<s>
It	O
was	O
discovered	O
repeatedly	O
in	O
very	O
diverse	O
fields	O
such	O
as	O
machine	O
learning	O
(	O
AdaBoost	B-Algorithm
,	O
Winnow	B-Algorithm
,	O
Hedge	O
)	O
,	O
optimization	O
(	O
solving	O
linear	B-Algorithm
programs	I-Algorithm
)	O
,	O
theoretical	O
computer	O
science	O
(	O
devising	O
fast	O
algorithm	O
for	O
LPs	B-Algorithm
and	O
SDPs	O
)	O
,	O
and	O
game	O
theory	O
.	O
</s>
<s>
"	O
Multiplicative	O
weights	O
"	O
implies	O
the	O
iterative	O
rule	O
used	O
in	O
algorithms	O
derived	O
from	O
the	O
multiplicative	B-Algorithm
weight	I-Algorithm
update	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
In	O
machine	O
learning	O
,	O
Littlestone	O
applied	O
the	O
earliest	O
form	O
of	O
the	O
multiplicative	O
weights	O
update	O
rule	O
in	O
his	O
famous	O
winnow	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
is	O
similar	O
to	O
Minsky	O
and	O
Papert	O
's	O
earlier	O
perceptron	B-Algorithm
learning	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Later	O
,	O
he	O
generalized	O
the	O
winnow	B-Algorithm
algorithm	I-Algorithm
to	O
weighted	O
majority	O
algorithm	O
.	O
</s>
<s>
Freund	O
and	O
Schapire	O
followed	O
his	O
steps	O
and	O
generalized	O
the	O
winnow	B-Algorithm
algorithm	I-Algorithm
in	O
the	O
form	O
of	O
hedge	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
multiplicative	O
weights	O
algorithm	O
is	O
also	O
widely	O
applied	O
in	O
computational	O
geometry	O
such	O
as	O
Kenneth	O
Clarkson	O
's	O
algorithm	O
for	O
linear	B-Algorithm
programming	I-Algorithm
(	O
LP	O
)	O
with	O
a	O
bounded	O
number	O
of	O
variables	O
in	O
linear	O
time	O
.	O
</s>
<s>
Later	O
,	O
Bronnimann	O
and	O
Goodrich	O
employed	O
analogous	O
methods	O
to	O
find	O
set	B-Algorithm
covers	I-Algorithm
for	O
hypergraphs	O
with	O
small	O
VC	O
dimension	O
.	O
</s>
<s>
Young	O
discovered	O
the	O
similarities	O
between	O
fast	O
LP	O
algorithms	O
and	O
Raghavan	O
's	O
method	O
of	O
pessimistic	O
estimators	O
for	O
derandomization	O
of	O
randomized	O
rounding	O
algorithms	O
;	O
Klivans	O
and	O
Servedio	O
linked	O
boosting	B-Algorithm
algorithms	O
in	O
learning	O
theory	O
to	O
proofs	O
of	O
Yao	O
's	O
XOR	O
Lemma	O
;	O
Garg	O
and	O
Khandekar	O
defined	O
a	O
common	O
framework	O
for	O
convex	O
optimization	O
problems	O
that	O
contains	O
Garg-Konemann	O
and	O
Plotkin-Shmoys-Tardos	O
as	O
subcases	O
.	O
</s>
<s>
The	O
Hedge	B-Algorithm
algorithm	I-Algorithm
is	O
a	O
special	O
case	O
of	O
mirror	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
this	B-Algorithm
algorithm	I-Algorithm
's	O
goal	O
is	O
to	O
limit	O
its	O
cumulative	O
losses	O
to	O
roughly	O
the	O
same	O
as	O
the	O
best	O
of	O
experts	O
.	O
</s>
<s>
The	O
number	O
of	O
mistakes	O
made	O
by	O
the	O
randomized	B-Algorithm
weighted	I-Algorithm
majority	I-Algorithm
algorithm	I-Algorithm
is	O
bounded	O
as	O
:	O
</s>
<s>
However	O
,	O
it	O
is	O
important	O
to	O
note	O
that	O
in	O
some	O
research	O
,	O
people	O
define	O
in	O
weighted	O
majority	O
algorithm	O
and	O
allow	O
in	O
randomized	B-Algorithm
weighted	I-Algorithm
majority	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Let	O
=	O
payoff	O
matrix	B-Architecture
of	O
a	O
finite	O
two-player	O
zero-sum	O
game	O
,	O
with	O
rows	O
.	O
</s>
<s>
In	O
machine	O
learning	O
,	O
Littlestone	O
and	O
Warmuth	O
generalized	O
the	O
winnow	B-Algorithm
algorithm	I-Algorithm
to	O
the	O
weighted	O
majority	O
algorithm	O
.	O
</s>
<s>
Later	O
,	O
Freund	O
and	O
Schapire	O
generalized	O
it	O
in	O
the	O
form	O
of	O
hedge	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
AdaBoost	B-Algorithm
Algorithm	O
formulated	O
by	O
Yoav	O
Freund	O
and	O
Robert	O
Schapire	O
also	O
employed	O
the	O
Multiplicative	B-Algorithm
Weight	I-Algorithm
Update	I-Algorithm
Method	I-Algorithm
.	O
</s>
<s>
Based	O
on	O
current	O
knowledge	O
in	O
algorithms	O
,	O
multiplicative	B-Algorithm
weight	I-Algorithm
update	I-Algorithm
method	I-Algorithm
was	O
first	O
used	O
in	O
Littlestone	O
's	O
winnow	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
It	O
is	O
used	O
in	O
machine	O
learning	O
to	O
solve	O
a	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
The	O
hedge	B-Algorithm
algorithm	I-Algorithm
is	O
similar	O
to	O
the	O
weighted	O
majority	O
algorithm	O
.	O
</s>
<s>
This	B-Algorithm
algorithm	I-Algorithm
maintains	O
a	O
set	O
of	O
weights	O
over	O
the	O
training	O
examples	O
.	O
</s>
<s>
This	O
distribution	O
is	O
fed	O
to	O
the	O
weak	B-Algorithm
learner	I-Algorithm
WeakLearn	O
which	O
generates	O
a	O
hypothesis	O
that	O
(	O
hopefully	O
)	O
has	O
small	O
error	O
with	O
respect	O
to	O
the	O
distribution	O
.	O
</s>
<s>
Using	O
the	O
new	O
hypothesis	O
,	O
AdaBoost	B-Algorithm
generates	O
the	O
next	O
weight	O
vector	O
.	O
</s>
<s>
Given	O
a	O
matrix	B-Architecture
and	O
,	O
is	O
there	O
a	O
such	O
that	O
?	O
</s>
<s>
Computational	O
geometry	O
The	O
multiplicative	O
weights	O
algorithm	O
is	O
also	O
widely	O
applied	O
in	O
computational	O
geometry	O
,	O
such	O
as	O
Clarkson	O
's	O
algorithm	O
for	O
linear	B-Algorithm
programming	I-Algorithm
(	O
LP	O
)	O
with	O
a	O
bounded	O
number	O
of	O
variables	O
in	O
linear	O
time	O
.	O
</s>
<s>
Later	O
,	O
Bronnimann	O
and	O
Goodrich	O
employed	O
analogous	O
methods	O
to	O
find	O
Set	B-Algorithm
Covers	I-Algorithm
for	O
hypergraphs	O
with	O
small	O
VC	O
dimension	O
.	O
</s>
