<s>
In	O
machine	O
learning	O
,	O
the	O
delta	B-Algorithm
rule	I-Algorithm
is	O
a	O
gradient	B-Algorithm
descent	I-Algorithm
learning	O
rule	O
for	O
updating	O
the	O
weights	O
of	O
the	O
inputs	O
to	O
artificial	B-Algorithm
neurons	I-Algorithm
in	O
a	O
single-layer	O
neural	O
network	O
.	O
</s>
<s>
It	O
is	O
a	O
special	O
case	O
of	O
the	O
more	O
general	O
backpropagation	B-Algorithm
algorithm	O
.	O
</s>
<s>
is	O
a	O
small	O
constant	O
called	O
learning	B-General_Concept
rate	I-General_Concept
is	O
the	O
neuron	O
's	O
activation	B-Algorithm
function	I-Algorithm
is	O
the	O
derivative	B-Algorithm
of	O
is	O
the	O
target	O
output	O
is	O
the	O
weighted	O
sum	O
of	O
the	O
neuron	O
's	O
inputs	O
is	O
the	O
actual	O
output	O
is	O
the	O
th	O
input	O
.	O
</s>
<s>
While	O
the	O
delta	B-Algorithm
rule	I-Algorithm
is	O
similar	O
to	O
the	O
perceptron	B-Algorithm
's	O
update	O
rule	O
,	O
the	O
derivation	B-Algorithm
is	O
different	O
.	O
</s>
<s>
The	O
perceptron	B-Algorithm
uses	O
the	O
Heaviside	O
step	O
function	O
as	O
the	O
activation	B-Algorithm
function	I-Algorithm
,	O
and	O
that	O
means	O
that	O
does	O
not	O
exist	O
at	O
zero	O
,	O
and	O
is	O
equal	O
to	O
zero	O
elsewhere	O
,	O
which	O
makes	O
the	O
direct	O
application	O
of	O
the	O
delta	B-Algorithm
rule	I-Algorithm
impossible	O
.	O
</s>
<s>
The	O
delta	B-Algorithm
rule	I-Algorithm
is	O
derived	O
by	O
attempting	O
to	O
minimize	O
the	O
error	O
in	O
the	O
output	O
of	O
the	O
neural	O
network	O
through	O
gradient	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
do	O
that	O
,	O
we	O
calculate	O
the	O
partial	O
derivative	B-Algorithm
of	O
the	O
error	O
with	O
respect	O
to	O
each	O
weight	O
.	O
</s>
<s>
Next	O
we	O
use	O
the	O
chain	O
rule	O
to	O
split	O
this	O
into	O
two	O
derivatives	B-Algorithm
:	O
</s>
<s>
To	O
find	O
the	O
left	O
derivative	B-Algorithm
,	O
we	O
simply	O
apply	O
the	O
chain	O
rule	O
:	O
</s>
<s>
To	O
find	O
the	O
right	O
derivative	B-Algorithm
,	O
we	O
again	O
apply	O
the	O
chain	O
rule	O
,	O
this	O
time	O
differentiating	O
with	O
respect	O
to	O
the	O
total	O
input	O
to	O
,	O
:	O
</s>
<s>
Note	O
that	O
the	O
output	O
of	O
the	O
th	O
neuron	O
,	O
,	O
is	O
just	O
the	O
neuron	O
's	O
activation	B-Algorithm
function	I-Algorithm
applied	O
to	O
the	O
neuron	O
's	O
input	O
.	O
</s>
<s>
We	O
can	O
therefore	O
write	O
the	O
derivative	B-Algorithm
of	O
with	O
respect	O
to	O
simply	O
as	O
'	O
s	O
first	B-Algorithm
derivative	I-Algorithm
:	O
</s>
<s>
As	O
noted	O
above	O
,	O
gradient	B-Algorithm
descent	I-Algorithm
tells	O
us	O
that	O
our	O
change	O
for	O
each	O
weight	O
should	O
be	O
proportional	O
to	O
the	O
gradient	O
.	O
</s>
