<s>
Linear	B-Algorithm
Programming	I-Algorithm
Boosting	B-Algorithm
(	O
LPBoost	B-Algorithm
)	O
is	O
a	O
supervised	B-General_Concept
classifier	I-General_Concept
from	O
the	O
boosting	B-Algorithm
family	O
of	O
classifiers	O
.	O
</s>
<s>
LPBoost	B-Algorithm
maximizes	O
a	O
margin	O
between	O
training	O
samples	O
of	O
different	O
classes	O
and	O
hence	O
also	O
belongs	O
to	O
the	O
class	O
of	O
margin-maximizing	O
supervised	B-General_Concept
classification	I-General_Concept
algorithms	O
.	O
</s>
<s>
LPBoost	B-Algorithm
is	O
an	O
algorithm	O
to	O
learn	O
such	O
a	O
classification	O
function	O
given	O
a	O
set	O
of	O
training	O
examples	O
with	O
known	O
class	O
labels	O
.	O
</s>
<s>
LPBoost	B-Algorithm
is	O
a	O
machine	O
learning	O
technique	O
and	O
especially	O
suited	O
for	O
applications	O
of	O
joint	O
classification	O
and	O
feature	O
selection	O
in	O
structured	O
domains	O
.	O
</s>
<s>
where	O
are	O
non-negative	O
weightings	O
for	O
weak	B-Algorithm
classifiers	I-Algorithm
.	O
</s>
<s>
Each	O
individual	O
weak	B-Algorithm
classifier	I-Algorithm
may	O
be	O
just	O
a	O
little	O
bit	O
better	O
than	O
random	O
,	O
but	O
the	O
resulting	O
linear	O
combination	O
of	O
many	O
weak	B-Algorithm
classifiers	I-Algorithm
can	O
perform	O
very	O
well	O
.	O
</s>
<s>
LPBoost	B-Algorithm
constructs	O
by	O
starting	O
with	O
an	O
empty	O
set	O
of	O
weak	B-Algorithm
classifiers	I-Algorithm
.	O
</s>
<s>
Iteratively	O
,	O
a	O
single	O
weak	B-Algorithm
classifier	I-Algorithm
to	O
add	O
to	O
the	O
set	O
of	O
considered	O
weak	B-Algorithm
classifiers	I-Algorithm
is	O
selected	O
,	O
added	O
and	O
all	O
the	O
weights	O
for	O
the	O
current	O
set	O
of	O
weak	B-Algorithm
classifiers	I-Algorithm
are	O
adjusted	O
.	O
</s>
<s>
This	O
is	O
repeated	O
until	O
no	O
weak	B-Algorithm
classifiers	I-Algorithm
to	O
add	O
remain	O
.	O
</s>
<s>
Early	O
boosting	B-Algorithm
methods	O
,	O
such	O
as	O
AdaBoost	B-Algorithm
do	O
not	O
have	O
this	O
property	O
and	O
converge	O
slower	O
.	O
</s>
<s>
More	O
generally	O
,	O
let	O
be	O
the	O
possibly	O
infinite	O
set	O
of	O
weak	B-Algorithm
classifiers	I-Algorithm
,	O
also	O
termed	O
hypotheses	O
.	O
</s>
<s>
One	O
way	O
to	O
write	O
down	O
the	O
problem	O
LPBoost	B-Algorithm
solves	O
is	O
as	O
a	O
linear	B-Algorithm
program	I-Algorithm
with	O
infinitely	O
many	O
variables	O
.	O
</s>
<s>
The	O
primal	O
linear	B-Algorithm
program	I-Algorithm
of	O
LPBoost	B-Algorithm
,	O
optimizing	O
over	O
the	O
non-negative	O
weight	O
vector	O
,	O
the	O
non-negative	O
vector	O
of	O
slack	O
variables	O
and	O
the	O
margin	O
is	O
the	O
following	O
.	O
</s>
<s>
Note	O
the	O
effects	O
of	O
slack	O
variables	O
:	O
their	O
one-norm	O
is	O
penalized	O
in	O
the	O
objective	O
function	O
by	O
a	O
constant	O
factor	O
,	O
which	O
—	O
if	O
small	O
enough	O
—	O
always	O
leads	O
to	O
a	O
primal	O
feasible	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Here	O
we	O
adopted	O
the	O
notation	O
of	O
a	O
parameter	O
space	O
,	O
such	O
that	O
for	O
a	O
choice	O
the	O
weak	B-Algorithm
classifier	I-Algorithm
is	O
uniquely	O
defined	O
.	O
</s>
<s>
When	O
the	O
above	O
linear	B-Algorithm
program	I-Algorithm
was	O
first	O
written	O
down	O
in	O
early	O
publications	O
about	O
boosting	B-Algorithm
methods	O
it	O
was	O
disregarded	O
as	O
intractable	O
due	O
to	O
the	O
large	O
number	O
of	O
variables	O
.	O
</s>
<s>
Only	O
later	O
it	O
was	O
discovered	O
that	O
such	O
linear	B-Algorithm
programs	I-Algorithm
can	O
indeed	O
be	O
solved	O
efficiently	O
using	O
the	O
classic	O
technique	O
of	O
column	B-Algorithm
generation	I-Algorithm
.	O
</s>
<s>
In	O
a	O
linear	B-Algorithm
program	I-Algorithm
a	O
column	O
corresponds	O
to	O
a	O
primal	O
variable	O
.	O
</s>
<s>
Column	B-Algorithm
generation	I-Algorithm
is	O
a	O
technique	O
to	O
solve	O
large	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
Columns	O
in	O
the	O
primal	O
linear	B-Algorithm
program	I-Algorithm
corresponds	O
to	O
rows	O
in	O
the	O
dual	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
The	O
equivalent	O
dual	O
linear	B-Algorithm
program	I-Algorithm
of	O
LPBoost	B-Algorithm
is	O
the	O
following	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
For	O
linear	B-Algorithm
programs	I-Algorithm
the	O
optimal	O
value	O
of	O
the	O
primal	O
and	O
dual	O
problem	O
are	O
equal	O
.	O
</s>
<s>
For	O
any	O
finite	O
subset	O
we	O
can	O
solve	O
the	O
linear	B-Algorithm
program	I-Algorithm
and	O
thus	O
satisfy	O
all	O
constraints	O
.	O
</s>
<s>
That	O
is	O
,	O
we	O
search	O
the	O
space	O
for	O
a	O
single	O
decision	B-Algorithm
stump	I-Algorithm
maximizing	O
the	O
left	O
hand	O
side	O
of	O
the	O
dual	O
constraint	O
.	O
</s>
<s>
If	O
the	O
constraint	O
cannot	O
be	O
violated	O
by	O
any	O
choice	O
of	O
decision	B-Algorithm
stump	I-Algorithm
,	O
none	O
of	O
the	O
corresponding	O
constraint	O
can	O
be	O
active	O
in	O
the	O
original	O
problem	O
and	O
the	O
restricted	O
problem	O
is	O
equivalent	O
.	O
</s>
<s>
Note	O
that	O
if	O
the	O
convergence	O
threshold	O
is	O
set	O
to	O
the	O
solution	O
obtained	O
is	O
the	O
global	O
optimal	O
solution	O
of	O
the	O
above	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
While	O
the	O
above	O
algorithm	O
is	O
proven	O
to	O
converge	O
,	O
in	O
contrast	O
to	O
other	O
boosting	B-Algorithm
formulations	O
,	O
such	O
as	O
AdaBoost	B-Algorithm
and	O
TotalBoost	O
,	O
there	O
are	O
no	O
known	O
convergence	O
bounds	O
for	O
LPBoost	B-Algorithm
.	O
</s>
<s>
In	O
practise	O
however	O
,	O
LPBoost	B-Algorithm
is	O
known	O
to	O
converge	O
quickly	O
,	O
often	O
faster	O
than	O
other	O
formulations	O
.	O
</s>
<s>
LPBoost	B-Algorithm
is	O
an	O
ensemble	B-Algorithm
learning	I-Algorithm
method	O
and	O
thus	O
does	O
not	O
dictate	O
the	O
choice	O
of	O
base	O
learners	O
,	O
the	O
space	O
of	O
hypotheses	O
.	O
</s>
<s>
If	O
the	O
base	O
learners	O
are	O
particularly	O
simple	O
,	O
they	O
are	O
often	O
referred	O
to	O
as	O
decision	B-Algorithm
stumps	I-Algorithm
.	O
</s>
<s>
The	O
number	O
of	O
base	O
learners	O
commonly	O
used	O
with	O
Boosting	B-Algorithm
in	O
the	O
literature	O
is	O
large	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
,	O
a	O
base	O
learner	O
could	O
be	O
a	O
linear	O
soft	O
margin	O
support	B-Algorithm
vector	I-Algorithm
machine	I-Algorithm
.	O
</s>
<s>
The	O
above	O
decision	B-Algorithm
stumps	I-Algorithm
looks	O
only	O
along	O
a	O
single	O
dimension	O
of	O
the	O
input	O
space	O
and	O
simply	O
thresholds	O
the	O
respective	O
column	O
of	O
the	O
sample	O
using	O
a	O
constant	O
threshold	O
.	O
</s>
<s>
Given	O
weights	O
for	O
the	O
training	O
samples	O
,	O
constructing	O
the	O
optimal	O
decision	B-Algorithm
stump	I-Algorithm
of	O
the	O
above	O
form	O
simply	O
involves	O
searching	O
along	O
all	O
sample	O
columns	O
and	O
determining	O
,	O
and	O
in	O
order	O
to	O
optimize	O
the	O
gain	O
function	O
.	O
</s>
