<s>
Weak	O
supervision	O
,	O
also	O
called	O
semi-supervised	B-General_Concept
learning	I-General_Concept
,	O
is	O
a	O
branch	O
of	O
machine	O
learning	O
that	O
combines	O
a	O
small	O
amount	O
of	O
labeled	B-General_Concept
data	I-General_Concept
with	O
a	O
large	O
amount	O
of	O
unlabeled	O
data	O
during	O
training	O
.	O
</s>
<s>
Semi-supervised	B-General_Concept
learning	I-General_Concept
falls	O
between	O
unsupervised	B-General_Concept
learning	I-General_Concept
(	O
with	O
no	O
labeled	O
training	O
data	O
)	O
and	O
supervised	B-General_Concept
learning	I-General_Concept
(	O
with	O
only	O
labeled	O
training	O
data	O
)	O
.	O
</s>
<s>
Semi-supervised	B-General_Concept
learning	I-General_Concept
aims	O
to	O
alleviate	O
the	O
issue	O
of	O
having	O
limited	O
amounts	O
of	O
labeled	B-General_Concept
data	I-General_Concept
available	O
for	O
training	O
.	O
</s>
<s>
Semi-supervised	B-General_Concept
learning	I-General_Concept
is	O
motivated	O
by	O
problem	O
settings	O
where	O
unlabeled	O
data	O
is	O
abundant	O
and	O
obtaining	O
labeled	B-General_Concept
data	I-General_Concept
is	O
expensive	O
.	O
</s>
<s>
Other	O
branches	O
of	O
machine	O
learning	O
that	O
share	O
the	O
same	O
motivation	O
but	O
follow	O
different	O
assumptions	O
and	O
methodologies	O
are	O
active	B-General_Concept
learning	I-General_Concept
and	O
weak	O
supervision	O
.	O
</s>
<s>
Unlabeled	O
data	O
,	O
when	O
used	O
in	O
conjunction	O
with	O
a	O
small	O
amount	O
of	O
labeled	B-General_Concept
data	I-General_Concept
,	O
can	O
produce	O
considerable	O
improvement	O
in	O
learning	O
accuracy	O
.	O
</s>
<s>
The	O
acquisition	O
of	O
labeled	B-General_Concept
data	I-General_Concept
for	O
a	O
learning	O
problem	O
often	O
requires	O
a	O
skilled	O
human	O
agent	O
(	O
e.g.	O
</s>
<s>
In	O
such	O
situations	O
,	O
semi-supervised	B-General_Concept
learning	I-General_Concept
can	O
be	O
of	O
great	O
practical	O
value	O
.	O
</s>
<s>
Semi-supervised	B-General_Concept
learning	I-General_Concept
is	O
also	O
of	O
theoretical	O
interest	O
in	O
machine	O
learning	O
and	O
as	O
a	O
model	O
for	O
human	O
learning	O
.	O
</s>
<s>
More	O
formally	O
,	O
semi-supervised	B-General_Concept
learning	I-General_Concept
assumes	O
a	O
set	O
of	O
independently	O
identically	O
distributed	O
examples	O
with	O
corresponding	O
labels	O
and	O
unlabeled	O
examples	O
are	O
processed	O
.	O
</s>
<s>
Semi-supervised	B-General_Concept
learning	I-General_Concept
combines	O
this	O
information	O
to	O
surpass	O
the	O
classification	B-General_Concept
performance	O
that	O
can	O
be	O
obtained	O
either	O
by	O
discarding	O
the	O
unlabeled	O
data	O
and	O
doing	O
supervised	B-General_Concept
learning	I-General_Concept
or	O
by	O
discarding	O
the	O
labels	O
and	O
doing	O
unsupervised	B-General_Concept
learning	I-General_Concept
.	O
</s>
<s>
Semi-supervised	B-General_Concept
learning	I-General_Concept
may	O
refer	O
to	O
either	O
transductive	B-General_Concept
learning	I-General_Concept
or	O
inductive	O
learning	O
.	O
</s>
<s>
The	O
goal	O
of	O
transductive	B-General_Concept
learning	I-General_Concept
is	O
to	O
infer	O
the	O
correct	O
labels	O
for	O
the	O
given	O
unlabeled	O
data	O
only	O
.	O
</s>
<s>
Intuitively	O
,	O
the	O
learning	O
problem	O
can	O
be	O
seen	O
as	O
an	O
exam	O
and	O
labeled	B-General_Concept
data	I-General_Concept
as	O
sample	O
problems	O
that	O
the	O
teacher	O
solves	O
for	O
the	O
class	O
as	O
an	O
aid	O
in	O
solving	O
another	O
set	O
of	O
problems	O
.	O
</s>
<s>
It	O
is	O
unnecessary	O
(	O
and	O
,	O
according	O
to	O
Vapnik	O
's	O
principle	O
,	O
imprudent	O
)	O
to	O
perform	O
transductive	B-General_Concept
learning	I-General_Concept
by	O
way	O
of	O
inferring	O
a	O
classification	B-General_Concept
rule	O
over	O
the	O
entire	O
input	O
space	O
;	O
however	O
,	O
in	O
practice	O
,	O
algorithms	O
formally	O
designed	O
for	O
transduction	B-General_Concept
or	O
induction	O
are	O
often	O
used	O
interchangeably	O
.	O
</s>
<s>
Semi-supervised	B-General_Concept
learning	I-General_Concept
algorithms	O
make	O
use	O
of	O
at	O
least	O
one	O
of	O
the	O
following	O
assumptions	O
:	O
</s>
<s>
This	O
is	O
also	O
generally	O
assumed	O
in	O
supervised	B-General_Concept
learning	I-General_Concept
and	O
yields	O
a	O
preference	O
for	O
geometrically	O
simple	O
decision	B-General_Concept
boundaries	I-General_Concept
.	O
</s>
<s>
In	O
the	O
case	O
of	O
semi-supervised	B-General_Concept
learning	I-General_Concept
,	O
the	O
smoothness	O
assumption	O
additionally	O
yields	O
a	O
preference	O
for	O
decision	B-General_Concept
boundaries	I-General_Concept
in	O
low-density	O
regions	O
,	O
so	O
few	O
points	O
are	O
close	O
to	O
each	O
other	O
but	O
in	O
different	O
classes	O
.	O
</s>
<s>
This	O
is	O
a	O
special	O
case	O
of	O
the	O
smoothness	O
assumption	O
and	O
gives	O
rise	O
to	O
feature	B-General_Concept
learning	I-General_Concept
with	O
clustering	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
The	O
data	O
lie	O
approximately	O
on	O
a	O
manifold	B-Architecture
of	O
much	O
lower	O
dimension	O
than	O
the	O
input	O
space	O
.	O
</s>
<s>
In	O
this	O
case	O
learning	O
the	O
manifold	B-Architecture
using	O
both	O
the	O
labeled	O
and	O
unlabeled	O
data	O
can	O
avoid	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
.	O
</s>
<s>
Then	O
learning	O
can	O
proceed	O
using	O
distances	O
and	O
densities	O
defined	O
on	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
The	O
manifold	B-Architecture
assumption	O
is	O
practical	O
when	O
high-dimensional	O
data	O
are	O
generated	O
by	O
some	O
process	O
that	O
may	O
be	O
hard	O
to	O
model	O
directly	O
,	O
but	O
which	O
has	O
only	O
a	O
few	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
The	O
heuristic	O
approach	O
of	O
self-training	O
(	O
also	O
known	O
as	O
self-learning	O
or	O
self-labeling	O
)	O
is	O
historically	O
the	O
oldest	O
approach	O
to	O
semi-supervised	B-General_Concept
learning	I-General_Concept
,	O
with	O
examples	O
of	O
applications	O
starting	O
in	O
the	O
1960s	O
.	O
</s>
<s>
The	O
transductive	B-General_Concept
learning	I-General_Concept
framework	O
was	O
formally	O
introduced	O
by	O
Vladimir	O
Vapnik	O
in	O
the	O
1970s	O
.	O
</s>
<s>
A	O
probably	O
approximately	O
correct	O
learning	O
bound	O
for	O
semi-supervised	B-General_Concept
learning	I-General_Concept
of	O
a	O
Gaussian	B-Application
mixture	O
was	O
demonstrated	O
by	O
Ratsaby	O
and	O
Venkatesh	O
in	O
1995	O
.	O
</s>
<s>
Semi-supervised	B-General_Concept
learning	I-General_Concept
has	O
recently	O
become	O
more	O
popular	O
and	O
practically	O
relevant	O
due	O
to	O
the	O
variety	O
of	O
problems	O
for	O
which	O
vast	O
quantities	O
of	O
unlabeled	O
data	O
are	O
available	O
—	O
e.g.	O
</s>
<s>
Semi-supervised	B-General_Concept
learning	I-General_Concept
with	O
generative	O
models	O
can	O
be	O
viewed	O
either	O
as	O
an	O
extension	O
of	O
supervised	B-General_Concept
learning	I-General_Concept
(	O
classification	B-General_Concept
plus	O
information	O
about	O
)	O
or	O
as	O
an	O
extension	O
of	O
unsupervised	B-General_Concept
learning	I-General_Concept
(	O
clustering	B-Algorithm
plus	O
some	O
labels	O
)	O
.	O
</s>
<s>
If	O
these	O
assumptions	O
are	O
incorrect	O
,	O
the	O
unlabeled	O
data	O
may	O
actually	O
decrease	O
the	O
accuracy	O
of	O
the	O
solution	O
relative	O
to	O
what	O
would	O
have	O
been	O
obtained	O
from	O
labeled	B-General_Concept
data	I-General_Concept
alone	O
.	O
</s>
<s>
Gaussian	B-Application
mixture	O
distributions	O
are	O
identifiable	O
and	O
commonly	O
used	O
for	O
generative	O
models	O
.	O
</s>
<s>
One	O
of	O
the	O
most	O
commonly	O
used	O
algorithms	O
is	O
the	O
transductive	B-Algorithm
support	I-Algorithm
vector	I-Algorithm
machine	I-Algorithm
,	O
or	O
TSVM	O
(	O
which	O
,	O
despite	O
its	O
name	O
,	O
may	O
be	O
used	O
for	O
inductive	O
learning	O
as	O
well	O
)	O
.	O
</s>
<s>
Whereas	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
for	O
supervised	B-General_Concept
learning	I-General_Concept
seek	O
a	O
decision	B-General_Concept
boundary	I-General_Concept
with	O
maximal	O
margin	B-Algorithm
over	O
the	O
labeled	B-General_Concept
data	I-General_Concept
,	O
the	O
goal	O
of	O
TSVM	O
is	O
a	O
labeling	O
of	O
the	O
unlabeled	O
data	O
such	O
that	O
the	O
decision	B-General_Concept
boundary	I-General_Concept
has	O
maximal	O
margin	B-Algorithm
over	O
all	O
of	O
the	O
data	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
standard	O
hinge	B-Algorithm
loss	I-Algorithm
for	O
labeled	B-General_Concept
data	I-General_Concept
,	O
a	O
loss	O
function	O
is	O
introduced	O
over	O
the	O
unlabeled	O
data	O
by	O
letting	O
.	O
</s>
<s>
TSVM	O
then	O
selects	O
from	O
a	O
reproducing	O
kernel	O
Hilbert	O
space	O
by	O
minimizing	O
the	O
regularized	O
empirical	B-General_Concept
risk	I-General_Concept
:	O
</s>
<s>
Other	O
approaches	O
that	O
implement	O
low-density	O
separation	O
include	O
Gaussian	B-Application
process	O
models	O
,	O
information	O
regularization	O
,	O
and	O
entropy	O
minimization	O
(	O
of	O
which	O
TSVM	O
is	O
a	O
special	O
case	O
)	O
.	O
</s>
<s>
Graph-based	O
methods	O
for	O
semi-supervised	B-General_Concept
learning	I-General_Concept
use	O
a	O
graph	O
representation	O
of	O
the	O
data	O
,	O
with	O
a	O
node	O
for	O
each	O
labeled	O
and	O
unlabeled	O
example	O
.	O
</s>
<s>
Within	O
the	O
framework	O
of	O
manifold	B-General_Concept
regularization	I-General_Concept
,	O
the	O
graph	O
serves	O
as	O
a	O
proxy	O
for	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
A	O
term	O
is	O
added	O
to	O
the	O
standard	O
Tikhonov	O
regularization	O
problem	O
to	O
enforce	O
smoothness	O
of	O
the	O
solution	O
relative	O
to	O
the	O
manifold	B-Architecture
(	O
in	O
the	O
intrinsic	O
space	O
of	O
the	O
problem	O
)	O
as	O
well	O
as	O
relative	O
to	O
the	O
ambient	O
input	O
space	O
.	O
</s>
<s>
where	O
is	O
a	O
reproducing	O
kernel	O
Hilbert	O
space	O
and	O
is	O
the	O
manifold	B-Architecture
on	O
which	O
the	O
data	O
lie	O
.	O
</s>
<s>
The	O
graph-based	O
approach	O
to	O
Laplacian	O
regularization	O
is	O
to	O
put	O
in	O
relation	O
with	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
Laplacian	O
can	O
also	O
be	O
used	O
to	O
extend	O
the	O
supervised	B-General_Concept
learning	I-General_Concept
algorithms	O
:	O
regularized	O
least	O
squares	O
and	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
(	O
SVM	B-Algorithm
)	O
to	O
semi-supervised	O
versions	O
Laplacian	O
regularized	O
least	O
squares	O
and	O
Laplacian	O
SVM	B-Algorithm
.	O
</s>
<s>
Some	O
methods	O
for	O
semi-supervised	B-General_Concept
learning	I-General_Concept
are	O
not	O
intrinsically	O
geared	O
to	O
learning	O
from	O
both	O
unlabeled	O
and	O
labeled	B-General_Concept
data	I-General_Concept
,	O
but	O
instead	O
make	O
use	O
of	O
unlabeled	O
data	O
within	O
a	O
supervised	B-General_Concept
learning	I-General_Concept
framework	O
.	O
</s>
<s>
Then	O
supervised	B-General_Concept
learning	I-General_Concept
proceeds	O
from	O
only	O
the	O
labeled	O
examples	O
.	O
</s>
<s>
Iteratively	O
refining	O
the	O
representation	O
and	O
then	O
performing	O
semi-supervised	B-General_Concept
learning	I-General_Concept
on	O
said	O
representation	O
may	O
further	O
improve	O
performance	O
.	O
</s>
<s>
Self-training	O
is	O
a	O
wrapper	O
method	O
for	O
semi-supervised	B-General_Concept
learning	I-General_Concept
.	O
</s>
<s>
First	O
a	O
supervised	B-General_Concept
learning	I-General_Concept
algorithm	O
is	O
trained	O
based	O
on	O
the	O
labeled	B-General_Concept
data	I-General_Concept
only	O
.	O
</s>
<s>
This	O
classifier	B-General_Concept
is	O
then	O
applied	O
to	O
the	O
unlabeled	O
data	O
to	O
generate	O
more	O
labeled	O
examples	O
as	O
input	O
for	O
the	O
supervised	B-General_Concept
learning	I-General_Concept
algorithm	O
.	O
</s>
<s>
Generally	O
only	O
the	O
labels	O
the	O
classifier	B-General_Concept
is	O
most	O
confident	O
in	O
are	O
added	O
at	O
each	O
step	O
.	O
</s>
<s>
Co-training	B-Algorithm
is	O
an	O
extension	O
of	O
self-training	O
in	O
which	O
multiple	O
classifiers	B-General_Concept
are	O
trained	O
on	O
different	O
(	O
ideally	O
disjoint	O
)	O
sets	O
of	O
features	O
and	O
generate	O
labeled	O
examples	O
for	O
one	O
another	O
.	O
</s>
<s>
Human	O
responses	O
to	O
formal	O
semi-supervised	B-General_Concept
learning	I-General_Concept
problems	O
have	O
yielded	O
varying	O
conclusions	O
about	O
the	O
degree	O
of	O
influence	O
of	O
the	O
unlabeled	O
data	O
.	O
</s>
<s>
More	O
natural	O
learning	O
problems	O
may	O
also	O
be	O
viewed	O
as	O
instances	O
of	O
semi-supervised	B-General_Concept
learning	I-General_Concept
.	O
</s>
