❛t✐♦♥s

❆♣♣❧✐❝❛t✐♦♥s

• ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♣❧❛②❡r ❞②♥❛♠✐❝s ✐♥ ❣❛♠❡s ✭❇❧❛❝❦✇❡❧❧ ❬✶❪✱ ❍❛♥♥❛♥❬✺❪✮

{✶, . . . , N} ✐s t❤❡ s❡t ♦❢ ❛❝t✐♦♥s✳

• ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣

{✶, . . . , N} ✐s t❤❡ tr❛✐♥✐♥❣ s❡t✳ ✏▼♦❞❡❧✲❢r❡❡✑ ♣♦rt❢♦❧✐♦ ♦♣t✐♠✐③❛t✐♦♥ ✭❈♦✈❡r ❬✹❪✱ ❇❧✉♠ ❬✷❪✮ ✶ ✐s t❤❡ s❡t ♦❢ st♦❝❦s✳ ▼❛♥② ♦t❤❡rs

❬✶❪❉❛✈✐❞ ❇❧❛❝❦✇❡❧❧✳ ❆♥ ❛♥❛❧♦❣ ♦❢ t❤❡ ♠✐♥✐♠❛① t❤❡♦r❡♠ ❢♦r ✈❡❝t♦r ♣❛②♦✛s✳ P❛❝✐✜❝ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✻✭✶✮✿✶✕✽✱ ✶✾✺✻ ❬✺❪❏❛♠❡s ❍❛♥♥❛♥✳ ❆♣♣r♦①✐♠❛t✐♦♥ t♦ ❇❛②❡s r✐s❦ ✐♥ r❡♣❡❛t❡❞ ♣❧❛②s✳ ❈♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ ❚❤❡♦r② ♦❢ ●❛♠❡s✱ ✸✿✾✼✕✶✸✾✱ ✶✾✺✼ ❬✹❪❚❤♦♠❛s ▼✳ ❈♦✈❡r✳ ❯♥✐✈❡rs❛❧ ♣♦rt❢♦❧✐♦s✳ ▼❛t❤❡♠❛t✐❝❛❧ ❋✐♥❛♥❝❡✱ ✶✭✶✮✿✶✕✷✾✱ ✶✾✾✶ ❬✷❪❆✈r✐♠ ❇❧✉♠ ❛♥❞ ❆❞❛♠ ❑❛❧❛✐✳ ❯♥✐✈❡rs❛❧ ♣♦rt❢♦❧✐♦s ✇✐t❤ ❛♥❞ ✇✐t❤♦✉t tr❛♥s❛❝t✐♦♥ ❝♦sts✳ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣✱ ✸✺✭✸✮✿✶✾✸✕✷✵✺✱ ✶✾✾✾

✸✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❆♣♣❧✐❝❛t✐♦♥s

❆♣♣❧✐❝❛t✐♦♥s

• ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♣❧❛②❡r ❞②♥❛♠✐❝s ✐♥ ❣❛♠❡s ✭❇❧❛❝❦✇❡❧❧ ❬✶❪✱ ❍❛♥♥❛♥❬✺❪✮

{✶, . . . , N} ✐s t❤❡ s❡t ♦❢ ❛❝t✐♦♥s✳

• ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣

{✶, . . . , N} ✐s t❤❡ tr❛✐♥✐♥❣ s❡t✳

• ✏▼♦❞❡❧✲❢r❡❡✑ ♣♦rt❢♦❧✐♦ ♦♣t✐♠✐③❛t✐♦♥ ✭❈♦✈❡r ❬✹❪✱ ❇❧✉♠ ❬✷❪✮

{✶, . . . , N} ✐s t❤❡ s❡t ♦❢ st♦❝❦s✳

• ▼❛♥② ♦t❤❡rs

❬✶❪❉❛✈✐❞ ❇❧❛❝❦✇❡❧❧✳ ❆♥ ❛♥❛❧♦❣ ♦❢ t❤❡ ♠✐♥✐♠❛① t❤❡♦r❡♠ ❢♦r ✈❡❝t♦r ♣❛②♦✛s✳ P❛❝✐✜❝ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✻✭✶✮✿✶✕✽✱ ✶✾✺✻ ❬✺❪❏❛♠❡s ❍❛♥♥❛♥✳ ❆♣♣r♦①✐♠❛t✐♦♥ t♦ ❇❛②❡s r✐s❦ ✐♥ r❡♣❡❛t❡❞ ♣❧❛②s✳ ❈♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ ❚❤❡♦r② ♦❢ ●❛♠❡s✱ ✸✿✾✼✕✶✸✾✱ ✶✾✺✼ ❬✹❪❚❤♦♠❛s ▼✳ ❈♦✈❡r✳ ❯♥✐✈❡rs❛❧ ♣♦rt❢♦❧✐♦s✳ ▼❛t❤❡♠❛t✐❝❛❧ ❋✐♥❛♥❝❡✱ ✶✭✶✮✿✶✕✷✾✱ ✶✾✾✶ ❬✷❪❆✈r✐♠ ❇❧✉♠ ❛♥❞ ❆❞❛♠ ❑❛❧❛✐✳ ❯♥✐✈❡rs❛❧ ♣♦rt❢♦❧✐♦s ✇✐t❤ ❛♥❞ ✇✐t❤♦✉t tr❛♥s❛❝t✐♦♥ ❝♦sts✳ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣✱ ✸✺✭✸✮✿✶✾✸✕✷✵✺✱ ✶✾✾✾

✹✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

▲❡❛r♥✐♥❣ ♦♥ ❛ ❝♦♥t✐♥✉✉♠

✏❲❤❛t ✐❢ t❤❡ ❛❝t✐♦♥ s❡t ✐s ✐♥✜♥✐t❡❄✑ Pr♦❜❧❡♠ ✶ ❖♥❧✐♥❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ♦♥ S✳

✶✿ ❢♦r t ∈ N ❞♦ ✷✿

❉❡❝✐s✐♦♥ ♠❛❦❡r ❝❤♦♦s❡s ❞✐str✐❜✉t✐♦♥ x(t) ♦✈❡r S✳

✸✿

❆ ❧♦ss ❢✉♥❝t✐♦♥ ℓ(t) : S → [✵, M] ✐s r❡✈❡❛❧❡❞✳

✹✿

❚❤❡ ❞❡❝✐s✐♦♥ ♠❛❦❡r ✐♥❝✉rs ❡①♣❡❝t❡❞ ❧♦ss

• x(t), ℓ(t)

=

• S

x(t)(s)ℓ(t)(s)λ(ds) = Es∼x(t)[ℓ(t)(s)]

✺✿ ❡♥❞ ❢♦r

❘❡❣r❡t

✶ ✶

s✉♣ s✉♣

✹✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

▲❡❛r♥✐♥❣ ♦♥ ❛ ❝♦♥t✐♥✉✉♠

✏❲❤❛t ✐❢ t❤❡ ❛❝t✐♦♥ s❡t ✐s ✐♥✜♥✐t❡❄✑ Pr♦❜❧❡♠ ✷ ❖♥❧✐♥❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ♦♥ S✳

✶✿ ❢♦r t ∈ N ❞♦ ✷✿

❉❡❝✐s✐♦♥ ♠❛❦❡r ❝❤♦♦s❡s ❞✐str✐❜✉t✐♦♥ x(t) ♦✈❡r S✳

✸✿

❆ ❧♦ss ❢✉♥❝t✐♦♥ ℓ(t) : S → [✵, M] ✐s r❡✈❡❛❧❡❞✳

✹✿

❚❤❡ ❞❡❝✐s✐♦♥ ♠❛❦❡r ✐♥❝✉rs ❡①♣❡❝t❡❞ ❧♦ss

• x(t), ℓ(t)

=

• S

x(t)(s)ℓ(t)(s)λ(ds) = Es∼x(t)[ℓ(t)(s)]

✺✿ ❡♥❞ ❢♦r

❘❡❣r❡t R(T)(x) =

T

• t=✶
• x(t), ℓ(t)

−

• x,

T

• t=✶

ℓ(t)

• s✉♣

(ℓ(t))

s✉♣

x∈∆N R(T)(x) = o(T)

✺✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❘❡s✉❧ts

❱❛r✐❛♥t ♦❢ t❤✐s ♣r♦❜❧❡♠✿ ❖♥❧✐♥❡ ♦♣t✐♠✐③❛t✐♦♥ ♦♥ ❝♦♥✈❡① s❡ts✳

❆ss✉♠♣t✐♦♥s ♦♥ ℓ(t) ❝♦♥✈❡① α✲❡①♣✲❝♦♥❝❛✈❡ ✉♥✐❢♦r♠❧② ▲✲▲✐♣s❝❤✐t③ ❆ss✉♠♣t✐♦♥s ♦♥ S ❝♦♥✈❡① ❝♦♥✈❡① v✲✉♥✐❢♦r♠❧② ❢❛t ▼❡t❤♦❞

• r❛❞✐❡♥t

✭❩✐♥❦❡✈✐❝❤ ❬✽❪✮ ❍❡❞❣❡✱ ❖◆❙✱ ❋❚❆▲ ✭❍❛③❛♥ ❡t ❛❧✳ ❬✻❪✮ ❍❡❞❣❡ ✭❚❤✐s t❛❧❦✮ ▲❡❛r♥✐♥❣ r❛t❡s ✶/√t α ✶/√t R(t) O √t

• O
• ❧♦❣ t
• O
• t ❧♦❣ t
• ❚❛❜❧❡✿ ❙♦♠❡ r❡❣r❡t ✉♣♣❡r ❜♦✉♥❞s ❢♦r ❞✐✛❡r❡♥t ❝❧❛ss❡s ♦❢ ❧♦ss❡s✳

❬✽❪▼❛rt✐♥ ❩✐♥❦❡✈✐❝❤✳ ❖♥❧✐♥❡ ❝♦♥✈❡① ♣r♦❣r❛♠♠✐♥❣ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ ✐♥✜♥✐t❡s✐♠❛❧ ❣r❛❞✐❡♥t ❛s❝❡♥t✳ ■♥ ■❈▼▲✱ ♣❛❣❡s ✾✷✽✕✾✸✻✱ ✷✵✵✸ ❬✻❪❊❧❛❞ ❍❛③❛♥✱ ❆♠✐t ❆❣❛r✇❛❧✱ ❛♥❞ ❙❛t②❡♥ ❑❛❧❡✳ ▲♦❣❛r✐t❤♠✐❝ r❡❣r❡t ❛❧❣♦r✐t❤♠s ❢♦r ♦♥❧✐♥❡ ❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✳ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣✱ ✻✾✭✷✲✸✮✿✶✻✾✕✶✾✷✱ ✷✵✵✼

✺✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❖✉t❧✐♥❡

✶ ❚❤❡ Pr♦❜❧❡♠ ✷ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ✸ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s

✻✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❍❡❞❣❡ ♦♥ ❛ ✜♥✐t❡ s❡t

❍❡❞❣❡ ❛❧❣♦r✐t❤♠ ✇✐t❤ ❧❡❛r♥✐♥❣ r❛t❡s (ηt)✳

✶✿ ❢♦r t ∈ N ❞♦ ✷✿

P❧❛② x(t)

✸✿

❘❡✈❡❛❧ ℓ(t) ∈ [✵, M]N✱ ❝❛❧❧ L(t) = t

τ=✶ ℓ(τ)

✹✿

❯♣❞❛t❡ x(t+✶)

n

∝ e−ηt+✶L(t)

n

✺✿ ❡♥❞ ❢♦r

❖♥❡ ✐♥t❡r♣r❡t❛t✐♦♥✿ ✐♥st❛♥❝❡ ♦❢ t❤❡ ❞✉❛❧ ❛✈❡r❛❣✐♥❣ ♠❡t❤♦❞ ❬✼❪ x(t+✶) ∈ ❛r❣ ♠✐♥

x∈∆N

• L(t), x
• +

✶ ηt+✶ ψ(x) ✇✐t❤ ψ(x) = N

n=✶ xn ❧♥ xn✳

❬✼❪❨✉r✐✐ ◆❡st❡r♦✈✳ Pr✐♠❛❧✲❞✉❛❧ s✉❜❣r❛❞✐❡♥t ♠❡t❤♦❞s ❢♦r ❝♦♥✈❡① ♣r♦❜❧❡♠s✳ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❣r❛♠♠✐♥❣✱ ✶✷✵✭✶✮✿✷✷✶✕✷✺✾✱ ✷✵✵✾

✼✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❍❡❞❣❡ ♦♥ ❛ ✜♥✐t❡ s❡t

❇❛s✐❝ ❘❡❣r❡t ❇♦✉♥❞ ❋♦r ❛❧❧ x ∈ ∆N✱ R(T)(x) ≤ M✷ ✷

t

• τ=✶

ητ+✶ + ψ(x) ηt+✶ ❚❛❦❡ ηt = θt− ✶

✷ ✱ t❤❡♥ t

✶ ητ = O(√t) ❛♥❞ ✶ t = O(√t)

■t s✉✣❝❡s t♦ ❜♦✉♥❞ ψ ♦♥ ∆N✳ ❲❤❡♥ ψ(x) =

i xi ❧♥ xi✱ ψ(x) ≤ ❧♥ N ♦♥ ∆N✳ ❙♦

s✉♣

x∈∆N R(T)(x) ≤

M✷θ ✷ + ❧♥ N θ √ T

✽✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❍❡❞❣❡ ♦♥ ❛ ❝♦♥t✐♥✉✉♠

❍❡❞❣❡ ♦♥ S ✇✐t❤ ❧❡❛r♥✐♥❣ r❛t❡s (ηt)✳

✶✿ ❢♦r t ∈ N ❞♦ ✷✿

P❧❛② ∼ x(t)

✸✿

❘❡✈❡❛❧ ℓ(t) : S → [✵, M]

✹✿

❯♣❞❛t❡ x(t+✶)(s) ∝ x(✵)(s)e−ηt+✶L(t)(s)

✺✿ ❡♥❞ ❢♦r

❖♥❡ ✐♥t❡r♣r❡t❛t✐♦♥✿ ✐♥st❛♥❝❡ ♦❢ t❤❡ ❞✉❛❧ ❛✈❡r❛❣✐♥❣ ♠❡t❤♦❞

✶

❛r❣ ♠✐♥ ✶

✶

✇✐t❤ ❍✐❧❜❡rt s♣❛❝❡

✷

✱

✷

✵

✶

✶ ❧♥

✽✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❍❡❞❣❡ ♦♥ ❛ ❝♦♥t✐♥✉✉♠

❍❡❞❣❡ ♦♥ S ✇✐t❤ ❧❡❛r♥✐♥❣ r❛t❡s (ηt)✳

✶✿ ❢♦r t ∈ N ❞♦ ✷✿

P❧❛② ∼ x(t)

✸✿

❘❡✈❡❛❧ ℓ(t) : S → [✵, M]

✹✿

❯♣❞❛t❡ x(t+✶)(s) ∝ x(✵)(s)e−ηt+✶L(t)(s)

✺✿ ❡♥❞ ❢♦r

❖♥❡ ✐♥t❡r♣r❡t❛t✐♦♥✿ ✐♥st❛♥❝❡ ♦❢ t❤❡ ❞✉❛❧ ❛✈❡r❛❣✐♥❣ ♠❡t❤♦❞ x(t+✶) ∈ ❛r❣ ♠✐♥

x∈∆(S)

• L(t), x
• +

✶ ηt+✶ ψ(x) ✇✐t❤

• ❍✐❧❜❡rt s♣❛❝❡ H = L✷(S)✱ ℓ, x =
• S ℓ(s)x(s)λ(ds)
• ∆(S) = {x ∈ L✷(S) : x ≥ ✵, x✶ = ✶}
• ψ(x) =
• S x(s) ❧♥ x(s)λ(ds)

✾✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❍❡❞❣❡ ♦♥ ❛ ❝♦♥t✐♥✉✉♠

❇❛s✐❝ ❘❡❣r❡t ❇♦✉♥❞ ❋♦r ❛❧❧ x ∈ ∆(S)✱ R(T)(x) ≤ M✷ ✷

t

• τ=✶

ητ+✶ + ψ(x) ηt+✶ ❇✉t ψ ✐s ✉♥❜♦✉♥❞❡❞ ♦♥ ∆(S)✳ ❚❛❦❡

✶ ✶

❢♦r s♦♠❡ ✳ ❚❤❡♥ ❧♥ ❧♥ ✶ ❝❛♥ ❜❡ ❛r❜✐tr❛r✐❧② ❧❛r❣❡ ❢♦r ❛r❜✐tr❛r✐❧② s♠❛❧❧ ✳

✾✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❍❡❞❣❡ ♦♥ ❛ ❝♦♥t✐♥✉✉♠

❇❛s✐❝ ❘❡❣r❡t ❇♦✉♥❞ ❋♦r ❛❧❧ x ∈ ∆(S)✱ R(T)(x) ≤ M✷ ✷

t

• τ=✶

ητ+✶ + ψ(x) ηt+✶ ❇✉t ψ ✐s ✉♥❜♦✉♥❞❡❞ ♦♥ ∆(S)✳ ❚❛❦❡ x =

✶ λ(A)✶A ❢♦r s♦♠❡ A ⊂ S✳ ❚❤❡♥

ψ(x) =

• S

x(s) ❧♥ x(s)λ(ds) = ❧♥ ✶ λ(A) ❝❛♥ ❜❡ ❛r❜✐tr❛r✐❧② ❧❛r❣❡ ❢♦r ❛r❜✐tr❛r✐❧② s♠❛❧❧ A✳

✶✵✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❲♦r❦✐♥❣ ❛r♦✉♥❞ ✉♥❜♦✉♥❞❡❞ r❡❣✉❧❛r✐③❡rs

■❞❡❛✿

• ❈❛❧❧ s⋆

t ∈ ❛r❣ ♠✐♥s∈S L(t)(s) ✭L s✉♣♣♦s❡❞ ❝♦♥t✐♥✉♦✉s✮✳

R(t)(x) =

t

• τ=✶
• ℓ(τ), x(τ) − x
• ≤

t

• τ=✶
• ℓ(τ), x(τ) − δs⋆

t

• =

t

• τ=✶
• ℓ(τ), x(τ) − y
• +

t

• τ=✶
• ℓ(τ), y − δs⋆

t

• = R(t)(y) +
• L(t), y − δs⋆

t

• ❚❛❦❡ y ∈ Bt✱ s❡t ♦❢ ❞✐str✐❜✉t✐♦♥s s✉♣♣♦rt❡❞ ♥❡❛r s⋆

t

❘❡✈✐s❡❞ r❡❣r❡t ❜♦✉♥❞ s✉♣

x∈∆(S)

R(t)(x) ≤ R(t)(y✵) + s✉♣

y∈Bt

• L(t), y − δs⋆

t

✶✶✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❲♦r❦✐♥❣ ❛r♦✉♥❞ ✉♥❜♦✉♥❞❡❞ r❡❣✉❧❛r✐③❡rs

❚❛❦❡ y✵ =

✶ λ(At)✶At

s✉♣

x∈∆(S)

R(t)(x) ≤ R(t)(y✵) + s✉♣

y∈Bt

• L(t), y − δs⋆

t

• R(t)(y✵)

≤ M✷ ✷

t

• τ=✶

ητ+✶ + ψ(y✵) ηt+✶ ≤ M✷ ✷

t

• τ=✶

ητ+✶ + ✶ ηt+✶ ❧♥ ✶ λ(At)

• L(t), y − δs⋆

t

• =
• At

y(s)(L(t)(s) − L(t)(s⋆

t ))λ(ds)

≤ Ltd(At)

✶✷✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❯♥✐❢♦r♠❧② ❢❛t s❡ts

❯♥✐❢♦r♠ ❢❛t♥❡ss S ✐s v✲✉♥✐❢♦r♠❧② ❢❛t ✭✇✳r✳t✳ t❤❡ ♠❡❛s✉r❡ λ✮ ✐❢ ∀s ∈ S✱ ∃ ❝♦♥✈❡① Ks ⊂ S✱ ✇✐t❤ s ∈ Ks ❛♥❞ λ(Ks) ≥ v✳

s Ks S

✶✷✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❯♥✐❢♦r♠❧② ❢❛t s❡ts

❯♥✐❢♦r♠ ❢❛t♥❡ss S ✐s v✲✉♥✐❢♦r♠❧② ❢❛t ✭✇✳r✳t✳ t❤❡ ♠❡❛s✉r❡ λ✮ ✐❢ ∀s ∈ S✱ ∃ ❝♦♥✈❡① Ks ⊂ S✱ ✇✐t❤ s ∈ Ks ❛♥❞ λ(Ks) ≥ v✳

s S Ks

✶✷✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❯♥✐❢♦r♠❧② ❢❛t s❡ts

❯♥✐❢♦r♠ ❢❛t♥❡ss S ✐s v✲✉♥✐❢♦r♠❧② ❢❛t ✭✇✳r✳t✳ t❤❡ ♠❡❛s✉r❡ λ✮ ✐❢ ∀s ∈ S✱ ∃ ❝♦♥✈❡① Ks ⊂ S✱ ✇✐t❤ s ∈ Ks ❛♥❞ λ(Ks) ≥ v✳

s S Ks

✶✷✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❯♥✐❢♦r♠❧② ❢❛t s❡ts

❯♥✐❢♦r♠ ❢❛t♥❡ss S ✐s v✲✉♥✐❢♦r♠❧② ❢❛t ✭✇✳r✳t✳ t❤❡ ♠❡❛s✉r❡ λ✮ ✐❢ ∀s ∈ S✱ ∃ ❝♦♥✈❡① Ks ⊂ S✱ ✇✐t❤ s ∈ Ks ❛♥❞ λ(Ks) ≥ v✳

s S Ks

✶✸✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❘❡❣r❡t ❜♦✉♥❞ ♦♥ ✉♥✐❢♦r♠❧② ❢❛t s❡ts

❋✐♥❛❧ ❜♦✉♥❞ s✉♣

x∈∆(S)

R(t)(x) ≤ M✷ ✷

t

• τ=✶

ητ+✶ + ❧♥ ✶

v

ηt+✶ + n ❧♥ t ηt+✶ + Ld(S)

❈❛♥ ♦♣t✐♠✐③❡ ♦✈❡r ηt t♦ ❣❡t s✉♣

x∈∆(S)

R(t)(x) ≤ Ld(S) + M √ t

• n ❧♥ t + ❧♥ ✶

v

✷

✶✹✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❇❡②♦♥❞ ❍❡❞❣❡

❉✉❛❧ ❛✈❡r❛❣✐♥❣ ✇✐t❤ ❧❡❛r♥✐♥❣ r❛t❡s (ηt)✱ str♦♥❣❧② ❝♦♥✈❡① r❡❣✉❧❛r✐③❡r ψ

✶✿ ❢♦r t ∈ N ❞♦ ✷✿

P❧❛② x(t)

✸✿

❉✐s❝♦✈❡r ℓ(t) ∈ H∗

✹✿

❯♣❞❛t❡ x(t+✶) = ❛r❣ ♠✐♥

x∈∆(S)

• L(t), x
• +

✶ ηt+✶ ψ(x) ✭✶✮

✺✿ ❡♥❞ ❢♦r

✐s ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧✳ ❈❛♥ ✇❡ s♦❧✈❡ ♠✐♥ ✶

✶

❈❛♥ ✇❡ ♦❜t❛✐♥ ❛ s✉❜❧✐♥❡❛r r❡❣r❡t ❜♦✉♥❞❄ s✉♣

✷

✷

✶ ✶

✶

✶

✶✹✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❇❡②♦♥❞ ❍❡❞❣❡

❉✉❛❧ ❛✈❡r❛❣✐♥❣ ✇✐t❤ ❧❡❛r♥✐♥❣ r❛t❡s (ηt)✱ str♦♥❣❧② ❝♦♥✈❡① r❡❣✉❧❛r✐③❡r ψ

✶✿ ❢♦r t ∈ N ❞♦ ✷✿

P❧❛② x(t)

✸✿

❉✐s❝♦✈❡r ℓ(t) ∈ H∗

✹✿

❯♣❞❛t❡ x(t+✶) = ❛r❣ ♠✐♥

x∈∆(S)

• L(t), x
• +

✶ ηt+✶ ψ(x) ✭✶✮

✺✿ ❡♥❞ ❢♦r

• H ✐s ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧✳ ❈❛♥ ✇❡ s♦❧✈❡

♠✐♥

x∈∆(S)

• L(t), x
• +

✶ ηt+✶ ψ(x) ❈❛♥ ✇❡ ♦❜t❛✐♥ ❛ s✉❜❧✐♥❡❛r r❡❣r❡t ❜♦✉♥❞❄ s✉♣

✷

✷

✶ ✶

✶

✶

✶✹✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❇❡②♦♥❞ ❍❡❞❣❡

❉✉❛❧ ❛✈❡r❛❣✐♥❣ ✇✐t❤ ❧❡❛r♥✐♥❣ r❛t❡s (ηt)✱ str♦♥❣❧② ❝♦♥✈❡① r❡❣✉❧❛r✐③❡r ψ

✶✿ ❢♦r t ∈ N ❞♦ ✷✿

P❧❛② x(t)

✸✿

❉✐s❝♦✈❡r ℓ(t) ∈ H∗

✹✿

❯♣❞❛t❡ x(t+✶) = ❛r❣ ♠✐♥

x∈∆(S)

• L(t), x
• +

✶ ηt+✶ ψ(x) ✭✶✮

✺✿ ❡♥❞ ❢♦r

• H ✐s ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧✳ ❈❛♥ ✇❡ s♦❧✈❡

♠✐♥

x∈∆(S)

• L(t), x
• +

✶ ηt+✶ ψ(x)

• ❈❛♥ ✇❡ ♦❜t❛✐♥ ❛ s✉❜❧✐♥❡❛r r❡❣r❡t ❜♦✉♥❞❄

s✉♣

x∈∆(S)

R(t)(x) ≤ M✷ ✷

t

• τ=✶

ητ+✶ + ✶ ηt+✶ ψ(y) + Ltd(At)

✶✹✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❖✉t❧✐♥❡

✶ ❚❤❡ Pr♦❜❧❡♠ ✷ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ✸ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s

✶✺✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡

❍❡❞❣❡ ❛❧❣♦r✐t❤♠ ♦♥ ❤♦❧❧♦✇ ❝✉❜❡ ✐♥ R✸✳

100 101 102 103

t

10-2 10-1 100 101

Rt/t log time-avg. cumulative regret

Quadratic Affine Polynomial regret bound

❋✐❣✉r❡✿ P❡r✲r♦✉♥❞ r❡❣r❡t

✶✻✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡

❍❡❞❣❡ ❛❧❣♦r✐t❤♠

• ♦♥ s❡t S
• ✇✐t❤ ▲✐♣s❝❤✐t③ ❧♦ss❡s
• ✇✐t❤ ηt = Θ
• ❧♥ t

t

100 101 102 103 104

t

10-1 100 101

Rt/t log time-avg. cumulative regret ηt =0.13 q

logt t

bound ηt =0.39 q

logt t

bound

❋✐❣✉r❡✿ ❊✛❡❝t ♦❢ t❤❡ ❧❡❛r♥✐♥❣ r❛t❡ ηt = θ

• ❧♥ t

t

✶✼✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡

❋✐❣✉r❡✿ ❊✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❍❡❞❣❡ ❞❡♥s✐t②

✶✽✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❈♦♥❝❧✉s✐♦♥

❙✉♠♠❛r②

• ❈❛♥ ❧❡❛r♥ ♦♥ ❛ ❝♦♥t✐♥✉✉♠✱ ✇❤❡♥ ❧♦ss❡s ❛r❡ ▲✐♣s❝❤✐t③ ❛♥❞ S ❤❛s r❡❛s♦♥❛❜❧❡

❣❡♦♠❡tr②✳

• ❙✐♠✐❧❛r ❣✉❛r❛♥t❡❡ t♦ ❧❡❛r♥✐♥❣ ♦♥ ❛ ❝♦✈❡r✱ ❜✉t ❞♦ ♥♦t ♥❡❡❞ t♦ ♠❛✐♥t❛✐♥ ❛

❝♦✈❡r✳

• ❈❛♥ ❣❡♥❡r❛❧✐③❡ t♦ t❤❡ ❞✉❛❧ ❛✈❡r❛❣✐♥❣ ♠❡t❤♦❞✳

❊①t❡♥s✐♦♥s ❛♥❞ ♦♣❡♥ q✉❡st✐♦♥s

• ❇❛♥❞✐t ❢♦r♠✉❧❛t✐♦♥✱ ❡✳❣✳ ❬✸❪✳
• ❘❡❣r❡t ❧♦✇❡r ❜♦✉♥❞✳
• ❲❤❡♥ ✐s ✐t ❡❛s② t♦ s❛♠♣❧❡ ❢r♦♠ t❤❡ ❍❡❞❣❡ ❞✐str✐❜✉t✐♦♥❄

❬✸❪❙é❜❛st✐❡♥ ❇✉❜❡❝❦✱ ❘é♠✐ ▼✉♥♦s✱ ●✐❧❧❡s ❙t♦❧t③✱ ❛♥❞ ❈s❛❜❛ ❙③❡♣❡s✈❛r✐✳ ❳✲❛r♠❡❞ ❜❛♥❞✐ts✳ ❏♦✉r♥❛❧ ♦❢ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ ❘❡s❡❛r❝❤ ✭❏▼▲❘✮✱ ✶✷✭✶✷✮✿✶✺✽✼✕✶✻✷✼✱ ✷✵✶✶

✶✾✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❚❤❛♥❦ ②♦✉

❚❤❛♥❦ ②♦✉✳ ④❦r✐❝❤❡♥❡✱ ❜❛❧❛♥❞❛t⑥❅❡❡❝s✳❜❡r❦❡❧❡②✳❡❞✉

✷✵✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s ■

❬✶❪ ❉❛✈✐❞ ❇❧❛❝❦✇❡❧❧✳ ❆♥ ❛♥❛❧♦❣ ♦❢ t❤❡ ♠✐♥✐♠❛① t❤❡♦r❡♠ ❢♦r ✈❡❝t♦r ♣❛②♦✛s✳ P❛❝✐✜❝ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✻✭✶✮✿✶✕✽✱ ✶✾✺✻✳ ❬✷❪ ❆✈r✐♠ ❇❧✉♠ ❛♥❞ ❆❞❛♠ ❑❛❧❛✐✳ ❯♥✐✈❡rs❛❧ ♣♦rt❢♦❧✐♦s ✇✐t❤ ❛♥❞ ✇✐t❤♦✉t tr❛♥s❛❝t✐♦♥ ❝♦sts✳ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣✱ ✸✺✭✸✮✿✶✾✸✕✷✵✺✱ ✶✾✾✾✳ ❬✸❪ ❙é❜❛st✐❡♥ ❇✉❜❡❝❦✱ ❘é♠✐ ▼✉♥♦s✱ ●✐❧❧❡s ❙t♦❧t③✱ ❛♥❞ ❈s❛❜❛ ❙③❡♣❡s✈❛r✐✳ ❳✲❛r♠❡❞ ❜❛♥❞✐ts✳ ❏♦✉r♥❛❧ ♦❢ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ ❘❡s❡❛r❝❤ ✭❏▼▲❘✮✱ ✶✷✭✶✷✮✿ ✶✺✽✼✕✶✻✷✼✱ ✷✵✶✶✳ ❬✹❪ ❚❤♦♠❛s ▼✳ ❈♦✈❡r✳ ❯♥✐✈❡rs❛❧ ♣♦rt❢♦❧✐♦s✳ ▼❛t❤❡♠❛t✐❝❛❧ ❋✐♥❛♥❝❡✱ ✶✭✶✮✿✶✕✷✾✱ ✶✾✾✶✳ ❬✺❪ ❏❛♠❡s ❍❛♥♥❛♥✳ ❆♣♣r♦①✐♠❛t✐♦♥ t♦ ❇❛②❡s r✐s❦ ✐♥ r❡♣❡❛t❡❞ ♣❧❛②s✳ ❈♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ ❚❤❡♦r② ♦❢ ●❛♠❡s✱ ✸✿✾✼✕✶✸✾✱ ✶✾✺✼✳ ❬✻❪ ❊❧❛❞ ❍❛③❛♥✱ ❆♠✐t ❆❣❛r✇❛❧✱ ❛♥❞ ❙❛t②❡♥ ❑❛❧❡✳ ▲♦❣❛r✐t❤♠✐❝ r❡❣r❡t ❛❧❣♦r✐t❤♠s ❢♦r ♦♥❧✐♥❡ ❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✳ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣✱ ✻✾✭✷✲✸✮✿✶✻✾✕✶✾✷✱ ✷✵✵✼✳ ❬✼❪ ❨✉r✐✐ ◆❡st❡r♦✈✳ Pr✐♠❛❧✲❞✉❛❧ s✉❜❣r❛❞✐❡♥t ♠❡t❤♦❞s ❢♦r ❝♦♥✈❡① ♣r♦❜❧❡♠s✳ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❣r❛♠♠✐♥❣✱ ✶✷✵✭✶✮✿✷✷✶✕✷✺✾✱ ✷✵✵✾✳ ❬✽❪ ▼❛rt✐♥ ❩✐♥❦❡✈✐❝❤✳ ❖♥❧✐♥❡ ❝♦♥✈❡① ♣r♦❣r❛♠♠✐♥❣ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ ✐♥✜♥✐t❡s✐♠❛❧ ❣r❛❞✐❡♥t ❛s❝❡♥t✳ ■♥ ■❈▼▲✱ ♣❛❣❡s ✾✷✽✕✾✸✻✱ ✷✵✵✸✳

✷✶✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❆❞❞✐t✐♦♥❛❧ s❧✐❞❡s✿ ❘❡❣r❡t ❜♦✉♥❞ ♦♥ ✉♥✐❢♦r♠❧② ❢❛t s❡ts

s✉♣

x∈∆(S)

R(t)(x) ≤ M✷ ✷

t

• τ=✶

ητ+✶ + ✶ ηt+✶ ❧♥ ✶ λ(At) + Ltd(At)

s⋆

t

S Ks⋆

t

At

❋✐❣✉r❡✿ At = s⋆

t + dt(Ks⋆

t − s⋆

t )✳ ❚❤❡♥ λ(At) ≥ dn t v ❛♥❞ d(At) ≤ dtd(S)✳

s✉♣

x∈∆(S)

R(t)(x) ≤ M✷ ✷

t

• τ=✶

ητ+✶ + ✶ ηt+✶ ❧♥ ✶ vdn

t

+ Ltdtd(S)

✷✷✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❆❞❞✐t✐♦♥❛❧ s❧✐❞❡s✿ ❍❡❞❣❡ ❱s✳ ❧❡❛r♥✐♥❣ ♦♥ ❛ ❝♦✈❡r

• ●✐✈❡♥ ❛ ❤♦r✐③♦♥ T ❛♥❞ ❛ ❝♦✈❡r AT ✇✐t❤ d(A) ≤ dTd(S) ❢♦r ❛❧❧ A ∈ AT✳
• ❘✉♥ ❞✐s❝r❡t❡ ❍❡❞❣❡ ♦♥ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❝♦✈❡r✳

❚❤❡♥

✷

✽ ❧♥

❉✐s❝r❡t❡ ❍❡❞❣❡ ❆❞❞✐t✐♦♥❛❧ r❡❣r❡t

❲✐t❤

✶ ✱ ✷

✽ ❧♥

✶

❍❛✈❡ t♦ ❡①♣❧✐❝✐t❧② ❝♦♠♣✉t❡ ❛ ✭❤✐❡r❛r❝❤✐❝❛❧✮ ❝♦✈❡r✳

✷✷✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❆❞❞✐t✐♦♥❛❧ s❧✐❞❡s✿ ❍❡❞❣❡ ❱s✳ ❧❡❛r♥✐♥❣ ♦♥ ❛ ❝♦✈❡r

• ●✐✈❡♥ ❛ ❤♦r✐③♦♥ T ❛♥❞ ❛ ❝♦✈❡r AT ✇✐t❤ d(A) ≤ dTd(S) ❢♦r ❛❧❧ A ∈ AT✳
• ❘✉♥ ❞✐s❝r❡t❡ ❍❡❞❣❡ ♦♥ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❝♦✈❡r✳
• ❚❤❡♥

R(T)(x) ≤ M✷Tη ✽ + ❧♥ |AT| η

• ❉✐s❝r❡t❡ ❍❡❞❣❡

+ Ld(S)dT

• ❆❞❞✐t✐♦♥❛❧ r❡❣r❡t
• ❲✐t❤ |AT| ≈

✶ dn

T ✱

R(T)(x) ≤ M✷Tη ✽ + ❧♥

✶ dn

T

η + LD(S)dT ❍❛✈❡ t♦ ❡①♣❧✐❝✐t❧② ❝♦♠♣✉t❡ ❛ ✭❤✐❡r❛r❝❤✐❝❛❧✮ ❝♦✈❡r✳

✷✷✴✶✾ ❚❤❡ Pr♦❜❧❡♠ ❍❡❞❣❡ ♦♥ ❛ ❈♦♥t✐♥✉✉♠ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❘❡❢❡r❡♥❝❡s

❆❞❞✐t✐♦♥❛❧ s❧✐❞❡s✿ ❍❡❞❣❡ ❱s✳ ❧❡❛r♥✐♥❣ ♦♥ ❛ ❝♦✈❡r

• ●✐✈❡♥ ❛ ❤♦r✐③♦♥ T ❛♥❞ ❛ ❝♦✈❡r AT ✇✐t❤ d(A) ≤ dTd(S) ❢♦r ❛❧❧ A ∈ AT✳
• ❘✉♥ ❞✐s❝r❡t❡ ❍❡❞❣❡ ♦♥ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❝♦✈❡r✳
• ❚❤❡♥

R(T)(x) ≤ M✷Tη ✽ + ❧♥ |AT| η

• ❉✐s❝r❡t❡ ❍❡❞❣❡

+ Ld(S)dT

• ❆❞❞✐t✐♦♥❛❧ r❡❣r❡t
• ❲✐t❤ |AT| ≈

✶ dn

T ✱

R(T)(x) ≤ M✷Tη ✽ + ❧♥

✶ dn

T

η + LD(S)dT

• ❍❛✈❡ t♦ ❡①♣❧✐❝✐t❧② ❝♦♠♣✉t❡ ❛ ✭❤✐❡r❛r❝❤✐❝❛❧✮ ❝♦✈❡r✳