<s>
The	O
sample	B-General_Concept
complexity	I-General_Concept
of	O
a	O
machine	O
learning	O
algorithm	O
represents	O
the	O
number	O
of	O
training-samples	O
that	O
it	O
needs	O
in	O
order	O
to	O
successfully	O
learn	O
a	O
target	O
function	O
.	O
</s>
<s>
More	O
precisely	O
,	O
the	O
sample	B-General_Concept
complexity	I-General_Concept
is	O
the	O
number	O
of	O
training-samples	O
that	O
we	O
need	O
to	O
supply	O
to	O
the	O
algorithm	O
,	O
so	O
that	O
the	O
function	O
returned	O
by	O
the	O
algorithm	O
is	O
within	O
an	O
arbitrarily	O
small	O
error	O
of	O
the	O
best	O
possible	O
function	O
,	O
with	O
probability	O
arbitrarily	O
close	O
to	O
1	O
.	O
</s>
<s>
There	O
are	O
two	O
variants	O
of	O
sample	B-General_Concept
complexity	I-General_Concept
:	O
</s>
<s>
The	O
strong	O
variant	O
takes	O
the	O
worst-case	O
sample	B-General_Concept
complexity	I-General_Concept
over	O
all	O
input-output	O
distributions	O
.	O
</s>
<s>
The	O
No	O
free	O
lunch	O
theorem	O
,	O
discussed	O
below	O
,	O
proves	O
that	O
,	O
in	O
general	O
,	O
the	O
strong	O
sample	B-General_Concept
complexity	I-General_Concept
is	O
infinite	O
,	O
i.e.	O
</s>
<s>
However	O
,	O
if	O
we	O
are	O
only	O
interested	O
in	O
a	O
particular	O
class	O
of	O
target	O
functions	O
(	O
e.g	O
,	O
only	O
linear	O
functions	O
)	O
then	O
the	O
sample	B-General_Concept
complexity	I-General_Concept
is	O
finite	O
,	O
and	O
it	O
depends	O
linearly	O
on	O
the	O
VC	O
dimension	O
on	O
the	O
class	O
of	O
target	O
functions	O
.	O
</s>
<s>
Typical	O
learning	O
algorithms	O
include	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
,	O
without	O
or	O
with	O
Tikhonov	O
regularization	O
.	O
</s>
<s>
The	O
sample	B-General_Concept
complexity	I-General_Concept
of	O
is	O
then	O
the	O
minimum	O
for	O
which	O
this	O
holds	O
,	O
as	O
a	O
function	O
of	O
,	O
and	O
.	O
</s>
<s>
We	O
write	O
the	O
sample	B-General_Concept
complexity	I-General_Concept
as	O
to	O
emphasize	O
that	O
this	O
value	O
of	O
depends	O
on	O
,	O
and	O
.	O
</s>
<s>
In	O
others	O
words	O
,	O
the	O
sample	B-General_Concept
complexity	I-General_Concept
defines	O
the	O
rate	O
of	O
consistency	O
of	O
the	O
algorithm	O
:	O
given	O
a	O
desired	O
accuracy	O
and	O
confidence	O
,	O
one	O
needs	O
to	O
sample	O
data	O
points	O
to	O
guarantee	O
that	O
the	O
risk	O
of	O
the	O
output	O
function	O
is	O
within	O
of	O
the	O
best	O
possible	O
,	O
with	O
probability	O
at	O
least	O
.	O
</s>
<s>
In	O
probably	O
approximately	O
correct	O
(	O
PAC	O
)	O
learning	O
,	O
one	O
is	O
concerned	O
with	O
whether	O
the	O
sample	B-General_Concept
complexity	I-General_Concept
is	O
polynomial	O
,	O
that	O
is	O
,	O
whether	O
is	O
bounded	O
by	O
a	O
polynomial	O
in	O
and	O
.	O
</s>
<s>
One	O
can	O
ask	O
whether	O
there	O
exists	O
a	O
learning	O
algorithm	O
so	O
that	O
the	O
sample	B-General_Concept
complexity	I-General_Concept
is	O
finite	O
in	O
the	O
strong	O
sense	O
,	O
that	O
is	O
,	O
there	O
is	O
a	O
bound	O
on	O
the	O
number	O
of	O
samples	O
needed	O
so	O
that	O
the	O
algorithm	O
can	O
learn	O
any	O
distribution	O
over	O
the	O
input-output	O
space	O
with	O
a	O
specified	O
target	O
error	O
.	O
</s>
<s>
The	O
No	O
Free	O
Lunch	O
Theorem	O
says	O
that	O
without	O
restrictions	O
on	O
the	O
hypothesis	O
space	O
,	O
this	O
is	O
not	O
the	O
case	O
,	O
i.e.	O
,	O
there	O
always	O
exist	O
"	O
bad	O
"	O
distributions	O
for	O
which	O
the	O
sample	B-General_Concept
complexity	I-General_Concept
is	O
arbitrarily	O
large	O
.	O
</s>
<s>
The	O
latter	O
approach	O
leads	O
to	O
concepts	O
such	O
as	O
VC	O
dimension	O
and	O
Rademacher	B-General_Concept
complexity	I-General_Concept
which	O
control	O
the	O
complexity	O
of	O
the	O
space	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
supervised	O
learning	O
setting	O
,	O
sample	B-General_Concept
complexity	I-General_Concept
is	O
relevant	O
to	O
semi-supervised	B-General_Concept
learning	I-General_Concept
problems	O
including	O
active	B-General_Concept
learning	I-General_Concept
,	O
where	O
the	O
algorithm	O
can	O
ask	O
for	O
labels	O
to	O
specifically	O
chosen	O
inputs	O
in	O
order	O
to	O
reduce	O
the	O
cost	O
of	O
obtaining	O
many	O
labels	O
.	O
</s>
<s>
The	O
concept	O
of	O
sample	B-General_Concept
complexity	I-General_Concept
also	O
shows	O
up	O
in	O
reinforcement	O
learning	O
,	O
online	B-Algorithm
learning	I-Algorithm
,	O
and	O
unsupervised	O
algorithms	O
,	O
e.g.	O
</s>
<s>
for	O
dictionary	B-General_Concept
learning	I-General_Concept
.	O
</s>
<s>
A	O
high	O
sample	B-General_Concept
complexity	I-General_Concept
means	O
,	O
that	O
many	O
calculations	O
are	O
needed	O
for	O
running	O
a	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
.	O
</s>
<s>
Its	O
equal	O
to	O
a	O
model	B-Algorithm
free	I-Algorithm
brute	O
force	O
search	O
in	O
the	O
state	O
space	O
.	O
</s>
<s>
In	O
contrast	O
,	O
a	O
high	O
efficiency	O
algorithm	O
has	O
a	O
low	O
sample	B-General_Concept
complexity	I-General_Concept
.	O
</s>
<s>
Possible	O
techniques	O
for	O
reducing	O
the	O
sample	B-General_Concept
complexity	I-General_Concept
are	O
metric	O
learning	O
and	O
model	O
based	O
reinforcement	O
learning	O
.	O
</s>
