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• r❡❡❞② ❢✉♥❝t✐♦♥ ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ❧❡❛r♥✐♥❣ t♦ r❛♥❦

➚♥❞r❡② ●✉❧✐♥✱ P❛✈❡❧ ❑❛r♣♦✈✐❝❤ P❡tr♦③❛✈♦❞s❦ ✷✵✵✾

❆♥♥♦t❛t✐♦♥

• r❡❡❞② ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♥❞ ❜♦♦st✐♥❣ ❛❧❣♦r✐t❤♠s ❛r❡ ✇❡❧❧

s✉✐t❡❞ ❢♦r s♦❧✈✐♥❣ ♣r❛❝t✐❝❛❧ ♠❛❝❤✐♥❡ ❧❡❛r♥✐♥❣ t❛s❦s✳ ❲❡ ✇✐❧❧ ❞❡s❝r✐❜❡ ✇❡❧❧✲❦♥♦✇♥ ❜♦♦st✐♥❣ ❛❧❣♦r✐t❤♠s ❛♥❞ t❤❡✐r ♠♦❞✐✜❝❛t✐♦♥s ✉s❡❞ ❢♦r s♦❧✈✐♥❣ ❧❡❛r♥✐♥❣ t♦ r❛♥❦ ♣r♦❜❧❡♠s✳

❈♦♥t❡♥t

❼ ❙❡❛r❝❤ ❡♥❣✐♥❡ r❛♥❦✐♥❣✳

❼ ❊✈❛❧✉❛t✐♦♥ ♠❡❛s✉r❡s✳ ❼ ❋❡❛t✉r❡ ❜❛s❡❞ r❛♥❦✐♥❣ ♠♦❞❡❧✳ ❼ ▲❡❛r♥✐♥❣ t♦ r❛♥❦✳ ❖♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✭❧✐st✇✐s❡✱ ♣♦✐t♥✇✐s❡✱

♣❛✐r✇✐s❡ ❛♣♣r♦❛❝❤❡s✮✳ ❼ P♦✐♥t✇✐s❡ ❛♣♣r♦❛❝❤✳ ❇♦♦st✐♥❣ ❛❧❣♦r✐t❤♠s ❛♥❞ ❣r❡❡❞② ❢✉♥❝t✐♦♥

❛♣♣r♦①✐♠❛t✐♦♥✳

❼ ▼♦❞✐✜❝❛t✐♦♥ ▼❛tr✐①◆❡t✳ ❼ ▲✐st✇✐s❡ ❛♣♣r♦❛❝❤✳ ❆♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❝♦♠♣❧❡① ❡✈❛❧✉❛t✐♦♥

♠❡❛s✉r❡s✭❉❈●✱ ♥❉❈●✮✳

❙❡❛r❝❤ ❡♥❣✐♥❡ r❛♥❦✐♥❣

▼❛✐♥ ❣♦❛❧✿ t♦ r❛♥❦ ❞♦❝✉♠❡♥ts ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r q✉❛❧✐t② ♦❢ ❝♦♥❢♦r♠❛♥❝❡ t♦ t❤❡ s❡❛r❝❤ q✉❡r②✳ ❍♦✇ t♦ ❡✈❛❧✉❛t❡ r❛♥❦✐♥❣❄ Pr❡r❡q✉✐s✐t❡s✿

❼ ❙❡t ♦❢ s❡❛r❝❤ q✉❡r✐❡s Q = {q1, .., qn}✳ ❼ ❙❡t ♦❢ ❞♦❝✉♠❡♥ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❡❛❝❤ q✉❡r② q ∈ Q ✳

q → {d1, d2, ...}

❼ ❘❡❧❡✈❛♥❝❡ ❥✉❞❣♠❡♥ts ❢♦r ❡❛❝❤ ♣❛✐r (query, document) ✭■♥ ♦✉r ♠♦❞❡❧ r❡❛❧ ♥✉♠❜❡rs rel(q, d) ∈ [0, 1]✮

❊✈❛❧✉❛t✐♦♥ ♠❡❛s✉r❡s

❊✈❛❧✉❛t✐♦♥ ♠❛r❦ ❢♦r r❛♥❦✐♥❣ ✇✐❧❧ ❜❡ ❛♥ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❡✈❛❧✉❛t✐♦♥ ♠❡❛s✉r❡ ♦✈❡r t❤❡ s❡t ♦❢ s❡❛r❝❤ q✉❡r✐❡s Q✿

• q∈Q

EvMeas(ranking for query q) n ❊①❛♠♣❧❡ ♦❢ ❡✈❛❧✉❛t✐♦♥ ♠❡❛s✉r❡ EvMeas✿

❼ Pr❡❝✐s✐♦♥✲✶✵ ✲ ♣❡r❝❡♥t ♦❢ ❞♦❝✉♠❡♥ts ✇✐t❤ r❡❧❡✈❛♥❝❡

❥✉❞❣♠❡♥ts ❣r❡❛t❡r t❤❛♥ ✵ ✐♥ t♦♣✲✶✵

❊✈❛❧✉❛t✐♦♥ ♠❡❛s✉r❡s

❊✈❛❧✉❛t✐♦♥ ♠❛r❦ ❢♦r r❛♥❦✐♥❣ ✇✐❧❧ ❜❡ ❛♥ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❡✈❛❧✉❛t✐♦♥ ♠❡❛s✉r❡ ♦✈❡r t❤❡ s❡t ♦❢ s❡❛r❝❤ q✉❡r✐❡s Q✿

• q∈Q

EvMeas(ranking for query q) n ❊①❛♠♣❧❡ ♦❢ ❡✈❛❧✉❛t✐♦♥ ♠❡❛s✉r❡ EvMeas✿

❼ Pr❡❝✐s✐♦♥✲✶✵ ✲ ♣❡r❝❡♥t ♦❢ ❞♦❝✉♠❡♥ts ✇✐t❤ r❡❧❡✈❛♥❝❡

❥✉❞❣♠❡♥ts ❣r❡❛t❡r t❤❛♥ ✵ ✐♥ t♦♣✲✶✵

❊✈❛❧✉❛t✐♦♥ ♠❡❛s✉r❡s

❼ ▼❆P ✲ ♠❡❛♥ ❛✈❡r❛❣❡ ♣r❡❝✐s✐♦♥

MAP(ranking for query q) = 1 k

k

• i=1

i nr(i) k ✲ ♥✉♠❜❡r ♦❢ ❞♦❝✉♠❡♥ts ✇✐t❤ ♣♦s✐t✐✈❡ r❡❧❡✈❛♥❝❡ ❥✉❞❣♠❡♥ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ q✉❡r② q✱ nr(i) ✲ ♣♦s✐t✐♦♥ ♦❢ t❤❡ i✲t❤ ❞♦❝✉♠❡♥t ✇✐t❤ r❡❧❡✈❛♥❝❡ ❥✉❞❣♠❡♥t ❣r❡❛t❡r t❤❛♥ ✵✳

❊✈❛❧✉❛t✐♦♥ ♠❡❛s✉r❡s

❼ ❉❈● ✲ ❞✐s❝♦✉♥t❡❞ ❝✉♠✉❧❛t✐✈❡ ❣❛✐♥

DCG(ranking for query q) =

Nq

• j=1

relj log2j + 1 Nq ✲ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❞♦❝✉♠❡♥ts ✐♥ r❛♥❦❡❞ ❧✐st✱ relj ✲ r❡❧❡✈❛♥❝❡ ❥✉❞❣♠❡♥t ❢♦r ❞♦❝✉♠❡♥t ♦♥ ♣♦s✐t✐♦♥ j✳

❼ ♥♦r♠❛❧✐③❡❞ ❉❈●✭♥❉❈●✮

nDCG(...) = DCG(ranking for query q) DCG(ideal ranking for query q)

❋❡❛t✉r❡ ❜❛s❡❞ r❛♥❦✐♥❣ ♠♦❞❡❧

❼ ❊❛❝❤ ♣❛✐r (query, document) ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ✈❡❝t♦r ♦❢

❢❡❛t✉r❡s✳ (q, d) → (f1(q, d), f2(q, d), ..)

❼ ❙❡❛r❝❤ r❛♥❦✐♥❣ ✐s t❤❡ s♦rt✐♥❣ ❜② t❤❡ ✈❛❧✉❡ ♦❢ ✧r❡❧❡✈❛♥❝❡

❢✉♥❝t✐♦♥✧✳ ❘❡❧❡✈❛♥❝❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❢❡❛t✉r❡s✿ fr(q, d) = 3.14 · log7(f9(q, d)) + ef66(q,d) + ...

❋❡❛t✉r❡ ❜❛s❡❞ r❛♥❦✐♥❣ ♠♦❞❡❧

❼ ❊❛❝❤ ♣❛✐r (query, document) ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ✈❡❝t♦r ♦❢

❢❡❛t✉r❡s✳ (q, d) → (f1(q, d), f2(q, d), ..)

❼ ❙❡❛r❝❤ r❛♥❦✐♥❣ ✐s t❤❡ s♦rt✐♥❣ ❜② t❤❡ ✈❛❧✉❡ ♦❢ ✧r❡❧❡✈❛♥❝❡

❢✉♥❝t✐♦♥✧✳ ❘❡❧❡✈❛♥❝❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❢❡❛t✉r❡s✿ fr(q, d) = 3.14 · log7(f9(q, d)) + ef66(q,d) + ...

❖♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s

❍♦✇ t♦ ❣❡t ❛ ❣♦♦❞ r❡❧❡✈❛♥❝❡ ❢✉♥❝t✐♦♥❄

• ❡t ❧❡❛r♥✐♥❣ s❡t ♦❢ ❡①❛♠♣❧❡s Pl ✲ s❡t ♦❢ ♣❛✐rs (q, d) ✇✐t❤ r❡❧❡✈❛♥❝❡

❥✉❞❣♠❡♥ts rel(q, d)✳ ❯s❡ ❧❡❛r♥✐♥❣ t♦ r❛♥❦ ♠❡t❤♦❞s t♦ ♦❜t❛✐♥ fr✳

❖♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✭❧✐st✇✐s❡ ❛♣♣r♦❛❝❤✮

❼ ❙♦❧✈❡ ❞✐r❡❝t ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✿

arg max

fr∈F =

• q∈Ql

EvMeas(ranking for query q with fr) n F ✲ s❡t ♦❢ ♣♦ss✐❜❧❡ r❛♥❦✐♥❣ ❢✉♥❝t✐♦♥s✳ Ql ✲ s❡t ♦❢ ❞✐✛❡r❡♥t q✉❡r✐❡s ✐♥ ❧❡❛r♥✐♥❣ s❡t Pl ❉✐✣❝✉❧t② ✐♥ s♦❧✈✐♥❣✿ ♠♦st ♦❢ ❡✈❛❧✉❛t✐♦♥ ♠❡❛s✉r❡s ❛r❡ ♥♦♥✲❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✳

❖♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✭♣♦✐♥t✇✐s❡ ❛♣♣r♦❛❝❤✮

❼ ❙✐♠♣❧✐❢② ♦♣t✐♠✐③❛t✐♦♥ t❛s❦ t♦ r❡❣r❡ss✐♦♥ ♣r♦❜❧❡♠ ❛♥❞ ♠✐♥✐♠✐③❡

s✉♠ ♦❢ ❧♦ss ❢✉♥❝t✐♦♥s✿ arg min

fr∈F Lt(fr) =

• (q,d)∈Pl

L(fr(q, d), rel(q, d)) n L(fr(q, d), rel(q, d)) ✲ ❧♦ss ❢✉♥❝t✐♦♥✱ F ✲ s❡t ♦❢ ♣♦ss✐❜❧❡ r❛♥❦✐♥❣ ❢✉♥❝t✐♦♥s✳ ❊①❛♠♣❧❡s ♦❢ ❧♦ss ❢✉♥❝t✐♦♥s✿

❼ L(fr, rel) = (fr − rel)2 ❼ L(fr, rel) = |fr − rel|

❖♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✭♣❛✐r✇✐s❡ ❛♣♣r♦❛❝❤✮

❼ ❚r② t♦ ✉s❡ ✇❡❧❧✲❦♥♦✇♥ ♠❛❝❤✐♥❡ ❧❡❛r♥✐♥❣ ❛❧❣♦r✐t❤♠s t♦ s♦❧✈❡ t❤❡

❢♦❧❧♦✇✐♥❣ ❝❧❛ss✐✜❝❛t✐♦♥ ♣r♦❜❧❡♠✿

❼ ❛♥ ♦r❞❡r❡❞ ♣❛✐r ♦❢ ❞♦❝✉♠❡♥ts (d1, d2)✭❝♦rr❡s♣♦♥❞✐♥❣ t♦ q✉❡r②

q✮ ❜❡❧♦♥❣s t♦ ✜rst ❝❧❛ss ✐✛ rel(q, d1) > rel(q, d2)

❼ ❛♥ ♦r❞❡r❡❞ ♣❛✐r ♦❢ ❞♦❝✉♠❡♥ts (d1, d2)✭❝♦rr❡s♣♦♥❞✐♥❣ t♦ q✉❡r②

q✮ ❜❡❧♦♥❣s t♦ s❡❝♦♥❞ ❝❧❛ss ✐✛ rel(q, d1) ≤ rel(q, d2)

❇♦♦st✐♥❣ ❛❧❣♦r✐t❤♠s ❛♥❞ ❣r❡❡❞② ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥

❲❡ ✇✐❧❧ s♦❧✈❡ r❡❣r❡ss✐♦♥ ♣r♦❜❧❡♠✿ arg min

fr∈F

• (q,d)∈Pl

L(fr(q, d), rel(q, d)) n ❲❡ ✇✐❧❧ s❡❛r❝❤ r❡❧❡✈❛♥❝❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ fr(q, d) =

M

• k=1

αkhk(q, d)

❘❡❧❡✈❛♥❝❡ ❢✉♥❝t✐♦♥ ✇✐❧❧ ❜❡ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s hk(q, d)✱ ❢✉♥❝t✐♦♥s hk(q, d) ❜❡❧♦♥❣ t♦ s✐♠♣❧❡ ❢❛♠✐❧② H ✭✇❡❛❦ ❧❡❛r♥❡rs ❢❛♠✐❧②✮ ✳

❇♦♦st✐♥❣ ❛❧❣♦r✐t❤♠s ❛♥❞ ❣r❡❡❞② ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥

❲❡ ✇✐❧❧ s♦❧✈❡ r❡❣r❡ss✐♦♥ ♣r♦❜❧❡♠✿ arg min

fr∈F

• (q,d)∈Pl

L(fr(q, d), rel(q, d)) n ❲❡ ✇✐❧❧ s❡❛r❝❤ r❡❧❡✈❛♥❝❡ ❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ fr(q, d) =

M

• k=1

αkhk(q, d)

❘❡❧❡✈❛♥❝❡ ❢✉♥❝t✐♦♥ ✇✐❧❧ ❜❡ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s hk(q, d)✱ ❢✉♥❝t✐♦♥s hk(q, d) ❜❡❧♦♥❣ t♦ s✐♠♣❧❡ ❢❛♠✐❧② H ✭✇❡❛❦ ❧❡❛r♥❡rs ❢❛♠✐❧②✮ ✳

❇♦♦st✐♥❣ ❛❧❣♦r✐t❤♠s ❛♥❞ ❣r❡❡❞② ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥

❲❡ ✇✐❧❧ ❝♦♥str✉❝t ✜♥❛❧ ❢✉♥❝t✐♦♥ ❜② ✐t❡r❛t✐♦♥s✳ ❖♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ ✇❡ ✇✐❧❧ ❛❞❞ ❛♥ ❛❞❞✐t✐♦♥❛❧ t❡r♠ αkhk(q, d) t♦ ♦✉r r❡❧❡✈❛♥❝❡ ❢✉♥❝t✐♦♥✿ frk(q, d) = frk−1(q, d) + αkhk(q, d) ❱❛❧✉❡s ♦❢ ♣❛r❛♠❡t❡r αk ❛♥❞ ✇❡❛❦ ❧❡❛r♥❡r hk(q, d) ❝❛♥ ❜❡ ❛ s♦❧✉t✐♦♥ ♦❢ ♥❛t✉r❛❧ ♦♣t✐♠✐③❛t✐♦♥ t❛s❦✿ arg min

α,h(q,d)

• (q,d)∈Pl

L(frk−1(q, d) + αh(q, d), rel(q, d)) n ❚❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s♦❧✈❡❞ ❞✐r❡❝t❧② ❢♦r q✉❛❞r❛t✐❝ ❧♦ss ❢✉♥❝t✐♦♥ ❛♥❞ s✐♠♣❧❡ ❝❧❛ss❡s H✱ ❜✉t ✐t ❝❛♥ ❜❡ ✈❡r② ❞✐✣❝✉❧t t♦ s♦❧✈❡ ❢♦r ♦t❤❡r ❧♦ss ❢✉♥❝t✐♦♥s✳

❇♦♦st✐♥❣ ❛❧❣♦r✐t❤♠s ❛♥❞ ❣r❡❡❞② ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥

❲❡ ✇✐❧❧ ❝♦♥str✉❝t ✜♥❛❧ ❢✉♥❝t✐♦♥ ❜② ✐t❡r❛t✐♦♥s✳ ❖♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ ✇❡ ✇✐❧❧ ❛❞❞ ❛♥ ❛❞❞✐t✐♦♥❛❧ t❡r♠ αkhk(q, d) t♦ ♦✉r r❡❧❡✈❛♥❝❡ ❢✉♥❝t✐♦♥✿ frk(q, d) = frk−1(q, d) + αkhk(q, d) ❱❛❧✉❡s ♦❢ ♣❛r❛♠❡t❡r αk ❛♥❞ ✇❡❛❦ ❧❡❛r♥❡r hk(q, d) ❝❛♥ ❜❡ ❛ s♦❧✉t✐♦♥ ♦❢ ♥❛t✉r❛❧ ♦♣t✐♠✐③❛t✐♦♥ t❛s❦✿ arg min

α,h(q,d)

• (q,d)∈Pl

L(frk−1(q, d) + αh(q, d), rel(q, d)) n ❚❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s♦❧✈❡❞ ❞✐r❡❝t❧② ❢♦r q✉❛❞r❛t✐❝ ❧♦ss ❢✉♥❝t✐♦♥ ❛♥❞ s✐♠♣❧❡ ❝❧❛ss❡s H✱ ❜✉t ✐t ❝❛♥ ❜❡ ✈❡r② ❞✐✣❝✉❧t t♦ s♦❧✈❡ ❢♦r ♦t❤❡r ❧♦ss ❢✉♥❝t✐♦♥s✳

❇♦♦st✐♥❣ ❛❧❣♦r✐t❤♠s ❛♥❞ ❣r❡❡❞② ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥

❲❡ ✇✐❧❧ ❝♦♥str✉❝t ❛❞❞✐t✐♦♥❛❧ t❡r♠ αkhk(q, d) ✐♥ t❤r❡❡ st❡♣s ✿

❼ ●r❛❞✐❡♥t ❛♣♣r♦①✐♠❛t✐♦♥✳ ❈♦♥s✐❞❡r r❡❧❡✈❛♥❝❡ ❢✉♥❝t✐♦♥ fr ❧✐❦❡

✈❡❝t♦r ♦❢ ✈❛❧✉❡s ✐♥❞❡①❡❞ ❜② ❧❡❛r♥✐♥❣ ❡①❛♠♣❧❡s✳ ●❡t ❣r❛❞✐❡♥t ✈❡❝t♦r g = {g(q,d)}(q,d)∈Pl ❢♦r ❡rr♦r ❢✉♥❝t✐♦♥ ✿ g(q,d) = ∂Lt(fr) ∂fr(q, d)

• fr=frk−1

❼ ❲❡❛❦ ❧❡❛r♥❡r s❡❧❡❝t✐♦♥✭✉♣ t♦ ❛ ❝♦♥st❛♥t✮✳ ❋✐♥❞ ♠♦st ❤✐❣❤❧②

❝♦rr❡❧❛t❡❞ ✇✐t❤ g ❢✉♥❝t✐♦♥ hk(q, d) ❜② s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣t✐♠✐③❛t✐♦♥ t❛s❦✿ arg min

β,h(q,d)∈H

• (q,d)∈Pl

(g(q,d) − βh(q, d))2

❇♦♦st✐♥❣ ❛❧❣♦r✐t❤♠s ❛♥❞ ❣r❡❡❞② ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥

❲❡ ✇✐❧❧ ❝♦♥str✉❝t ❛❞❞✐t✐♦♥❛❧ t❡r♠ αkhk(q, d) ✐♥ t❤r❡❡ st❡♣s ✿

❼ ●r❛❞✐❡♥t ❛♣♣r♦①✐♠❛t✐♦♥✳ ❈♦♥s✐❞❡r r❡❧❡✈❛♥❝❡ ❢✉♥❝t✐♦♥ fr ❧✐❦❡

✈❡❝t♦r ♦❢ ✈❛❧✉❡s ✐♥❞❡①❡❞ ❜② ❧❡❛r♥✐♥❣ ❡①❛♠♣❧❡s✳ ●❡t ❣r❛❞✐❡♥t ✈❡❝t♦r g = {g(q,d)}(q,d)∈Pl ❢♦r ❡rr♦r ❢✉♥❝t✐♦♥ ✿ g(q,d) = ∂Lt(fr) ∂fr(q, d)

• fr=frk−1

❼ ❲❡❛❦ ❧❡❛r♥❡r s❡❧❡❝t✐♦♥✭✉♣ t♦ ❛ ❝♦♥st❛♥t✮✳ ❋✐♥❞ ♠♦st ❤✐❣❤❧②

❝♦rr❡❧❛t❡❞ ✇✐t❤ g ❢✉♥❝t✐♦♥ hk(q, d) ❜② s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣t✐♠✐③❛t✐♦♥ t❛s❦✿ arg min

β,h(q,d)∈H

• (q,d)∈Pl

(g(q,d) − βh(q, d))2

❇♦♦st✐♥❣ ❛❧❣♦r✐t❤♠s ❛♥❞ ❣r❡❡❞② ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥

❼ ❙❡❧❡❝t✐♦♥ ♦❢ αk✳ ❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢ αk ❢r♦♠ ♦♥❡✲♣❛r❛♠❡t❡r

♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✿ arg min

α

• (q,d)∈Pl

L(frk−1(q, d) + αhk(q, d), rel(q, d)) n ■t❡r❛t❡✳✳✳ ■t❡r❛t❡✳✳✳ ■t❡r❛t❡✳✳✳

❲❡❛❦ ❧❡❛r♥❡r s❡❧❡❝t✐♦♥

▲❡t ♦✉r ❝❧❛ss ♦❢ ✇❡❛❦ ❧❡❛r♥❡rs H ✇✐❧❧ ❜❡ ❛ s❡t ♦❢ ❞❡❝✐s✐♦♥✲tr❡❡ ❢✉♥❝t✐♦♥s✿

f3(q, d) > 0.5

❩❩❩ ⑦ ✚ ✚ ✚ ❂

❨❡s ◆♦ res = β1 f65(q, d) > 0.78

❩❩❩ ⑦ ✚ ✚ ✚ ❂

❨❡s ◆♦ res = β2 res = β3

❊①❛♠♣❧❡ ♦❢ ✸✲r❡❣✐♦♥ ❞❡❝✐s✐♦♥✲tr❡❡ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❢✉♥❝t✐♦♥ s♣❧✐ts ❢❡❛t✉r❡ s♣❛❝❡ ♦♥ ✸ r❡❣✐♦♥s ❜② ❝♦♥❞✐t✐♦♥s ✐♥ t❤❡ ❢♦r♠ fj(q, d) > α ✭fj ✲ s♣❧✐t ❢❡❛t✉r❡✱ α ✲ s♣❧✐t ❜♦✉♥❞✮✳ ■t ❤❛s ❛ ❝♦♥st❛♥t ✈❛❧✉❡ ❢♦r ❢❡❛t✉r❡ ✈❡❝t♦rs ✐♥ ♦♥❡ r❡❣✐♦♥✳

❲❡❛❦ ❧❡❛r♥❡r s❡❧❡❝t✐♦♥ ✭❢✉♥❝t✐♦♥ ✈❛❧✉❡s✮

❖✉r ✇❡❛❦ ❧❡❛r♥❡rs ❢❛♠✐❧② ✇✐❧❧ ❜❡ ✻✲r❡❣✐♦♥✭❡①❛♠♣❧❡✱ ❝♦♥st✲r❡❣✐♦♥s✮ ❞❡❝✐s✐♦♥✲tr❡❡ ❢✉♥❝t✐♦♥s✳ ❲❡ ✇✐❧❧ tr② t♦ s♦❧✈❡✿ arg min

h(q,d)∈H

• (q,d)∈Pl

(g(q,d) − βh(q, d))2 ❙✉♣♣♦s❡ ✇❡ ❦♥♦✇ tr❡❡✲str✉❝t✉r❡ ♦❢ ✇❡❛❦ ❧❡❛r♥❡r h(q, d) ✲ ✇❡ ❦♥♦✇ s♣❧✐t ❝♦♥❞✐t✐♦♥s ❛♥❞ r❡❣✐♦♥s✳ ❲❡ s❤♦✉❧❞ ✜♥❞ ✧r❡❣✐♦♥ ❝♦♥st❛♥t ✈❛❧✉❡s✧✳ ❖♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ r❡❞✉❝❡s t♦ ♦r❞✐♥❛r② r❡❣r❡ss✐♦♥ ♣r♦❜❧❡♠✿ arg min

h(q,d)∈H,β

• (q,d)∈Pl

(g(q,d) − ββind(q,d))2 ind(q, d) ✲ ♥✉♠❜❡r ♦❢ r❡❣✐♦♥✱ ✇❤✐❝❤ ❝♦♥t❛✐♥s ❢❡❛t✉r❡s ✈❡❝t♦r ❢♦r ♣❛✐r (q, d) ✭ind(q, d) ∈ {1, .., 6}✮✳

❲❡❛❦ ❧❡❛r♥❡r s❡❧❡❝t✐♦♥ ✭❢✉♥❝t✐♦♥ ✈❛❧✉❡s✮

❖✉r ✇❡❛❦ ❧❡❛r♥❡rs ❢❛♠✐❧② ✇✐❧❧ ❜❡ ✻✲r❡❣✐♦♥✭❡①❛♠♣❧❡✱ ❝♦♥st✲r❡❣✐♦♥s✮ ❞❡❝✐s✐♦♥✲tr❡❡ ❢✉♥❝t✐♦♥s✳ ❲❡ ✇✐❧❧ tr② t♦ s♦❧✈❡✿ arg min

h(q,d)∈H

• (q,d)∈Pl

(g(q,d) − βh(q, d))2 ❙✉♣♣♦s❡ ✇❡ ❦♥♦✇ tr❡❡✲str✉❝t✉r❡ ♦❢ ✇❡❛❦ ❧❡❛r♥❡r h(q, d) ✲ ✇❡ ❦♥♦✇ s♣❧✐t ❝♦♥❞✐t✐♦♥s ❛♥❞ r❡❣✐♦♥s✳ ❲❡ s❤♦✉❧❞ ✜♥❞ ✧r❡❣✐♦♥ ❝♦♥st❛♥t ✈❛❧✉❡s✧✳ ❖♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ r❡❞✉❝❡s t♦ ♦r❞✐♥❛r② r❡❣r❡ss✐♦♥ ♣r♦❜❧❡♠✿ arg min

h(q,d)∈H,β

• (q,d)∈Pl

(g(q,d) − ββind(q,d))2 ind(q, d) ✲ ♥✉♠❜❡r ♦❢ r❡❣✐♦♥✱ ✇❤✐❝❤ ❝♦♥t❛✐♥s ❢❡❛t✉r❡s ✈❡❝t♦r ❢♦r ♣❛✐r (q, d) ✭ind(q, d) ∈ {1, .., 6}✮✳

❲❡❛❦ ❧❡❛r♥❡r s❡❧❡❝t✐♦♥ ✭tr❡❡ str✉❝t✉r❡✮

• r❡❡❞② tr❡❡ s❡❧❡❝t✐♦♥✿

❼ bestTree ❂ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ✭✶✲r❡❣✐♦♥ tr❡❡✮✳ ❼ ●r❡❡❞② s♣❧✐t✳ ❚r② t♦ s♣❧✐t r❡❣✐♦♥s ♦❢ bestTree ❛♥❞ ✜♥❞ t❤❡

❜❡st s♣❧✐t✳

f3(q, d) > 0.5

❩❩❩ ⑦ ✚ ✚ ✚ ❂

❨❡s ◆♦ f?(q, d) >? f?(q, d) >?

❩ ✚ ❩ ✚ ❄

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❝♦♥st❛♥t s❡t ♦❢ ♣♦ss✐❜❧❡ s♣❧✐t ❜♦✉♥❞s✳ ◆✉♠❜❡r ♦❢ ♣♦ss✐❜❧❡ s♣❧✐ts ✐s ❜♦✉♥❞❡❞ ❜② t❤❡ ✈❛❧✉❡✿ #{regions} · #{features} · #{split bounds}

❼ ❘❡♣❡❛t ♣r❡✈✐♦✉s st❡♣✳

❲❡❛❦ ❧❡❛r♥❡r s❡❧❡❝t✐♦♥ ✭tr❡❡ str✉❝t✉r❡✮

• r❡❡❞② tr❡❡ s❡❧❡❝t✐♦♥✿

❼ bestTree ❂ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ✭✶✲r❡❣✐♦♥ tr❡❡✮✳ ❼ ●r❡❡❞② s♣❧✐t✳ ❚r② t♦ s♣❧✐t r❡❣✐♦♥s ♦❢ bestTree ❛♥❞ ✜♥❞ t❤❡

❜❡st s♣❧✐t✳

f3(q, d) > 0.5

❩❩❩ ⑦ ✚ ✚ ✚ ❂

❨❡s ◆♦ f?(q, d) >? f?(q, d) >?

❩ ✚ ❩ ✚ ❄

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❝♦♥st❛♥t s❡t ♦❢ ♣♦ss✐❜❧❡ s♣❧✐t ❜♦✉♥❞s✳ ◆✉♠❜❡r ♦❢ ♣♦ss✐❜❧❡ s♣❧✐ts ✐s ❜♦✉♥❞❡❞ ❜② t❤❡ ✈❛❧✉❡✿ #{regions} · #{features} · #{split bounds}

❼ ❘❡♣❡❛t ♣r❡✈✐♦✉s st❡♣✳

❲❡❛❦ ❧❡❛r♥❡r s❡❧❡❝t✐♦♥ ✭tr❡❡ str✉❝t✉r❡✮

• r❡❡❞② tr❡❡ s❡❧❡❝t✐♦♥✿

❼ bestTree ❂ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ✭✶✲r❡❣✐♦♥ tr❡❡✮✳ ❼ ●r❡❡❞② s♣❧✐t✳ ❚r② t♦ s♣❧✐t r❡❣✐♦♥s ♦❢ bestTree ❛♥❞ ✜♥❞ t❤❡

❜❡st s♣❧✐t✳

f3(q, d) > 0.5

❩❩❩ ⑦ ✚ ✚ ✚ ❂

❨❡s ◆♦ f?(q, d) >? f?(q, d) >?

❩ ✚ ❩ ✚ ❄

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❝♦♥st❛♥t s❡t ♦❢ ♣♦ss✐❜❧❡ s♣❧✐t ❜♦✉♥❞s✳ ◆✉♠❜❡r ♦❢ ♣♦ss✐❜❧❡ s♣❧✐ts ✐s ❜♦✉♥❞❡❞ ❜② t❤❡ ✈❛❧✉❡✿ #{regions} · #{features} · #{split bounds}

❼ ❘❡♣❡❛t ♣r❡✈✐♦✉s st❡♣✳

▼❛tr✐①◆❡t

❲❡❛❦ ❧❡❛r♥❡rs s❡t✲ ❢✉❧❧ ❞❡❝✐s✐♦♥ tr❡❡s ✇✐t❤ ❞❡♣t❤ k ❛♥❞ 2k r❡❣✐♦♥s✳

❼ ❈♦♥st❛♥t ♥✉♠❜❡r ♦❢ ❧❛②❡rs ✭❝♦♥st❛♥t ❞❡♣t❤✮✳ ❼ ❚❤❡ s❛♠❡ s♣❧✐t ❝♦♥❞✐t✐♦♥s ❢♦r ♦♥❡ ❧❛②❡r✳

f3(q, d) > 0.5

❩❩❩ ⑦ ✚ ✚ ✚ ❂

❨❡s ◆♦ f56(q, d) > 0.34 f56(q, d) > 0.34

❩❩❩ ⑦ ✚ ✚ ✚ ❂ ❩❩❩ ⑦ ✚ ✚ ✚ ❂

❨❡s ◆♦ ❨❡s ◆♦ β1 β2 β3 β4

❲❡ ❞♦♥✬t ♥❡❡❞ ❝♦♠♣❧❡① str✉❝t✉r❡✿ ❞❡♣t❤ ✐s t❤❡ ♠❛✐♥ t❤✐♥❣✳

▼❛tr✐①◆❡t

❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① ❡✈❛❧✉❛t✐♦♥ ♠❡❛s✉r❡s ✭❉❈●✮

❈❤❛♥❣❡ r❛♥❦✐♥❣ t♦ ✧♣r♦❜❛❜✐❧✐t② r❛♥❦✐♥❣✧✳ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❉❈● ❢♦r q✉❡r② q✱ s❡t ♦❢ ❞♦❝✉♠❡♥ts {d1, .., dn}✱ ❛♥❞ r❛♥❦✐♥❣ ❢✉♥❝t✐♦♥ fr(q, d)✿ apxDCG =

• r∈all permutations of docs

P(fr, r)DCG(r) P(fr, r) ✲ ♣r♦❜❛❜✐❧✐t② t♦ ❣❡t r❛♥❦✐♥❣ r ✐♥ ▲✉❝❡✲P❧❛❝❦❡tt ♠♦❞❡❧✳ DCG(r) ✲ ❉❈● s❝♦r❡ ❢♦r ♣❡r♠✉❛t✐♦♥ r✳

▲✉❝❡✲P❧❛❝❦❡tt ♠♦❞❡❧

❲❡ ❤❛✈❡ s❡t ♦❢ ❞♦❝✉♠❡♥ts {d1, .., dn} ❛♥❞ s❡t ♦❢ r❡❧❡✈❛♥❝❡s {fr(q, d1), .., fr(q, dn)} ❝♦rr❡s♣♦♥❞✐♥❣ t❤❡♠✳ Pr♦❝❡ss ♦❢ r❛♥❦✐♥❣ s❡❧❡❝t✐♦♥ ✐♥ ▲✉❝❡✲P❧❛❝❦❡tt ♠♦❞❡❧✿

❼ ❙❡❧❡❝t ❞♦❝✉♠❡♥t ❢♦r ✜rst ♣♦s✐t✐♦♥✳ Pr♦❜❛❜✐❧✐t② ♦❢ s❡❧❡❝t✐♦♥ ♦❢

❞♦❝✉♠❡♥t di ✐s ❡q✉❛❧ t♦

fr(q,di)

n

• i=1

fr(q,di)

✳ ❙✉♣♣♦s❡ ✇❡ s❡❧❡❝t ❞♦❝✉♠❡♥t dx✳

❼ ❙❡❧❡❝t ❞♦❝✉♠❡♥t ❢♦r s❡❝♦♥❞ ♣♦s✐t✐♦♥ ❢r♦♠ t❤❡ r❡st✳ Pr♦❜❛❜✐❧✐t②

♦❢ s❡❧❡❝t✐♦♥ ♦❢ ❞♦❝✉♠❡♥t di ✐s ❡q✉❛❧ t♦

fr(q,di)

n

• i=1

fr(q,di)−fr(q,dx)

❼ ✳✳✳

❋♦r ❡❛❝❤ s❡❧❡❝t✐♦♥✱ ✐❢ t✇♦ ❞♦❝✉♠❡♥ts di ❛♥❞ dj t❛❦❡ ♣❛rt ✐♥ ✐t✱ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡✐r s❡❧❡❝t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s s❤♦✉❧❞ ❜❡ ❡q✉❡❛❧ t♦ t❤❡ ✈❛❧✉❡

fr(q,di) fr(q,dj)

▲✉❝❡✲P❧❛❝❦❡tt ♠♦❞❡❧

{ ´ d1, .., ´ dn} ✲ s♦♠❡ ♣❡r♠✉t❛t✐♦♥ ♦❢ {d1, .., dn} P(fr, { ´ d1, .., ´ dn}) =

n

• j=1

fr(q, ´ dj)

n

• k=j

fr(q, ´ dk)

❚❤❡ ❡♥❞✳ ❚❤❛♥❦ ②♦✉✳ ❚✐❡✲❨❛♥ ▲✐✉✳ ▲❡❛r♥✐♥❣ t♦ ❘❛♥❦ ❢♦r ■♥❢♦r♠❛t✐♦♥ ❘❡tr✐❡✈❛❧✳ ❚✉t♦r✐❛❧ ♦♥ ❲❲❲✷✵✵✽✳ ❋r✐❡❞♠❛♥✱ ❏✳ ❍✳ ✭✷✵✵✶✮✳ ●r❡❡❞② ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥✿ ❆ ❣r❛❞✐❡♥t ❜♦♦st✐♥❣ ♠❛❝❤✐♥❡✳ ❆♥♥❛❧s ♦❢ ❙t❛t✐st✐❝s✱ ✷✾✭✺✮✱ ✶✶✽✾✲✶✷✸✷✳ ❋r✐❡❞♠❛♥✱ ❏✳ ❍✳ ✭✶✾✾✾✮✳ ❙t♦❝❤❛st✐❝ ❣r❛❞✐❡♥t ❜♦♦st✐♥❣ ✭❚❡❝❤✳ ❘❡♣✳✮✳ P❛❧♦✳ ❆❧t♦✱ ❈❆✿ ❙t❛♥❢♦r❞ ❯♥✐✈❡rs✐t②✱ ❙t❛t✐st✐❝s ❉❡♣❛rt♠❡♥t✳ P❧❛❝❦❡tt✱ ❘✳ ▲✳ ✭✶✾✼✺✮✳ ❚❤❡ ❛♥❛❧②s✐s ♦❢ ♣❡r♠✉t❛t✐♦♥s✳ ❆♣♣❧✐❡❞ ❙t❛t✐st✐❝s✱ ✷✹✱ ✶✾✸✲✷✵✷