<s>
In	O
machine	O
learning	O
,	O
the	O
kernel	B-General_Concept
perceptron	I-General_Concept
is	O
a	O
variant	O
of	O
the	O
popular	O
perceptron	B-Algorithm
learning	I-Algorithm
algorithm	I-Algorithm
that	O
can	O
learn	O
kernel	O
machines	O
,	O
i.e.	O
</s>
<s>
The	O
perceptron	B-Algorithm
algorithm	I-Algorithm
is	O
an	O
online	B-Algorithm
learning	I-Algorithm
algorithm	O
that	O
operates	O
by	O
a	O
principle	O
called	O
"	O
error-driven	O
learning	O
"	O
.	O
</s>
<s>
It	O
iteratively	O
improves	O
a	O
model	O
by	O
running	O
it	O
on	O
training	O
samples	O
,	O
then	O
updating	O
the	O
model	O
whenever	O
it	O
finds	O
it	O
has	O
made	O
an	O
incorrect	O
classification	O
with	O
respect	O
to	O
a	O
supervised	B-General_Concept
signal	O
.	O
</s>
<s>
In	O
pseudocode	B-Language
,	O
the	O
perceptron	B-Algorithm
algorithm	I-Algorithm
is	O
given	O
by	O
:	O
</s>
<s>
To	O
derive	O
a	O
kernelized	O
version	O
of	O
the	O
perceptron	B-Algorithm
algorithm	I-Algorithm
,	O
we	O
must	O
first	O
formulate	O
it	O
in	O
dual	B-Algorithm
form	I-Algorithm
,	O
starting	O
from	O
the	O
observation	O
that	O
the	O
weight	O
vector	O
can	O
be	O
expressed	O
as	O
a	O
linear	B-General_Concept
combination	O
of	O
the	O
training	O
samples	O
.	O
</s>
<s>
Using	O
this	O
result	O
,	O
we	O
can	O
formulate	O
the	O
dual	O
perceptron	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
loops	O
through	O
the	O
samples	O
as	O
before	O
,	O
making	O
predictions	O
,	O
but	O
instead	O
of	O
storing	O
and	O
updating	O
a	O
weight	O
vector	O
,	O
it	O
updates	O
a	O
"	O
mistake	O
counter	O
"	O
vector	O
.	O
</s>
<s>
Plugging	O
these	O
two	O
equations	O
into	O
the	O
training	O
loop	O
turn	O
it	O
into	O
the	O
dual	O
perceptron	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Finally	O
,	O
we	O
can	O
replace	O
the	O
dot	O
product	O
in	O
the	O
dual	O
perceptron	B-Algorithm
by	O
an	O
arbitrary	O
kernel	O
function	O
,	O
to	O
get	O
the	O
effect	O
of	O
a	O
feature	O
map	O
without	O
computing	O
explicitly	O
for	O
any	O
samples	O
.	O
</s>
<s>
Doing	O
this	O
yields	O
the	O
kernel	B-General_Concept
perceptron	I-General_Concept
algorithm	O
:	O
</s>
<s>
One	O
problem	O
with	O
the	O
kernel	B-General_Concept
perceptron	I-General_Concept
,	O
as	O
presented	O
above	O
,	O
is	O
that	O
it	O
does	O
not	O
learn	O
sparse	B-Algorithm
kernel	O
machines	O
.	O
</s>
<s>
Moreover	O
,	O
when	O
the	O
kernel	B-General_Concept
perceptron	I-General_Concept
is	O
used	O
in	O
an	O
online	B-Algorithm
setting	O
,	O
the	O
number	O
of	O
non-zero	O
and	O
thus	O
the	O
evaluation	O
cost	O
grow	O
linearly	O
in	O
the	O
number	O
of	O
examples	O
presented	O
to	O
the	O
algorithm	O
.	O
</s>
<s>
The	O
forgetron	O
variant	O
of	O
the	O
kernel	B-General_Concept
perceptron	I-General_Concept
was	O
suggested	O
to	O
deal	O
with	O
this	O
problem	O
.	O
</s>
<s>
It	O
maintains	O
an	O
active	B-Algorithm
set	I-Algorithm
of	O
examples	O
with	O
non-zero	O
,	O
removing	O
(	O
"	O
forgetting	O
"	O
)	O
examples	O
from	O
the	O
active	B-Algorithm
set	I-Algorithm
when	O
it	O
exceeds	O
a	O
pre-determined	O
budget	O
and	O
"	O
shrinking	O
"	O
(	O
lowering	O
the	O
weight	O
of	O
)	O
old	O
examples	O
as	O
new	O
ones	O
are	O
promoted	O
to	O
non-zero	O
.	O
</s>
<s>
Another	O
problem	O
with	O
the	O
kernel	B-General_Concept
perceptron	I-General_Concept
is	O
that	O
it	O
does	O
not	O
regularize	O
,	O
making	O
it	O
vulnerable	O
to	O
overfitting	B-Error_Name
.	O
</s>
<s>
The	O
NORMA	O
online	B-Algorithm
kernel	O
learning	O
algorithm	O
can	O
be	O
regarded	O
as	O
a	O
generalization	O
of	O
the	O
kernel	B-General_Concept
perceptron	I-General_Concept
algorithm	O
with	O
regularization	O
.	O
</s>
<s>
The	O
sequential	B-Algorithm
minimal	I-Algorithm
optimization	I-Algorithm
(	O
SMO	O
)	O
algorithm	O
used	O
to	O
learn	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
can	O
also	O
be	O
regarded	O
as	O
a	O
generalization	O
of	O
the	O
kernel	B-General_Concept
perceptron	I-General_Concept
.	O
</s>
<s>
The	O
voted	O
perceptron	B-Algorithm
algorithm	I-Algorithm
of	O
Freund	O
and	O
Schapire	O
also	O
extends	O
to	O
the	O
kernelized	O
case	O
,	O
giving	O
generalization	O
bounds	O
comparable	O
to	O
the	O
kernel	O
SVM	B-Algorithm
.	O
</s>
