<s>
Within	O
mathematical	O
analysis	O
,	O
Regularization	B-Algorithm
perspectives	I-Algorithm
on	I-Algorithm
support-vector	I-Algorithm
machines	I-Algorithm
provide	O
a	O
way	O
of	O
interpreting	O
support-vector	B-Algorithm
machines	I-Algorithm
(	O
SVMs	B-Algorithm
)	O
in	O
the	O
context	O
of	O
other	O
regularization-based	O
machine-learning	O
algorithms	O
.	O
</s>
<s>
SVM	B-Algorithm
algorithms	O
categorize	O
binary	O
data	O
,	O
with	O
the	O
goal	O
of	O
fitting	O
the	O
training	O
set	O
data	O
in	O
a	O
way	O
that	O
minimizes	O
the	O
average	O
of	O
the	O
hinge-loss	O
function	O
and	O
L2	O
norm	O
of	O
the	O
learned	O
weights	O
.	O
</s>
<s>
This	O
strategy	O
avoids	O
overfitting	B-Error_Name
via	O
Tikhonov	O
regularization	O
and	O
in	O
the	O
L2	O
norm	O
sense	O
and	O
also	O
corresponds	O
to	O
minimizing	O
the	O
bias	O
and	O
variance	O
of	O
our	O
estimator	O
of	O
the	O
weights	O
.	O
</s>
<s>
Estimators	O
with	O
lower	O
Mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
predict	O
better	O
or	O
generalize	O
better	O
when	O
given	O
unseen	O
data	O
.	O
</s>
<s>
Specifically	O
,	O
Tikhonov	O
regularization	O
algorithms	O
produce	O
a	O
decision	B-General_Concept
boundary	I-General_Concept
that	O
minimizes	O
the	O
average	O
training-set	O
error	O
and	O
constrain	O
the	O
Decision	B-General_Concept
boundary	I-General_Concept
not	O
to	O
be	O
excessively	O
complicated	O
or	O
overfit	B-Error_Name
the	O
training	O
data	O
via	O
a	O
L2	O
norm	O
of	O
the	O
weights	O
term	O
.	O
</s>
<s>
Regularization	B-Algorithm
perspectives	I-Algorithm
on	I-Algorithm
support-vector	I-Algorithm
machines	I-Algorithm
interpret	O
SVM	B-Algorithm
as	O
a	O
special	O
case	O
of	O
Tikhonov	O
regularization	O
,	O
specifically	O
Tikhonov	O
regularization	O
with	O
the	O
hinge	B-Algorithm
loss	I-Algorithm
for	O
a	O
loss	O
function	O
.	O
</s>
<s>
This	O
provides	O
a	O
theoretical	O
framework	O
with	O
which	O
to	O
analyze	O
SVM	B-Algorithm
algorithms	O
and	O
compare	O
them	O
to	O
other	O
algorithms	O
with	O
the	O
same	O
goals	O
:	O
to	O
generalize	O
without	O
overfitting	B-Error_Name
.	O
</s>
<s>
SVM	B-Algorithm
was	O
first	O
proposed	O
in	O
1995	O
by	O
Corinna	O
Cortes	O
and	O
Vladimir	O
Vapnik	O
,	O
and	O
framed	O
geometrically	O
as	O
a	O
method	O
for	O
finding	O
hyperplanes	O
that	O
can	O
separate	O
multidimensional	O
data	O
into	O
two	O
categories	O
.	O
</s>
<s>
This	O
traditional	O
geometric	O
interpretation	O
of	O
SVMs	B-Algorithm
provides	O
useful	O
intuition	O
about	O
how	O
SVMs	B-Algorithm
work	O
,	O
but	O
is	O
difficult	O
to	O
relate	O
to	O
other	O
machine-learning	O
techniques	O
for	O
avoiding	O
overfitting	B-Error_Name
,	O
like	O
regularization	O
,	O
early	B-Algorithm
stopping	I-Algorithm
,	O
sparsity	B-Algorithm
and	O
Bayesian	O
inference	O
.	O
</s>
<s>
However	O
,	O
once	O
it	O
was	O
discovered	O
that	O
SVM	B-Algorithm
is	O
also	O
a	O
special	O
case	O
of	O
Tikhonov	O
regularization	O
,	O
regularization	O
perspectives	O
on	O
SVM	B-Algorithm
provided	O
the	O
theory	O
necessary	O
to	O
fit	O
SVM	B-Algorithm
within	O
a	O
broader	O
class	O
of	O
algorithms	O
.	O
</s>
<s>
This	O
has	O
enabled	O
detailed	O
comparisons	O
between	O
SVM	B-Algorithm
and	O
other	O
forms	O
of	O
Tikhonov	O
regularization	O
,	O
and	O
theoretical	O
grounding	O
for	O
why	O
it	O
is	O
beneficial	O
to	O
use	O
SVM	B-Algorithm
's	O
loss	O
function	O
,	O
the	O
hinge	B-Algorithm
loss	I-Algorithm
.	O
</s>
<s>
In	O
the	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
framework	O
,	O
an	O
algorithm	O
is	O
a	O
strategy	O
for	O
choosing	O
a	O
function	O
given	O
a	O
training	O
set	O
of	O
inputs	O
and	O
their	O
labels	O
(	O
the	O
labels	O
are	O
usually	O
)	O
.	O
</s>
<s>
Regularization	O
strategies	O
avoid	O
overfitting	B-Error_Name
by	O
choosing	O
a	O
function	O
that	O
fits	O
the	O
data	O
,	O
but	O
is	O
not	O
too	O
complex	O
.	O
</s>
<s>
When	O
is	O
a	O
reproducing	O
kernel	O
Hilbert	O
space	O
,	O
there	O
exists	O
a	O
kernel	O
function	O
that	O
can	O
be	O
written	O
as	O
an	O
symmetric	B-Algorithm
positive-definite	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
hinge	B-Algorithm
loss	I-Algorithm
,	O
,	O
where	O
,	O
provides	O
such	O
a	O
convex	O
relaxation	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
hinge	B-Algorithm
loss	I-Algorithm
is	O
the	O
tightest	O
convex	O
upper	O
bound	O
to	O
the	O
0	O
–	O
1	O
misclassification	O
loss	O
function	O
,	O
and	O
with	O
infinite	O
data	O
returns	O
the	O
Bayes-optimal	O
solution	O
:	O
</s>
<s>
The	O
Tikhonov	O
regularization	O
problem	O
can	O
be	O
shown	O
to	O
be	O
equivalent	O
to	O
traditional	O
formulations	O
of	O
SVM	B-Algorithm
by	O
expressing	O
it	O
in	O
terms	O
of	O
the	O
hinge	B-Algorithm
loss	I-Algorithm
.	O
</s>
<s>
with	O
,	O
which	O
is	O
equivalent	O
to	O
the	O
standard	O
SVM	B-Algorithm
minimization	O
problem	O
.	O
</s>
