<s>
In	O
the	O
context	O
of	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
,	O
the	O
rectifier	B-Algorithm
or	O
ReLU	B-Algorithm
(	O
rectified	B-Algorithm
linear	I-Algorithm
unit	I-Algorithm
)	O
activation	B-Algorithm
function	I-Algorithm
is	O
an	O
activation	B-Algorithm
function	I-Algorithm
defined	O
as	O
the	O
positive	O
part	O
of	O
its	O
argument	O
:	O
</s>
<s>
This	O
activation	B-Algorithm
function	I-Algorithm
was	O
introduced	O
by	O
Kunihiko	O
Fukushima	O
in	O
1969	O
in	O
the	O
context	O
of	O
visual	O
feature	O
extraction	O
in	O
hierarchical	O
neural	B-Architecture
networks	I-Architecture
.	O
</s>
<s>
In	O
2011	O
it	O
was	O
found	O
to	O
enable	O
better	O
training	O
of	O
deeper	O
networks	O
,	O
compared	O
to	O
the	O
widely	O
used	O
activation	B-Algorithm
functions	I-Algorithm
prior	O
to	O
2011	O
,	O
e.g.	O
,	O
the	O
logistic	O
sigmoid	O
(	O
which	O
is	O
inspired	O
by	O
probability	O
theory	O
;	O
see	O
logistic	O
regression	O
)	O
and	O
its	O
more	O
practical	O
counterpart	O
,	O
the	O
hyperbolic	O
tangent	O
.	O
</s>
<s>
The	O
rectifier	B-Algorithm
is	O
,	O
,	O
the	O
most	O
popular	O
activation	B-Algorithm
function	I-Algorithm
for	O
deep	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
Rectified	B-Algorithm
linear	I-Algorithm
units	I-Algorithm
find	O
applications	O
in	O
computer	B-Application
vision	I-Application
and	O
speech	B-Application
recognition	I-Application
using	O
deep	B-Algorithm
neural	I-Algorithm
nets	I-Algorithm
and	O
computational	O
neuroscience	O
.	O
</s>
<s>
Better	O
gradient	O
propagation	O
:	O
Fewer	O
vanishing	B-Algorithm
gradient	I-Algorithm
problems	I-Algorithm
compared	O
to	O
sigmoidal	O
activation	B-Algorithm
functions	I-Algorithm
that	O
saturate	O
in	O
both	O
directions	O
.	O
</s>
<s>
Rectifying	O
activation	B-Algorithm
functions	I-Algorithm
were	O
used	O
to	O
separate	O
specific	O
excitation	O
and	O
unspecific	O
inhibition	O
in	O
the	O
neural	O
abstraction	O
pyramid	O
,	O
which	O
was	O
trained	O
in	O
a	O
supervised	B-General_Concept
way	O
to	O
learn	O
several	O
computer	B-Application
vision	I-Application
tasks	O
.	O
</s>
<s>
In	O
2011	O
,	O
the	O
use	O
of	O
the	O
rectifier	B-Algorithm
as	O
a	O
non-linearity	O
has	O
been	O
shown	O
to	O
enable	O
training	O
deep	O
supervised	B-General_Concept
neural	B-Architecture
networks	I-Architecture
without	O
requiring	O
unsupervised	B-General_Concept
pre-training	O
.	O
</s>
<s>
Rectified	B-Algorithm
linear	I-Algorithm
units	I-Algorithm
,	O
compared	O
to	O
sigmoid	B-Algorithm
function	I-Algorithm
or	O
similar	O
activation	B-Algorithm
functions	I-Algorithm
,	O
allow	O
faster	O
and	O
effective	O
training	O
of	O
deep	O
neural	O
architectures	O
on	O
large	O
and	O
complex	O
datasets	O
.	O
</s>
<s>
Non-differentiable	O
at	O
zero	O
;	O
however	O
,	O
it	O
is	O
differentiable	O
anywhere	O
else	O
,	O
and	O
the	O
value	O
of	O
the	O
derivative	B-Algorithm
at	O
zero	O
can	O
be	O
arbitrarily	O
chosen	O
to	O
be	O
0	O
or1	O
.	O
</s>
<s>
Dying	O
ReLU	B-Algorithm
problem	O
:	O
ReLU	B-Algorithm
(	O
rectified	B-Algorithm
linear	I-Algorithm
unit	I-Algorithm
)	O
neurons	O
can	O
sometimes	O
be	O
pushed	O
into	O
states	O
in	O
which	O
they	O
become	O
inactive	O
for	O
essentially	O
all	O
inputs	O
.	O
</s>
<s>
This	O
is	O
a	O
form	O
of	O
the	O
vanishing	B-Algorithm
gradient	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
It	O
may	O
be	O
mitigated	O
by	O
using	O
leaky	O
ReLUs	B-Algorithm
instead	O
,	O
which	O
assign	O
a	O
small	O
positive	O
slope	O
for	O
x	O
<	O
0	O
;	O
however	O
,	O
the	O
performance	O
is	O
reduced	O
.	O
</s>
<s>
Leaky	O
ReLUs	B-Algorithm
allow	O
a	O
small	O
,	O
positive	O
gradient	O
when	O
the	O
unit	O
is	O
not	O
active	O
.	O
</s>
<s>
Parametric	O
ReLUs	B-Algorithm
(	O
PReLUs	O
)	O
take	O
this	O
idea	O
further	O
by	O
making	O
the	O
coefficient	O
of	O
leakage	O
into	O
a	O
parameter	O
that	O
is	O
learned	O
along	O
with	O
the	O
other	O
neural-network	O
parameters	O
.	O
</s>
<s>
GELU	O
is	O
a	O
smooth	O
approximation	O
to	O
the	O
rectifier	B-Algorithm
:	O
</s>
<s>
This	O
activation	B-Algorithm
function	I-Algorithm
is	O
illustrated	O
in	O
the	O
figure	O
at	O
the	O
start	O
of	O
this	O
article	O
.	O
</s>
<s>
The	O
SiLU	O
(	O
sigmoid	O
linear	O
unit	O
)	O
or	O
swish	B-Algorithm
function	I-Algorithm
is	O
another	O
smooth	O
approximation	O
,	O
first	O
coined	O
in	O
the	O
GELU	O
paper	O
:	O
</s>
<s>
where	O
is	O
the	O
sigmoid	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
The	O
derivative	B-Algorithm
of	O
softplus	O
is	O
the	O
logistic	O
function	O
.	O
</s>
<s>
The	O
logistic	O
sigmoid	B-Algorithm
function	I-Algorithm
is	O
a	O
smooth	O
approximation	O
of	O
the	O
derivative	B-Algorithm
of	O
the	O
rectifier	B-Algorithm
,	O
the	O
Heaviside	O
step	O
function	O
.	O
</s>
<s>
and	O
its	O
gradient	O
is	O
the	O
softmax	B-Algorithm
;	O
the	O
softmax	B-Algorithm
with	O
the	O
first	O
argument	O
set	O
to	O
zero	O
is	O
the	O
multivariable	O
generalization	O
of	O
the	O
logistic	O
function	O
.	O
</s>
<s>
Both	O
LogSumExp	O
and	O
softmax	B-Algorithm
are	O
used	O
in	O
machine	O
learning	O
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
that	O
ELUs	O
can	O
obtain	O
higher	O
classification	O
accuracy	O
than	O
ReLUs	B-Algorithm
.	O
</s>
<s>
In	O
these	O
formulas	O
,	O
is	O
a	O
hyper-parameter	B-General_Concept
to	O
be	O
tuned	O
with	O
the	O
constraint	O
.	O
</s>
<s>
The	O
ELU	O
can	O
be	O
viewed	O
as	O
a	O
smoothed	O
version	O
of	O
a	O
shifted	O
ReLU	B-Algorithm
(	O
SReLU	O
)	O
,	O
which	O
has	O
the	O
form	O
,	O
given	O
the	O
same	O
interpretation	O
of	O
.	O
</s>
<s>
The	O
mish	O
function	O
could	O
also	O
be	O
used	O
as	O
a	O
smooth	O
approximation	O
of	O
the	O
rectifier	B-Algorithm
.	O
</s>
<s>
It	O
was	O
inspired	O
by	O
Swish	B-Algorithm
,	O
itself	O
a	O
variant	O
of	O
ReLU	B-Algorithm
.	O
</s>
<s>
The	O
function	O
describing	O
the	O
metallic	O
means	O
and	O
its	O
derivative	B-Algorithm
is	O
:	O
</s>
