4

The Metasearch Problem

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β ∈ 0,1[ ];N strategies (systems)

initial weights w1 ∈ 0,1[ ]N ; wi1

i=1

N

∑ =1

∑=

= N

i

ti

tit

i

w

wp

1

€

lt ∈ 0,1[ ]N tttH lpL ⋅=

tilt

iti ww β⋅=+1

∑ =⋅=

T

ttt

HEDGE lpL1

β

β

−

+

≤1

ln1ln NLL

SYSTEMBEST

HEDGE

11

+=

−=

TNONRELEVANRELEVANT

rZ

Zrrrvalue ln1...

111)( ≈+++

+=

rZdlabelrankvaluedlabelsdLOSS sd ln)()()(),( , ⋅≈⋅=

•  map ranks to values

total loss vs. total precision vs. average precision

)(

)(_

,1

1sd

N

s

ts

t

rankvaluew

dvalueaverage

⋅=

=

∑=

−

),(1 sdLOSSts

ts

ttww β⋅=+

)()(),( ,sdtt trankvaluedlabelsdLOSS ⋅=

( ))(_maxarg dvalueaveraged dt =