<s>
In	O
computational	O
learning	O
theory	O
(	O
machine	O
learning	O
and	O
theory	O
of	O
computation	O
)	O
,	O
Rademacher	B-General_Concept
complexity	I-General_Concept
,	O
named	O
after	O
Hans	O
Rademacher	O
,	O
measures	O
richness	O
of	O
a	O
class	O
of	O
real-valued	O
functions	O
with	O
respect	O
to	O
a	O
probability	O
distribution	O
.	O
</s>
<s>
Given	O
a	O
set	O
,	O
the	O
Rademacher	B-General_Concept
complexity	I-General_Concept
of	O
A	O
is	O
defined	O
as	O
follows	O
:	O
</s>
<s>
Then	O
,	O
the	O
empirical	O
Rademacher	B-General_Concept
complexity	I-General_Concept
of	O
given	O
is	O
defined	O
as	O
:	O
</s>
<s>
where	O
denotes	O
function	B-Application
composition	I-Application
,	O
i.e.	O
</s>
<s>
The	O
Rademacher	B-General_Concept
complexity	I-General_Concept
of	O
the	O
function	O
class	O
with	O
respect	O
to	O
for	O
sample	O
size	O
is	O
:	O
</s>
<s>
The	O
Rademacher	B-General_Concept
complexity	I-General_Concept
is	O
typically	O
applied	O
on	O
a	O
function	O
class	O
of	O
models	O
that	O
are	O
used	O
for	O
classification	O
,	O
with	O
the	O
goal	O
of	O
measuring	O
their	O
ability	O
to	O
classify	O
points	O
drawn	O
from	O
a	O
probability	O
space	O
under	O
arbitrary	O
labellings	O
.	O
</s>
<s>
The	O
Rademacher	B-General_Concept
complexity	I-General_Concept
can	O
be	O
used	O
to	O
derive	O
data-dependent	O
upper-bounds	O
on	O
the	O
learnability	O
of	O
function	O
classes	O
.	O
</s>
<s>
Intuitively	O
,	O
a	O
function-class	O
with	O
smaller	O
Rademacher	B-General_Concept
complexity	I-General_Concept
is	O
easier	O
to	O
learn	O
.	O
</s>
<s>
Smaller	O
representativeness	O
is	O
better	O
,	O
since	O
it	O
provides	O
a	O
way	O
to	O
avoid	O
overfitting	B-Error_Name
:	O
it	O
means	O
that	O
the	O
true	O
error	O
of	O
a	O
classifier	O
is	O
not	O
much	O
higher	O
than	O
its	O
estimated	O
error	O
,	O
and	O
so	O
selecting	O
a	O
classifier	O
that	O
has	O
low	O
estimated	O
error	O
will	O
ensure	O
that	O
the	O
true	O
error	O
is	O
also	O
low	O
.	O
</s>
<s>
The	O
expected	O
representativeness	O
of	O
a	O
sample	O
can	O
be	O
bounded	O
above	O
by	O
the	O
Rademacher	B-General_Concept
complexity	I-General_Concept
of	O
the	O
function	O
class	O
:	O
</s>
<s>
When	O
the	O
Rademacher	B-General_Concept
complexity	I-General_Concept
is	O
small	O
,	O
it	O
is	O
possible	O
to	O
learn	O
the	O
hypothesis	O
class	O
H	O
using	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
.	O
</s>
<s>
Since	O
smaller	O
Rademacher	B-General_Concept
complexity	I-General_Concept
is	O
better	O
,	O
it	O
is	O
useful	O
to	O
have	O
upper	O
bounds	O
on	O
the	O
Rademacher	B-General_Concept
complexity	I-General_Concept
of	O
various	O
function	O
sets	O
.	O
</s>
<s>
The	O
following	O
rules	O
can	O
be	O
used	O
to	O
upper-bound	O
the	O
Rademacher	B-General_Concept
complexity	I-General_Concept
of	O
a	O
set	O
.	O
</s>
<s>
The	O
Rademacher	B-General_Concept
complexity	I-General_Concept
of	O
the	O
convex	O
hull	O
of	O
equals	O
Rad(A )	O
.	O
</s>
<s>
(	O
Massart	O
Lemma	O
)	O
The	O
Rademacher	B-General_Concept
complexity	I-General_Concept
of	O
a	O
finite	O
set	O
grows	O
logarithmically	O
with	O
the	O
set	O
size	O
.	O
</s>
<s>
With	O
more	O
advanced	O
techniques	O
(	O
Dudley	O
's	O
entropy	O
bound	O
and	O
Haussler	O
's	O
upper	O
bound	O
)	O
one	O
can	O
show	O
,	O
for	O
example	O
,	O
that	O
there	O
exists	O
a	O
constant	O
,	O
such	O
that	O
any	O
class	O
of	O
-indicator	O
functions	O
with	O
Vapnik	O
–	O
Chervonenkis	O
dimension	O
has	O
Rademacher	B-General_Concept
complexity	I-General_Concept
upper-bounded	O
by	O
.	O
</s>
<s>
The	O
following	O
bound	O
relates	O
the	O
Rademacher	B-General_Concept
complexity	I-General_Concept
of	O
a	O
set	O
to	O
its	O
external	O
covering	O
number	O
–	O
the	O
number	O
of	O
balls	O
of	O
a	O
given	O
radius	O
whose	O
union	O
contains	O
.	O
</s>
<s>
Substituting	O
this	O
in	O
the	O
previous	O
bound	O
gives	O
the	O
following	O
bound	O
on	O
the	O
Rademacher	B-General_Concept
complexity	I-General_Concept
:	O
</s>
<s>
Gaussian	O
complexity	O
is	O
a	O
similar	O
complexity	O
with	O
similar	O
physical	O
meanings	O
,	O
and	O
can	O
be	O
obtained	O
from	O
the	O
Rademacher	B-General_Concept
complexity	I-General_Concept
using	O
the	O
random	O
variables	O
instead	O
of	O
,	O
where	O
are	O
Gaussian	O
i.i.d.	O
</s>
<s>
Gaussian	O
and	O
Rademacher	B-General_Concept
complexities	I-General_Concept
are	O
known	O
to	O
be	O
equivalent	O
up	O
to	O
logarithmic	O
factors	O
.	O
</s>
