<s>
Statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
is	O
a	O
framework	O
for	O
machine	O
learning	O
drawing	O
from	O
the	O
fields	O
of	O
statistics	O
and	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
Statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
deals	O
with	O
the	O
statistical	O
inference	O
problem	O
of	O
finding	O
a	O
predictive	O
function	O
based	O
on	O
data	O
.	O
</s>
<s>
Statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
has	O
led	O
to	O
successful	O
applications	O
in	O
fields	O
such	O
as	O
computer	B-Application
vision	I-Application
,	O
speech	B-Application
recognition	I-Application
,	O
and	O
bioinformatics	O
.	O
</s>
<s>
Learning	O
falls	O
into	O
many	O
categories	O
,	O
including	O
supervised	B-General_Concept
learning	I-General_Concept
,	O
unsupervised	B-General_Concept
learning	I-General_Concept
,	O
online	B-Algorithm
learning	I-Algorithm
,	O
and	O
reinforcement	O
learning	O
.	O
</s>
<s>
From	O
the	O
perspective	O
of	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
,	O
supervised	B-General_Concept
learning	I-General_Concept
is	O
best	O
understood	O
.	O
</s>
<s>
Supervised	B-General_Concept
learning	I-General_Concept
involves	O
learning	O
from	O
a	O
training	O
set	O
of	O
data	O
.	O
</s>
<s>
Depending	O
on	O
the	O
type	O
of	O
output	O
,	O
supervised	B-General_Concept
learning	I-General_Concept
problems	O
are	O
either	O
problems	O
of	O
regression	O
or	O
problems	O
of	O
classification	B-General_Concept
.	O
</s>
<s>
Classification	B-General_Concept
problems	O
are	O
those	O
for	O
which	O
the	O
output	O
will	O
be	O
an	O
element	O
from	O
a	O
discrete	O
set	O
of	O
labels	O
.	O
</s>
<s>
Classification	B-General_Concept
is	O
very	O
common	O
for	O
machine	O
learning	O
applications	O
.	O
</s>
<s>
Statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
takes	O
the	O
perspective	O
that	O
there	O
is	O
some	O
unknown	O
probability	O
distribution	O
over	O
the	O
product	O
space	O
,	O
i.e.	O
</s>
<s>
A	O
learning	O
algorithm	O
that	O
chooses	O
the	O
function	O
that	O
minimizes	O
the	O
empirical	O
risk	O
is	O
called	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
.	O
</s>
<s>
Different	O
loss	O
functions	O
are	O
used	O
depending	O
on	O
whether	O
the	O
problem	O
is	O
one	O
of	O
regression	O
or	O
one	O
of	O
classification	B-General_Concept
.	O
</s>
<s>
This	O
familiar	O
loss	O
function	O
is	O
used	O
in	O
Ordinary	B-General_Concept
Least	I-General_Concept
Squares	I-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
In	O
some	O
sense	O
the	O
0-1	O
indicator	O
function	O
is	O
the	O
most	O
natural	O
loss	O
function	O
for	O
classification	B-General_Concept
.	O
</s>
<s>
For	O
binary	O
classification	B-General_Concept
with	O
,	O
this	O
is	O
:	O
</s>
<s>
In	O
machine	O
learning	O
problems	O
,	O
a	O
major	O
problem	O
that	O
arises	O
is	O
that	O
of	O
overfitting	B-Error_Name
.	O
</s>
<s>
Empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
runs	O
this	O
risk	O
of	O
overfitting	B-Error_Name
:	O
finding	O
a	O
function	O
that	O
matches	O
the	O
data	O
exactly	O
but	O
does	O
not	O
predict	O
future	O
output	O
well	O
.	O
</s>
<s>
Overfitting	B-Error_Name
is	O
symptomatic	O
of	O
unstable	O
solutions	O
;	O
a	O
small	O
perturbation	O
in	O
the	O
training	O
set	O
data	O
would	O
cause	O
a	O
large	O
variation	O
in	O
the	O
learned	O
function	O
.	O
</s>
<s>
A	O
common	O
example	O
would	O
be	O
restricting	O
to	O
linear	O
functions	O
:	O
this	O
can	O
be	O
seen	O
as	O
a	O
reduction	O
to	O
the	O
standard	O
problem	O
of	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
Restriction	O
of	O
the	O
hypothesis	O
space	O
avoids	O
overfitting	B-Error_Name
because	O
the	O
form	O
of	O
the	O
potential	O
functions	O
are	O
limited	O
,	O
and	O
so	O
does	O
not	O
allow	O
for	O
the	O
choice	O
of	O
a	O
function	O
that	O
gives	O
empirical	O
risk	O
arbitrarily	O
close	O
to	O
zero	O
.	O
</s>
