<s>
In	O
machine	O
learning	O
,	O
the	O
perceptron	B-Algorithm
(	O
or	O
McCulloch-Pitts	B-Algorithm
neuron	I-Algorithm
)	O
is	O
an	O
algorithm	O
for	O
supervised	B-General_Concept
learning	I-General_Concept
of	O
binary	B-General_Concept
classifiers	I-General_Concept
.	O
</s>
<s>
A	O
binary	B-General_Concept
classifier	I-General_Concept
is	O
a	O
function	O
which	O
can	O
decide	O
whether	O
or	O
not	O
an	O
input	O
,	O
represented	O
by	O
a	O
vector	O
of	O
numbers	O
,	O
belongs	O
to	O
some	O
specific	O
class	O
.	O
</s>
<s>
It	O
is	O
a	O
type	O
of	O
linear	B-General_Concept
classifier	I-General_Concept
,	O
i.e.	O
</s>
<s>
a	O
classification	O
algorithm	O
that	O
makes	O
its	O
predictions	O
based	O
on	O
a	O
linear	B-General_Concept
predictor	I-General_Concept
function	I-General_Concept
combining	O
a	O
set	O
of	O
weights	B-General_Concept
with	O
the	O
feature	B-Algorithm
vector	I-Algorithm
.	O
</s>
<s>
The	O
perceptron	B-Algorithm
was	O
invented	O
in	O
1943	O
by	O
Warren	O
McCulloch	O
and	O
Walter	O
Pitts	O
.	O
</s>
<s>
The	O
perceptron	B-Algorithm
was	O
intended	O
to	O
be	O
a	O
machine	O
,	O
rather	O
than	O
a	O
program	O
,	O
and	O
while	O
its	O
first	O
implementation	O
was	O
in	O
software	O
for	O
the	O
IBM	B-Device
704	I-Device
,	O
it	O
was	O
subsequently	O
implemented	O
in	O
custom-built	O
hardware	O
as	O
the	O
"	O
Mark	O
1	O
perceptron	B-Algorithm
"	O
.	O
</s>
<s>
Weights	B-General_Concept
were	O
encoded	O
in	O
potentiometers	B-Device
,	O
and	O
weight	O
updates	O
during	O
learning	O
were	O
performed	O
by	O
electric	O
motors	O
.	O
</s>
<s>
In	O
a	O
1958	O
press	O
conference	O
organized	O
by	O
the	O
US	O
Navy	O
,	O
Rosenblatt	O
made	O
statements	O
about	O
the	O
perceptron	B-Algorithm
that	O
caused	O
a	O
heated	O
controversy	O
among	O
the	O
fledgling	O
AI	B-Application
community	O
;	O
based	O
on	O
Rosenblatt	O
's	O
statements	O
,	O
The	O
New	O
York	O
Times	O
reported	O
the	O
perceptron	B-Algorithm
to	O
be	O
"	O
the	O
embryo	O
of	O
an	O
electronic	O
computer	O
that	O
[	O
the	O
Navy ]	O
expects	O
will	O
be	O
able	O
to	O
walk	O
,	O
talk	O
,	O
see	O
,	O
write	O
,	O
reproduce	O
itself	O
and	O
be	O
conscious	O
of	O
its	O
existence.	O
"	O
</s>
<s>
Although	O
the	O
perceptron	B-Algorithm
initially	O
seemed	O
promising	O
,	O
it	O
was	O
quickly	O
proved	O
that	O
perceptrons	B-Algorithm
could	O
not	O
be	O
trained	O
to	O
recognise	O
many	O
classes	O
of	O
patterns	O
.	O
</s>
<s>
This	O
caused	O
the	O
field	O
of	O
neural	B-Architecture
network	I-Architecture
research	O
to	O
stagnate	O
for	O
many	O
years	O
,	O
before	O
it	O
was	O
recognised	O
that	O
a	O
feedforward	B-Algorithm
neural	I-Algorithm
network	I-Algorithm
with	O
two	O
or	O
more	O
layers	O
(	O
also	O
called	O
a	O
multilayer	B-Algorithm
perceptron	I-Algorithm
)	O
had	O
greater	O
processing	O
power	O
than	O
perceptrons	B-Algorithm
with	O
one	O
layer	O
(	O
also	O
called	O
a	O
single-layer	O
perceptron	B-Algorithm
)	O
.	O
</s>
<s>
Single-layer	O
perceptrons	B-Algorithm
are	O
only	O
capable	O
of	O
learning	O
linearly	O
separable	O
patterns	O
.	O
</s>
<s>
A	O
second	O
layer	O
of	O
perceptrons	B-Algorithm
,	O
or	O
even	O
linear	O
nodes	O
,	O
are	O
sufficient	O
to	O
solve	O
a	O
lot	O
of	O
otherwise	O
non-separable	O
problems	O
.	O
</s>
<s>
In	O
1969	O
,	O
a	O
famous	O
book	O
entitled	O
Perceptrons	B-Algorithm
by	O
Marvin	O
Minsky	O
and	O
Seymour	O
Papert	O
showed	O
that	O
it	O
was	O
impossible	O
for	O
these	O
classes	O
of	O
network	O
to	O
learn	O
an	O
XOR	B-Application
function	O
.	O
</s>
<s>
It	O
is	O
often	O
believed	O
(	O
incorrectly	O
)	O
that	O
they	O
also	O
conjectured	O
that	O
a	O
similar	O
result	O
would	O
hold	O
for	O
a	O
multi-layer	B-Algorithm
perceptron	I-Algorithm
network	O
.	O
</s>
<s>
However	O
,	O
this	O
is	O
not	O
true	O
,	O
as	O
both	O
Minsky	O
and	O
Papert	O
already	O
knew	O
that	O
multi-layer	B-Algorithm
perceptrons	I-Algorithm
were	O
capable	O
of	O
producing	O
an	O
XOR	B-Application
function	O
.	O
</s>
<s>
(	O
See	O
the	O
page	O
on	O
Perceptrons	B-Algorithm
(	O
book	O
)	O
for	O
more	O
information	O
.	O
)	O
</s>
<s>
Nevertheless	O
,	O
the	O
often-miscited	O
Minsky/Papert	O
text	O
caused	O
a	O
significant	O
decline	O
in	O
interest	O
and	O
funding	O
of	O
neural	B-Architecture
network	I-Architecture
research	O
.	O
</s>
<s>
It	O
took	O
ten	O
more	O
years	O
until	O
neural	B-Architecture
network	I-Architecture
research	O
experienced	O
a	O
resurgence	O
in	O
the	O
1980s	O
.	O
</s>
<s>
This	O
text	O
was	O
reprinted	O
in	O
1987	O
as	O
"	O
Perceptrons	B-Algorithm
-	O
Expanded	O
Edition	O
"	O
where	O
some	O
errors	O
in	O
the	O
original	O
text	O
are	O
shown	O
and	O
corrected	O
.	O
</s>
<s>
A	O
2022	O
article	O
states	O
that	O
the	O
Mark	O
1	O
Perceptron	B-Algorithm
was	O
"	O
part	O
of	O
a	O
previously	O
secret	O
four-year	O
NPIC	O
[	O
the	O
US	O
 '	O
National	O
Photographic	O
Interpretation	O
Center ]	O
effort	O
from	O
1963	O
through	O
1966	O
to	O
develop	O
this	O
algorithm	O
into	O
a	O
useful	O
tool	O
for	O
photo-interpreters	O
"	O
.	O
</s>
<s>
The	O
kernel	B-General_Concept
perceptron	I-General_Concept
algorithm	O
was	O
already	O
introduced	O
in	O
1964	O
by	O
Aizerman	O
et	O
al	O
.	O
</s>
<s>
Margin	O
bounds	O
guarantees	O
were	O
given	O
for	O
the	O
Perceptron	B-Algorithm
algorithm	I-Algorithm
in	O
the	O
general	O
non-separable	O
case	O
first	O
by	O
Freund	O
and	O
Schapire	O
(	O
1998	O
)	O
,	O
and	O
more	O
recently	O
by	O
Mohri	O
and	O
Rostamizadeh	O
(	O
2013	O
)	O
who	O
extend	O
previous	O
results	O
and	O
give	O
new	O
L1	O
bounds	O
.	O
</s>
<s>
The	O
perceptron	B-Algorithm
is	O
a	O
simplified	O
model	O
of	O
a	O
biological	O
neuron	O
.	O
</s>
<s>
While	O
the	O
complexity	O
of	O
biological	O
neuron	O
models	O
is	O
often	O
required	O
to	O
fully	O
understand	O
neural	O
behavior	O
,	O
research	O
suggests	O
a	O
perceptron-like	O
linear	O
model	O
can	O
produce	O
some	O
behavior	O
seen	O
in	O
real	O
neurons	O
.	O
</s>
<s>
In	O
the	O
modern	O
sense	O
,	O
the	O
perceptron	B-Algorithm
is	O
an	O
algorithm	O
for	O
learning	O
a	O
binary	B-General_Concept
classifier	I-General_Concept
called	O
a	O
threshold	O
function	O
:	O
a	O
function	O
that	O
maps	O
its	O
input	O
(	O
a	O
real-valued	O
vector	O
)	O
to	O
an	O
output	O
value	O
(	O
a	O
single	O
binary	O
value	O
)	O
:	O
</s>
<s>
where	O
is	O
a	O
vector	O
of	O
real-valued	O
weights	B-General_Concept
,	O
is	O
the	O
dot	O
product	O
,	O
where	O
is	O
the	O
number	O
of	O
inputs	O
to	O
the	O
perceptron	B-Algorithm
,	O
and	O
is	O
the	O
bias	O
.	O
</s>
<s>
The	O
bias	O
shifts	O
the	O
decision	B-General_Concept
boundary	I-General_Concept
away	O
from	O
the	O
origin	O
and	O
does	O
not	O
depend	O
on	O
any	O
input	O
value	O
.	O
</s>
<s>
The	O
value	O
of	O
(	O
0	O
or	O
1	O
)	O
is	O
used	O
to	O
classify	O
as	O
either	O
a	O
positive	O
or	O
a	O
negative	O
instance	O
,	O
in	O
the	O
case	O
of	O
a	O
binary	B-General_Concept
classification	I-General_Concept
problem	O
.	O
</s>
<s>
Spatially	O
,	O
the	O
bias	O
alters	O
the	O
position	O
(	O
though	O
not	O
the	O
orientation	O
)	O
of	O
the	O
decision	B-General_Concept
boundary	I-General_Concept
.	O
</s>
<s>
The	O
perceptron	B-Algorithm
learning	I-Algorithm
algorithm	I-Algorithm
does	O
not	O
terminate	O
if	O
the	O
learning	O
set	O
is	O
not	O
linearly	O
separable	O
.	O
</s>
<s>
The	O
most	O
famous	O
example	O
of	O
the	O
perceptron	B-Algorithm
's	O
inability	O
to	O
solve	O
problems	O
with	O
linearly	O
nonseparable	O
vectors	O
is	O
the	O
Boolean	O
exclusive-or	O
problem	O
.	O
</s>
<s>
The	O
solution	O
spaces	O
of	O
decision	B-General_Concept
boundaries	I-General_Concept
for	O
all	O
binary	O
functions	O
and	O
learning	O
behaviors	O
are	O
studied	O
in	O
the	O
reference	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
neural	B-Architecture
networks	I-Architecture
,	O
a	O
perceptron	B-Algorithm
is	O
an	O
artificial	B-Algorithm
neuron	I-Algorithm
using	O
the	O
Heaviside	O
step	O
function	O
as	O
the	O
activation	O
function	O
.	O
</s>
<s>
The	O
perceptron	B-Algorithm
algorithm	I-Algorithm
is	O
also	O
termed	O
the	O
single-layer	O
perceptron	B-Algorithm
,	O
to	O
distinguish	O
it	O
from	O
a	O
multilayer	B-Algorithm
perceptron	I-Algorithm
,	O
which	O
is	O
a	O
misnomer	O
for	O
a	O
more	O
complicated	O
neural	B-Architecture
network	I-Architecture
.	O
</s>
<s>
As	O
a	O
linear	B-General_Concept
classifier	I-General_Concept
,	O
the	O
single-layer	O
perceptron	B-Algorithm
is	O
the	O
simplest	O
feedforward	B-Algorithm
neural	I-Algorithm
network	I-Algorithm
.	O
</s>
<s>
Below	O
is	O
an	O
example	O
of	O
a	O
learning	O
algorithm	O
for	O
a	O
single-layer	O
perceptron	B-Algorithm
.	O
</s>
<s>
For	O
multilayer	B-Algorithm
perceptrons	I-Algorithm
,	O
where	O
a	O
hidden	O
layer	O
exists	O
,	O
more	O
sophisticated	O
algorithms	O
such	O
as	O
backpropagation	B-Algorithm
must	O
be	O
used	O
.	O
</s>
<s>
If	O
the	O
activation	O
function	O
or	O
the	O
underlying	O
process	O
being	O
modeled	O
by	O
the	O
perceptron	B-Algorithm
is	O
nonlinear	O
,	O
alternative	O
learning	O
algorithms	O
such	O
as	O
the	O
delta	B-Algorithm
rule	I-Algorithm
can	O
be	O
used	O
as	O
long	O
as	O
the	O
activation	O
function	O
is	O
differentiable	O
.	O
</s>
<s>
Nonetheless	O
,	O
the	O
learning	O
algorithm	O
described	O
in	O
the	O
steps	O
below	O
will	O
often	O
work	O
,	O
even	O
for	O
multilayer	B-Algorithm
perceptrons	I-Algorithm
with	O
nonlinear	O
activation	O
functions	O
.	O
</s>
<s>
When	O
multiple	O
perceptrons	B-Algorithm
are	O
combined	O
in	O
an	O
artificial	O
neural	B-Architecture
network	I-Architecture
,	O
each	O
output	O
neuron	O
operates	O
independently	O
of	O
all	O
the	O
others	O
;	O
thus	O
,	O
learning	O
each	O
output	O
can	O
be	O
considered	O
in	O
isolation	O
.	O
</s>
<s>
r	O
is	O
the	O
learning	B-General_Concept
rate	I-General_Concept
of	O
the	O
perceptron	B-Algorithm
.	O
</s>
<s>
Learning	B-General_Concept
rate	I-General_Concept
is	O
between	O
0	O
and	O
1	O
.	O
</s>
<s>
denotes	O
the	O
output	O
from	O
the	O
perceptron	B-Algorithm
for	O
an	O
input	O
vector	O
.	O
</s>
<s>
is	O
the	O
desired	O
output	O
value	O
of	O
the	O
perceptron	B-Algorithm
for	O
that	O
input	O
.	O
</s>
<s>
To	O
represent	O
the	O
weights	B-General_Concept
:	O
</s>
<s>
The	O
algorithm	O
updates	O
the	O
weights	B-General_Concept
after	O
steps	O
2a	O
and	O
2b	O
.	O
</s>
<s>
These	O
weights	B-General_Concept
are	O
immediately	O
applied	O
to	O
a	O
pair	O
in	O
the	O
training	O
set	O
,	O
and	O
subsequently	O
updated	O
,	O
rather	O
than	O
waiting	O
until	O
all	O
pairs	O
in	O
the	O
training	O
set	O
have	O
undergone	O
these	O
steps	O
.	O
</s>
<s>
The	O
perceptron	B-Algorithm
is	O
a	O
linear	B-General_Concept
classifier	I-General_Concept
,	O
therefore	O
it	O
will	O
never	O
get	O
to	O
the	O
state	O
with	O
all	O
the	O
input	O
vectors	O
classified	O
correctly	O
if	O
the	O
training	O
set	O
is	O
not	O
linearly	O
separable	O
,	O
i.e.	O
</s>
<s>
If	O
the	O
training	O
set	O
is	O
linearly	O
separable	O
,	O
then	O
the	O
perceptron	B-Algorithm
is	O
guaranteed	O
to	O
converge	O
.	O
</s>
<s>
Furthermore	O
,	O
there	O
is	O
an	O
upper	O
bound	O
on	O
the	O
number	O
of	O
times	O
the	O
perceptron	B-Algorithm
will	O
adjust	O
its	O
weights	B-General_Concept
during	O
the	O
training	O
.	O
</s>
<s>
there	O
exists	O
a	O
weight	O
vector	O
,	O
and	O
a	O
bias	O
term	O
such	O
that	O
for	O
all	O
with	O
and	O
for	O
all	O
with	O
,	O
where	O
is	O
the	O
desired	O
output	O
value	O
of	O
the	O
perceptron	B-Algorithm
for	O
input	O
.	O
</s>
<s>
Novikoff	O
(	O
1962	O
)	O
proved	O
that	O
in	O
this	O
case	O
the	O
perceptron	B-Algorithm
algorithm	I-Algorithm
converges	O
after	O
making	O
updates	O
.	O
</s>
<s>
While	O
the	O
perceptron	B-Algorithm
algorithm	I-Algorithm
is	O
guaranteed	O
to	O
converge	O
on	O
some	O
solution	O
in	O
the	O
case	O
of	O
a	O
linearly	O
separable	O
training	O
set	O
,	O
it	O
may	O
still	O
pick	O
any	O
solution	O
and	O
problems	O
may	O
admit	O
many	O
solutions	O
of	O
varying	O
quality	O
.	O
</s>
<s>
The	O
perceptron	B-Algorithm
of	O
optimal	O
stability	O
,	O
nowadays	O
better	O
known	O
as	O
the	O
linear	O
support-vector	B-Algorithm
machine	I-Algorithm
,	O
was	O
designed	O
to	O
solve	O
this	O
problem	O
(	O
Krauth	O
and	O
Mezard	O
,	O
1987	O
)	O
.	O
</s>
<s>
The	O
pocket	O
algorithm	O
with	O
ratchet	O
(	O
Gallant	O
,	O
1990	O
)	O
solves	O
the	O
stability	O
problem	O
of	O
perceptron	B-Algorithm
learning	O
by	O
keeping	O
the	O
best	O
solution	O
seen	O
so	O
far	O
"	O
in	O
its	O
pocket	O
"	O
.	O
</s>
<s>
It	O
can	O
be	O
used	O
also	O
for	O
non-separable	O
data	O
sets	O
,	O
where	O
the	O
aim	O
is	O
to	O
find	O
a	O
perceptron	B-Algorithm
with	O
a	O
small	O
number	O
of	O
misclassifications	O
.	O
</s>
<s>
The	O
Maxover	O
algorithm	O
(	O
Wendemuth	O
,	O
1995	O
)	O
is	O
"	B-Application
robust	I-Application
"	I-Application
in	O
the	O
sense	O
that	O
it	O
will	O
converge	O
regardless	O
of	O
(	O
prior	O
)	O
knowledge	O
of	O
linear	O
separability	O
of	O
the	O
data	O
set	O
.	O
</s>
<s>
The	O
Voted	O
Perceptron	B-Algorithm
(	O
Freund	O
and	O
Schapire	O
,	O
1999	O
)	O
,	O
is	O
a	O
variant	O
using	O
multiple	O
weighted	O
perceptrons	B-Algorithm
.	O
</s>
<s>
The	O
algorithm	O
starts	O
a	O
new	O
perceptron	B-Algorithm
every	O
time	O
an	O
example	O
is	O
wrongly	O
classified	O
,	O
initializing	O
the	O
weights	B-General_Concept
vector	O
with	O
the	O
final	O
weights	B-General_Concept
of	O
the	O
last	O
perceptron	B-Algorithm
.	O
</s>
<s>
Each	O
perceptron	B-Algorithm
will	O
also	O
be	O
given	O
another	O
weight	O
corresponding	O
to	O
how	O
many	O
examples	O
do	O
they	O
correctly	O
classify	O
before	O
wrongly	O
classifying	O
one	O
,	O
and	O
at	O
the	O
end	O
the	O
output	O
will	O
be	O
a	O
weighted	O
vote	O
on	O
all	O
perceptrons	B-Algorithm
.	O
</s>
<s>
In	O
separable	O
problems	O
,	O
perceptron	B-Algorithm
training	O
can	O
also	O
aim	O
at	O
finding	O
the	O
largest	O
separating	O
margin	O
between	O
the	O
classes	O
.	O
</s>
<s>
The	O
so-called	O
perceptron	B-Algorithm
of	O
optimal	O
stability	O
can	O
be	O
determined	O
by	O
means	O
of	O
iterative	O
training	O
and	O
optimization	O
schemes	O
,	O
such	O
as	O
the	O
Min-Over	O
algorithm	O
(	O
Krauth	O
and	O
Mezard	O
,	O
1987	O
)	O
or	O
the	O
AdaTron	O
(	O
Anlauf	O
and	O
Biehl	O
,	O
1989	O
)	O
)	O
.	O
</s>
<s>
The	O
perceptron	B-Algorithm
of	O
optimal	O
stability	O
,	O
together	O
with	O
the	O
kernel	O
trick	O
,	O
are	O
the	O
conceptual	O
foundations	O
of	O
the	O
support-vector	B-Algorithm
machine	I-Algorithm
.	O
</s>
<s>
The	O
-perceptron	O
further	O
used	O
a	O
pre-processing	O
layer	O
of	O
fixed	O
random	O
weights	B-General_Concept
,	O
with	O
thresholded	O
output	O
units	O
.	O
</s>
<s>
This	O
enabled	O
the	O
perceptron	B-Algorithm
to	O
classify	O
analogue	O
patterns	O
,	O
by	O
projecting	O
them	O
into	O
a	O
binary	O
space	O
.	O
</s>
<s>
Indeed	O
,	O
if	O
we	O
had	O
the	O
prior	O
constraint	O
that	O
the	O
data	O
come	O
from	O
equi-variant	O
Gaussian	O
distributions	O
,	O
the	O
linear	O
separation	O
in	O
the	O
input	O
space	O
is	O
optimal	O
,	O
and	O
the	O
nonlinear	O
solution	O
is	O
overfitted	B-Error_Name
.	O
</s>
<s>
Other	O
linear	B-General_Concept
classification	I-General_Concept
algorithms	O
include	O
Winnow	B-Algorithm
,	O
support-vector	B-Algorithm
machine	I-Algorithm
,	O
and	O
logistic	O
regression	O
.	O
</s>
<s>
Like	O
most	O
other	O
techniques	O
for	O
training	O
linear	B-General_Concept
classifiers	I-General_Concept
,	O
the	O
perceptron	B-Algorithm
generalizes	O
naturally	O
to	O
multiclass	B-General_Concept
classification	I-General_Concept
.	O
</s>
<s>
A	O
feature	O
representation	O
function	O
maps	O
each	O
possible	O
input/output	O
pair	O
to	O
a	O
finite-dimensional	O
real-valued	O
feature	B-Algorithm
vector	I-Algorithm
.	O
</s>
<s>
As	O
before	O
,	O
the	O
feature	B-Algorithm
vector	I-Algorithm
is	O
multiplied	O
by	O
a	O
weight	O
vector	O
,	O
but	O
now	O
the	O
resulting	O
score	O
is	O
used	O
to	O
choose	O
among	O
many	O
possible	O
outputs	O
:	O
</s>
<s>
Learning	O
again	O
iterates	O
over	O
the	O
examples	O
,	O
predicting	O
an	O
output	O
for	O
each	O
,	O
leaving	O
the	O
weights	B-General_Concept
unchanged	O
when	O
the	O
predicted	O
output	O
matches	O
the	O
target	O
,	O
and	O
changing	O
them	O
when	O
it	O
does	O
not	O
.	O
</s>
<s>
This	O
multiclass	O
feedback	O
formulation	O
reduces	O
to	O
the	O
original	O
perceptron	B-Algorithm
when	O
is	O
a	O
real-valued	O
vector	O
,	O
is	O
chosen	O
from	O
,	O
and	O
.	O
</s>
<s>
Since	O
2002	O
,	O
perceptron	B-Algorithm
training	O
has	O
become	O
popular	O
in	O
the	O
field	O
of	O
natural	B-Language
language	I-Language
processing	I-Language
for	O
such	O
tasks	O
as	O
part-of-speech	O
tagging	O
and	O
syntactic	B-Language
parsing	I-Language
(	O
Collins	O
,	O
2002	O
)	O
.	O
</s>
<s>
It	O
has	O
also	O
been	O
applied	O
to	O
large-scale	O
machine	O
learning	O
problems	O
in	O
a	O
distributed	B-Architecture
computing	I-Architecture
setting	O
.	O
</s>
