<s>
For	O
supervised	B-General_Concept
learning	I-General_Concept
applications	O
in	O
machine	O
learning	O
and	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
,	O
generalization	B-Algorithm
error	I-Algorithm
(	O
also	O
known	O
as	O
the	O
out-of-sample	O
error	O
or	O
the	O
risk	O
)	O
is	O
a	O
measure	O
of	O
how	O
accurately	O
an	O
algorithm	O
is	O
able	O
to	O
predict	O
outcome	O
values	O
for	O
previously	O
unseen	O
data	O
.	O
</s>
<s>
Generalization	B-Algorithm
error	I-Algorithm
can	O
be	O
minimized	O
by	O
avoiding	O
overfitting	B-Error_Name
in	O
the	O
learning	O
algorithm	O
.	O
</s>
<s>
The	O
performance	O
of	O
a	O
machine	O
learning	O
algorithm	O
is	O
visualized	O
by	O
plots	O
that	O
show	O
values	O
of	O
estimates	O
of	O
the	O
generalization	B-Algorithm
error	I-Algorithm
through	O
the	O
learning	O
process	O
,	O
which	O
are	O
called	O
learning	B-General_Concept
curves	I-General_Concept
.	O
</s>
<s>
The	O
generalization	B-Algorithm
error	I-Algorithm
or	O
expected	O
loss	O
or	O
risk	O
of	O
a	O
particular	O
function	O
over	O
all	O
possible	O
values	O
of	O
and	O
is	O
the	O
expected	O
value	O
of	O
the	O
loss	O
function	O
:	O
</s>
<s>
Of	O
particular	O
importance	O
is	O
the	O
generalization	B-Algorithm
error	I-Algorithm
of	O
the	O
data-dependent	O
function	O
that	O
is	O
found	O
by	O
a	O
learning	O
algorithm	O
based	O
on	O
the	O
sample	O
.	O
</s>
<s>
Instead	O
,	O
the	O
aim	O
of	O
many	O
problems	O
in	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
is	O
to	O
bound	O
or	O
characterize	O
the	O
difference	O
of	O
the	O
generalization	B-Algorithm
error	I-Algorithm
and	O
the	O
empirical	O
error	O
in	O
probability	O
:	O
</s>
<s>
That	O
is	O
,	O
the	O
goal	O
is	O
to	O
characterize	O
the	O
probability	O
that	O
the	O
generalization	B-Algorithm
error	I-Algorithm
is	O
less	O
than	O
the	O
empirical	O
error	O
plus	O
some	O
error	O
bound	O
(	O
generally	O
dependent	O
on	O
and	O
)	O
.	O
</s>
<s>
For	O
many	O
types	O
of	O
algorithms	O
,	O
it	O
has	O
been	O
shown	O
that	O
an	O
algorithm	O
has	O
generalization	O
bounds	O
if	O
it	O
meets	O
certain	O
stability	B-General_Concept
criteria	O
.	O
</s>
<s>
Specifically	O
,	O
if	O
an	O
algorithm	O
is	O
symmetric	O
(	O
the	O
order	O
of	O
inputs	O
does	O
not	O
affect	O
the	O
result	O
)	O
,	O
has	O
bounded	O
loss	O
and	O
meets	O
two	O
stability	B-General_Concept
conditions	O
,	O
it	O
will	O
generalize	O
.	O
</s>
<s>
The	O
first	O
stability	B-General_Concept
condition	O
,	O
leave-one-out	O
cross-validation	B-Application
stability	B-General_Concept
,	O
says	O
that	O
to	O
be	O
stable	O
,	O
the	O
prediction	O
error	O
for	O
each	O
data	O
point	O
when	O
leave-one-out	O
cross	O
validation	O
is	O
used	O
must	O
converge	O
to	O
zero	O
as	O
.	O
</s>
<s>
The	O
second	O
condition	O
,	O
expected-to-leave-one-out	O
error	O
stability	B-General_Concept
(	O
also	O
known	O
as	O
hypothesis	O
stability	B-General_Concept
if	O
operating	O
in	O
the	O
norm	O
)	O
is	O
met	O
if	O
the	O
prediction	O
on	O
a	O
left-out	O
datapoint	O
does	O
not	O
change	O
when	O
a	O
single	O
data	O
point	O
is	O
removed	O
from	O
the	O
training	O
dataset	O
.	O
</s>
<s>
An	O
algorithm	O
has	O
stability	B-General_Concept
if	O
for	O
each	O
,	O
there	O
exists	O
a	O
and	O
such	O
that	O
:	O
</s>
<s>
An	O
algorithm	O
has	O
stability	B-General_Concept
if	O
for	O
each	O
there	O
exists	O
a	O
and	O
a	O
such	O
that	O
:	O
</s>
<s>
For	O
leave-one-out	O
stability	B-General_Concept
in	O
the	O
norm	O
,	O
this	O
is	O
the	O
same	O
as	O
hypothesis	O
stability	B-General_Concept
:	O
</s>
<s>
A	O
number	O
of	O
algorithms	O
have	O
been	O
proven	O
to	O
be	O
stable	O
and	O
as	O
a	O
result	O
have	O
bounds	O
on	O
their	O
generalization	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
A	O
list	O
of	O
these	O
algorithms	O
and	O
the	O
papers	O
that	O
proved	O
stability	B-General_Concept
is	O
available	O
here	O
.	O
</s>
<s>
The	O
concepts	O
of	O
generalization	B-Algorithm
error	I-Algorithm
and	O
overfitting	B-Error_Name
are	O
closely	O
related	O
.	O
</s>
<s>
Overfitting	B-Error_Name
occurs	O
when	O
the	O
learned	O
function	O
becomes	O
sensitive	O
to	O
the	O
noise	O
in	O
the	O
sample	O
.	O
</s>
<s>
Thus	O
,	O
the	O
more	O
overfitting	B-Error_Name
occurs	O
,	O
the	O
larger	O
the	O
generalization	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
The	O
amount	O
of	O
overfitting	B-Error_Name
can	O
be	O
tested	O
using	O
cross-validation	B-Application
methods	O
,	O
that	O
split	O
the	O
sample	O
into	O
simulated	O
training	O
samples	O
and	O
testing	O
samples	O
.	O
</s>
<s>
This	O
test	O
sample	O
allows	O
us	O
to	O
approximate	O
the	O
expected	O
error	O
and	O
as	O
a	O
result	O
approximate	O
a	O
particular	O
form	O
of	O
the	O
generalization	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
Many	O
algorithms	O
exist	O
to	O
prevent	O
overfitting	B-Error_Name
.	O
</s>
<s>
The	O
approach	O
to	O
finding	O
a	O
function	O
that	O
does	O
not	O
overfit	B-Error_Name
is	O
at	O
odds	O
with	O
the	O
goal	O
of	O
finding	O
a	O
function	O
that	O
is	O
sufficiently	O
complex	O
to	O
capture	O
the	O
particular	O
characteristics	O
of	O
the	O
data	O
.	O
</s>
<s>
This	O
is	O
known	O
as	O
the	O
bias	B-General_Concept
–	I-General_Concept
variance	I-General_Concept
tradeoff	I-General_Concept
.	O
</s>
<s>
Keeping	O
a	O
function	O
simple	O
to	O
avoid	O
overfitting	B-Error_Name
may	O
introduce	O
a	O
bias	O
in	O
the	O
resulting	O
predictions	O
,	O
while	O
allowing	O
it	O
to	O
be	O
more	O
complex	O
leads	O
to	O
overfitting	B-Error_Name
and	O
a	O
higher	O
variance	O
in	O
the	O
predictions	O
.	O
</s>
