<s>
In	O
machine	O
learning	O
,	O
early	B-Algorithm
stopping	I-Algorithm
is	O
a	O
form	O
of	O
regularization	O
used	O
to	O
avoid	O
overfitting	B-Error_Name
when	O
training	O
a	O
learner	O
with	O
an	O
iterative	O
method	O
,	O
such	O
as	O
gradient	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
Past	O
that	O
point	O
,	O
however	O
,	O
improving	O
the	O
learner	O
's	O
fit	O
to	O
the	O
training	O
data	O
comes	O
at	O
the	O
expense	O
of	O
increased	O
generalization	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
Early	B-Algorithm
stopping	I-Algorithm
rules	O
provide	O
guidance	O
as	O
to	O
how	O
many	O
iterations	O
can	O
be	O
run	O
before	O
the	O
learner	O
begins	O
to	O
over-fit	O
.	O
</s>
<s>
Early	B-Algorithm
stopping	I-Algorithm
rules	O
have	O
been	O
employed	O
in	O
many	O
different	O
machine	O
learning	O
methods	O
,	O
with	O
varying	O
amounts	O
of	O
theoretical	O
foundation	O
.	O
</s>
<s>
This	O
section	O
presents	O
some	O
of	O
the	O
basic	O
machine-learning	O
concepts	O
required	O
for	O
a	O
description	O
of	O
early	B-Algorithm
stopping	I-Algorithm
methods	O
.	O
</s>
<s>
Overfitting	B-Error_Name
occurs	O
when	O
a	O
model	O
fits	O
the	O
data	O
in	O
the	O
training	O
set	O
well	O
,	O
while	O
incurring	O
larger	O
generalization	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
Regularization	O
,	O
in	O
the	O
context	O
of	O
machine	O
learning	O
,	O
refers	O
to	O
the	O
process	O
of	O
modifying	O
a	O
learning	O
algorithm	O
so	O
as	O
to	O
prevent	O
overfitting	B-Error_Name
.	O
</s>
<s>
Tikhonov	O
regularization	O
,	O
along	O
with	O
principal	B-Algorithm
component	I-Algorithm
regression	I-Algorithm
and	O
many	O
other	O
regularization	O
schemes	O
,	O
fall	O
under	O
the	O
umbrella	O
of	O
spectral	O
regularization	O
,	O
regularization	O
characterized	O
by	O
the	O
application	O
of	O
a	O
filter	O
.	O
</s>
<s>
Early	B-Algorithm
stopping	I-Algorithm
also	O
belongs	O
to	O
this	O
class	O
of	O
methods	O
.	O
</s>
<s>
Gradient	B-Algorithm
descent	I-Algorithm
methods	I-Algorithm
are	O
first-order	O
,	O
iterative	O
,	O
optimization	O
methods	O
.	O
</s>
<s>
Gradient	B-Algorithm
descent	I-Algorithm
is	O
used	O
in	O
machine-learning	O
by	O
defining	O
a	O
loss	O
function	O
that	O
reflects	O
the	O
error	O
of	O
the	O
learner	O
on	O
the	O
training	O
set	O
and	O
then	O
minimizing	O
that	O
function	O
.	O
</s>
<s>
These	O
spaces	O
can	O
be	O
infinite	O
dimensional	O
,	O
in	O
which	O
they	O
can	O
supply	O
solutions	O
that	O
overfit	B-Error_Name
training	O
sets	O
of	O
arbitrary	O
size	O
.	O
</s>
<s>
One	O
way	O
to	O
regularize	O
non-parametric	O
regression	O
problems	O
is	O
to	O
apply	O
an	O
early	B-Algorithm
stopping	I-Algorithm
rule	O
to	O
an	O
iterative	O
procedure	O
such	O
as	O
gradient	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
The	O
early	B-Algorithm
stopping	I-Algorithm
rules	O
proposed	O
for	O
these	O
problems	O
are	O
based	O
on	O
analysis	O
of	O
upper	O
bounds	O
on	O
the	O
generalization	B-Algorithm
error	I-Algorithm
as	O
a	O
function	O
of	O
the	O
iteration	O
number	O
.	O
</s>
<s>
Let	O
and	O
be	O
the	O
t-th	O
iterates	O
of	O
gradient	B-Algorithm
descent	I-Algorithm
applied	O
to	O
the	O
expected	O
and	O
empirical	O
risks	O
,	O
respectively	O
,	O
where	O
both	O
iterations	O
are	O
initialized	O
at	O
the	O
origin	O
,	O
and	O
both	O
use	O
the	O
step	O
size	O
.	O
</s>
<s>
The	O
form	O
the	O
population	O
iteration	O
,	O
which	O
converges	O
to	O
,	O
but	O
cannot	O
be	O
used	O
in	O
computation	O
,	O
while	O
the	O
form	O
the	O
sample	O
iteration	O
which	O
usually	O
converges	O
to	O
an	O
overfitting	B-Error_Name
solution	O
.	O
</s>
<s>
This	O
equation	O
presents	O
a	O
bias-variance	B-General_Concept
tradeoff	I-General_Concept
,	O
which	O
is	O
then	O
solved	O
to	O
give	O
an	O
optimal	O
stopping	O
rule	O
that	O
may	O
depend	O
on	O
the	O
unknown	O
probability	O
distribution	O
.	O
</s>
<s>
That	O
rule	O
has	O
associated	O
probabilistic	O
bounds	O
on	O
the	O
generalization	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
For	O
the	O
analysis	O
leading	O
to	O
the	O
early	B-Algorithm
stopping	I-Algorithm
rule	O
and	O
bounds	O
,	O
the	O
reader	O
is	O
referred	O
to	O
the	O
original	O
article	O
.	O
</s>
<s>
cross-validation	B-Application
can	O
be	O
used	O
to	O
obtain	O
an	O
adaptive	O
stopping	O
rule	O
.	O
</s>
<s>
Boosting	B-Algorithm
refers	O
to	O
a	O
family	O
of	O
algorithms	O
in	O
which	O
a	O
set	O
of	O
weak	B-Algorithm
learners	I-Algorithm
(	O
learners	O
that	O
are	O
only	O
slightly	O
correlated	O
with	O
the	O
true	O
process	O
)	O
are	O
combined	O
to	O
produce	O
a	O
strong	O
learner	O
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
,	O
for	O
several	O
boosting	B-Algorithm
algorithms	O
(	O
including	O
AdaBoost	B-Algorithm
)	O
,	O
that	O
regularization	O
via	O
early	B-Algorithm
stopping	I-Algorithm
can	O
provide	O
guarantees	O
of	O
consistency	O
,	O
that	O
is	O
,	O
that	O
the	O
result	O
of	O
the	O
algorithm	O
approaches	O
the	O
true	O
solution	O
as	O
the	O
number	O
of	O
samples	O
goes	O
to	O
infinity	O
.	O
</s>
<s>
Boosting	B-Algorithm
methods	O
have	O
close	O
ties	O
to	O
the	O
gradient	B-Algorithm
descent	I-Algorithm
methods	I-Algorithm
described	O
above	O
can	O
be	O
regarded	O
as	O
a	O
boosting	B-Algorithm
method	O
based	O
on	O
the	O
loss	O
:	O
LBoost	O
.	O
</s>
<s>
These	O
early	B-Algorithm
stopping	I-Algorithm
rules	O
work	O
by	O
splitting	O
the	O
original	O
training	O
set	O
into	O
a	O
new	O
training	O
set	O
and	O
a	O
validation	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
The	O
error	O
on	O
the	O
validation	B-General_Concept
set	I-General_Concept
is	O
used	O
as	O
a	O
proxy	O
for	O
the	O
generalization	B-Algorithm
error	I-Algorithm
in	O
determining	O
when	O
overfitting	B-Error_Name
has	O
begun	O
.	O
</s>
<s>
These	O
methods	O
are	O
most	O
commonly	O
employed	O
in	O
the	O
training	O
of	O
neural	B-Architecture
networks	I-Architecture
.	O
</s>
<s>
Prechelt	O
gives	O
the	O
following	O
summary	O
of	O
a	O
naive	O
implementation	O
of	O
holdout-based	O
early	B-Algorithm
stopping	I-Algorithm
as	O
follows	O
:	O
</s>
<s>
More	O
sophisticated	O
forms	O
use	O
cross-validation	B-Application
–	O
multiple	O
partitions	O
of	O
the	O
data	O
into	O
training	O
set	O
and	O
validation	B-General_Concept
set	I-General_Concept
–	O
instead	O
of	O
a	O
single	O
partition	O
into	O
a	O
training	O
set	O
and	O
validation	B-General_Concept
set	I-General_Concept
.	O
</s>
<s>
This	O
complication	O
has	O
led	O
to	O
the	O
creation	O
of	O
many	O
ad	O
hoc	O
rules	O
for	O
deciding	O
when	O
overfitting	B-Error_Name
has	O
truly	O
begun	O
.	O
</s>
