<s>
Stability	O
,	O
also	O
known	O
as	O
algorithmic	B-General_Concept
stability	I-General_Concept
,	O
is	O
a	O
notion	O
in	O
computational	O
learning	O
theory	O
of	O
how	O
a	O
machine	O
learning	O
algorithm	O
output	O
is	O
changed	O
with	O
small	O
perturbations	O
to	O
its	O
inputs	O
.	O
</s>
<s>
For	O
instance	O
,	O
consider	O
a	O
machine	O
learning	O
algorithm	O
that	O
is	O
being	O
trained	O
to	O
recognize	B-Application
handwritten	I-Application
letters	I-Application
of	O
the	O
alphabet	O
,	O
using	O
1000	O
examples	O
of	O
handwritten	O
letters	O
and	O
their	O
labels	O
(	O
"	O
A	O
"	O
to	O
"	O
Z	O
"	O
)	O
as	O
a	O
training	O
set	O
.	O
</s>
<s>
A	O
stable	O
learning	O
algorithm	O
would	O
produce	O
a	O
similar	O
classifier	B-General_Concept
with	O
both	O
the	O
1000-element	O
and	O
999-element	O
training	O
sets	O
.	O
</s>
<s>
Stability	O
can	O
be	O
studied	O
for	O
many	O
types	O
of	O
learning	O
problems	O
,	O
from	O
language	B-Language
learning	I-Language
to	O
inverse	O
problems	O
in	O
physics	O
and	O
engineering	O
,	O
as	O
it	O
is	O
a	O
property	O
of	O
the	O
learning	O
process	O
rather	O
than	O
the	O
type	O
of	O
information	O
being	O
learned	O
.	O
</s>
<s>
It	O
was	O
shown	O
that	O
for	O
large	O
classes	O
of	O
learning	O
algorithms	O
,	O
notably	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
algorithms	O
,	O
certain	O
types	O
of	O
stability	O
ensure	O
good	O
generalization	O
.	O
</s>
<s>
In	O
the	O
1990s	O
,	O
milestones	O
were	O
reached	O
in	O
obtaining	O
generalization	O
bounds	O
for	O
supervised	B-General_Concept
learning	I-General_Concept
algorithms	I-General_Concept
.	O
</s>
<s>
The	O
technique	O
historically	O
used	O
to	O
prove	O
generalization	O
was	O
to	O
show	O
that	O
an	O
algorithm	O
was	O
consistent	O
,	O
using	O
the	O
uniform	B-Algorithm
convergence	I-Algorithm
properties	O
of	O
empirical	O
quantities	O
to	O
their	O
means	O
.	O
</s>
<s>
This	O
technique	O
was	O
used	O
to	O
obtain	O
generalization	O
bounds	O
for	O
the	O
large	O
class	O
of	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
(	O
ERM	O
)	O
algorithms	O
.	O
</s>
<s>
Another	O
example	O
is	O
language	B-Language
learning	I-Language
algorithms	O
that	O
can	O
produce	O
sentences	O
of	O
arbitrary	O
length	O
.	O
</s>
<s>
A	O
measure	O
of	O
Leave	B-General_Concept
one	I-General_Concept
out	I-General_Concept
error	I-General_Concept
is	O
used	O
in	O
a	O
Cross	O
Validation	O
Leave	O
One	O
Out	O
(	O
CVloo	O
)	O
algorithm	O
to	O
evaluate	O
a	O
learning	O
algorithm	O
's	O
stability	O
with	O
respect	O
to	O
the	O
loss	O
function	O
.	O
</s>
<s>
Early	O
1900s	O
-	O
Stability	B-General_Concept
in	I-General_Concept
learning	I-General_Concept
theory	O
was	O
earliest	O
described	O
in	O
terms	O
of	O
continuity	O
of	O
the	O
learning	O
map	O
,	O
traced	O
to	O
Andrey	O
Nikolayevich	O
Tikhonov	O
.	O
</s>
<s>
k-NN	O
classifier	B-General_Concept
with	O
a	O
 { 0-1 } 	O
loss	O
function	O
.	O
</s>
<s>
Support	B-Algorithm
Vector	I-Algorithm
Machine	I-Algorithm
(	O
SVM	B-Algorithm
)	O
classification	O
with	O
a	O
bounded	O
kernel	O
and	O
where	O
the	O
regularizer	O
is	O
a	O
norm	O
in	O
a	O
Reproducing	O
Kernel	O
Hilbert	O
Space	O
.	O
</s>
<s>
Soft	O
margin	O
SVM	B-Algorithm
classification	O
.	O
</s>
<s>
A	O
version	O
of	O
bagging	B-Algorithm
regularizers	O
with	O
the	O
number	O
of	O
regressors	O
increasing	O
with	O
.	O
</s>
<s>
Multi-class	O
SVM	B-Algorithm
classification	O
.	O
</s>
