<s>
In	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
,	O
the	O
activation	B-Algorithm
function	I-Algorithm
of	O
a	O
node	O
defines	O
the	O
output	O
of	O
that	O
node	O
given	O
an	O
input	O
or	O
set	O
of	O
inputs	O
.	O
</s>
<s>
A	O
standard	O
integrated	O
circuit	O
can	O
be	O
seen	O
as	O
a	O
digital	O
network	O
of	O
activation	B-Algorithm
functions	I-Algorithm
that	O
can	O
be	O
"	O
ON	O
"	O
(	O
1	O
)	O
or	O
"	O
OFF	O
"	O
(	O
0	O
)	O
,	O
depending	O
on	O
input	O
.	O
</s>
<s>
This	O
is	O
similar	O
to	O
the	O
linear	B-Algorithm
perceptron	I-Algorithm
in	O
neural	B-General_Concept
networks	I-General_Concept
.	O
</s>
<s>
However	O
,	O
only	O
nonlinear	O
activation	B-Algorithm
functions	I-Algorithm
allow	O
such	O
networks	O
to	O
compute	O
nontrivial	O
problems	O
using	O
only	O
a	O
small	O
number	O
of	O
nodes	O
,	O
and	O
such	O
activation	B-Algorithm
functions	I-Algorithm
are	O
called	O
nonlinearities	O
.	O
</s>
<s>
The	O
most	O
common	O
activation	B-Algorithm
functions	I-Algorithm
can	O
be	O
divided	O
in	O
three	O
categories	O
:	O
ridge	B-Algorithm
functions	I-Algorithm
,	O
radial	O
functions	O
and	O
fold	B-Application
functions	I-Application
.	O
</s>
<s>
An	O
activation	B-Algorithm
function	I-Algorithm
is	O
saturating	O
if	O
.	O
</s>
<s>
Non-saturating	O
activation	B-Algorithm
functions	I-Algorithm
,	O
such	O
as	O
ReLU	B-Algorithm
,	O
may	O
be	O
better	O
than	O
saturating	O
activation	B-Algorithm
functions	I-Algorithm
,	O
as	O
they	O
do	O
n't	O
suffer	O
from	O
vanishing	O
gradient	O
.	O
</s>
<s>
Ridge	B-Algorithm
functions	I-Algorithm
are	O
multivariate	O
functions	O
acting	O
on	O
a	O
linear	O
combination	O
of	O
the	O
input	O
variables	O
.	O
</s>
<s>
ReLU	B-Algorithm
activation	O
:	O
,	O
</s>
<s>
In	O
biologically	B-General_Concept
inspired	I-General_Concept
neural	I-General_Concept
networks	I-General_Concept
,	O
the	O
activation	B-Algorithm
function	I-Algorithm
is	O
usually	O
an	O
abstraction	O
representing	O
the	O
rate	O
of	O
action	B-Algorithm
potential	I-Algorithm
firing	O
in	O
the	O
cell	O
.	O
</s>
<s>
Neurons	O
also	O
cannot	O
fire	O
faster	O
than	O
a	O
certain	O
rate	O
,	O
motivating	O
sigmoid	B-Algorithm
activation	B-Algorithm
functions	I-Algorithm
whose	O
range	O
is	O
a	O
finite	O
interval	O
.	O
</s>
<s>
A	O
special	O
class	O
of	O
activation	B-Algorithm
functions	I-Algorithm
known	O
as	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
(	O
RBFs	O
)	O
are	O
used	O
in	O
RBF	B-Algorithm
networks	I-Algorithm
,	O
which	O
are	O
extremely	O
efficient	O
as	O
universal	O
function	O
approximators	O
.	O
</s>
<s>
These	O
activation	B-Algorithm
functions	I-Algorithm
can	O
take	O
many	O
forms	O
,	O
but	O
they	O
are	O
usually	O
found	O
as	O
one	O
of	O
the	O
following	O
functions	O
:	O
</s>
<s>
Folding	O
activation	B-Algorithm
functions	I-Algorithm
are	O
extensively	O
used	O
in	O
the	O
pooling	O
layers	O
in	O
convolutional	B-Architecture
neural	I-Architecture
networks	I-Architecture
,	O
and	O
in	O
output	O
layers	O
of	O
multiclass	O
classification	O
networks	O
.	O
</s>
<s>
In	O
multiclass	O
classification	O
the	O
softmax	B-Algorithm
activation	O
is	O
often	O
used	O
.	O
</s>
<s>
There	O
are	O
numerous	O
activation	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
'	O
s	O
seminal	O
2012	O
paper	O
on	O
automatic	O
speech	O
recognition	O
uses	O
a	O
logistic	O
sigmoid	B-Algorithm
activation	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
The	O
seminal	O
2012	O
AlexNet	B-Algorithm
computer	O
vision	O
architecture	O
uses	O
the	O
ReLU	B-Algorithm
activation	B-Algorithm
function	I-Algorithm
,	O
as	O
did	O
the	O
seminal	O
2015	O
computer	O
vision	O
architecture	O
ResNet	B-Algorithm
.	O
</s>
<s>
The	O
seminal	O
2018	O
language	O
processing	O
model	O
BERT	B-General_Concept
uses	O
a	O
smooth	O
version	O
of	O
the	O
ReLU	B-Algorithm
,	O
the	O
GELU	O
.	O
</s>
<s>
Aside	O
from	O
their	O
empirical	O
performance	O
,	O
activation	B-Algorithm
functions	I-Algorithm
also	O
have	O
different	O
mathematical	O
properties	O
:	O
</s>
<s>
Nonlinear	O
When	O
the	O
activation	B-Algorithm
function	I-Algorithm
is	O
non-linear	O
,	O
then	O
a	O
two-layer	O
neural	B-General_Concept
network	I-General_Concept
can	O
be	O
proven	O
to	O
be	O
a	O
universal	O
function	O
approximator	O
.	O
</s>
<s>
This	O
is	O
known	O
as	O
the	O
Universal	B-Algorithm
Approximation	I-Algorithm
Theorem	I-Algorithm
.	O
</s>
<s>
The	O
identity	O
activation	B-Algorithm
function	I-Algorithm
does	O
not	O
satisfy	O
this	O
property	O
.	O
</s>
<s>
When	O
multiple	O
layers	O
use	O
the	O
identity	O
activation	B-Algorithm
function	I-Algorithm
,	O
the	O
entire	O
network	O
is	O
equivalent	O
to	O
a	O
single-layer	O
model	O
.	O
</s>
<s>
Range	O
When	O
the	O
range	O
of	O
the	O
activation	B-Algorithm
function	I-Algorithm
is	O
finite	O
,	O
gradient-based	O
training	O
methods	O
tend	O
to	O
be	O
more	O
stable	O
,	O
because	O
pattern	O
presentations	O
significantly	O
affect	O
only	O
limited	O
weights	O
.	O
</s>
<s>
In	O
the	O
latter	O
case	O
,	O
smaller	O
learning	B-General_Concept
rates	I-General_Concept
are	O
typically	O
necessary	O
.	O
</s>
<s>
Continuously	O
differentiable	O
This	O
property	O
is	O
desirable	O
(	O
ReLU	B-Algorithm
is	O
not	O
continuously	O
differentiable	O
and	O
has	O
some	O
issues	O
with	O
gradient-based	O
optimization	O
,	O
but	O
it	O
is	O
still	O
possible	O
)	O
for	O
enabling	O
gradient-based	O
optimization	O
methods	O
.	O
</s>
<s>
The	O
binary	O
step	O
activation	B-Algorithm
function	I-Algorithm
is	O
not	O
differentiable	O
at	O
0	O
,	O
and	O
it	O
differentiates	O
to	O
0	O
for	O
all	O
other	O
values	O
,	O
so	O
gradient-based	O
methods	O
can	O
make	O
no	O
progress	O
with	O
it	O
.	O
</s>
<s>
The	O
following	O
table	O
compares	O
the	O
properties	O
of	O
several	O
activation	B-Algorithm
functions	I-Algorithm
that	O
are	O
functions	O
of	O
one	O
fold	B-Application
from	O
the	O
previous	O
layer	O
or	O
layers	O
:	O
</s>
<s>
The	O
following	O
table	O
lists	O
activation	B-Algorithm
functions	I-Algorithm
that	O
are	O
not	O
functions	O
of	O
a	O
single	O
fold	B-Application
from	O
the	O
previous	O
layer	O
or	O
layers	O
:	O
</s>
<s>
For	O
instance	O
,	O
could	O
be	O
iterating	O
through	O
the	O
number	O
of	O
kernels	O
of	O
the	O
previous	O
neural	B-General_Concept
network	I-General_Concept
layer	O
while	O
iterates	O
through	O
the	O
number	O
of	O
kernels	O
of	O
the	O
current	O
layer	O
.	O
</s>
