<s>
A	O
Boltzmann	B-Algorithm
machine	I-Algorithm
(	O
also	O
called	O
Sherrington	O
–	O
Kirkpatrick	O
model	O
with	O
external	O
field	O
or	O
stochastic	O
Ising	O
–	O
Lenz	O
–	O
Little	O
model	O
)	O
is	O
a	O
stochastic	O
spin-glass	O
model	O
with	O
an	O
external	O
field	O
,	O
i.e.	O
,	O
a	O
Sherrington	O
–	O
Kirkpatrick	O
model	O
,	O
that	O
is	O
a	O
stochastic	O
Ising	O
model	O
.	O
</s>
<s>
Boltzmann	B-Algorithm
machines	I-Algorithm
are	O
theoretically	O
intriguing	O
because	O
of	O
the	O
locality	O
and	O
Hebbian	O
nature	O
of	O
their	O
training	O
algorithm	O
(	O
being	O
trained	O
by	O
Hebb	O
's	O
rule	O
)	O
,	O
and	O
because	O
of	O
their	O
parallelism	B-Operating_System
and	O
the	O
resemblance	O
of	O
their	O
dynamics	O
to	O
simple	O
physical	O
processes	O
.	O
</s>
<s>
Boltzmann	B-Algorithm
machines	I-Algorithm
with	O
unconstrained	O
connectivity	O
have	O
not	O
been	O
proven	O
useful	O
for	O
practical	O
problems	O
in	O
machine	O
learning	O
or	O
inference	O
,	O
but	O
if	O
the	O
connectivity	O
is	O
properly	O
constrained	O
,	O
the	O
learning	O
can	O
be	O
made	O
efficient	O
enough	O
to	O
be	O
useful	O
for	O
practical	O
problems	O
.	O
</s>
<s>
A	O
Boltzmann	B-Algorithm
machine	I-Algorithm
,	O
like	O
a	O
Sherrington	O
–	O
Kirkpatrick	O
model	O
,	O
is	O
a	O
network	O
of	O
units	O
with	O
a	O
total	O
"	O
energy	O
"	O
(	O
Hamiltonian	O
)	O
defined	O
for	O
the	O
overall	O
network	O
.	O
</s>
<s>
Boltzmann	B-Algorithm
machine	I-Algorithm
weights	O
are	O
stochastic	O
.	O
</s>
<s>
The	O
global	O
energy	O
in	O
a	O
Boltzmann	B-Algorithm
machine	I-Algorithm
is	O
identical	O
in	O
form	O
to	O
that	O
of	O
Hopfield	B-Algorithm
networks	I-Algorithm
and	O
Ising	O
models	O
:	O
</s>
<s>
This	O
relation	O
is	O
the	O
source	O
of	O
the	O
logistic	O
function	O
found	O
in	O
probability	O
expressions	O
in	O
variants	O
of	O
the	O
Boltzmann	B-Algorithm
machine	I-Algorithm
.	O
</s>
<s>
This	O
process	O
is	O
called	O
simulated	B-Algorithm
annealing	I-Algorithm
.	O
</s>
<s>
The	O
units	O
in	O
the	O
Boltzmann	B-Algorithm
machine	I-Algorithm
are	O
divided	O
into	O
'	O
visible	O
 '	O
units	O
,	O
V	O
,	O
and	O
'	O
hidden	O
 '	O
units	O
,	O
H	O
.	O
The	O
visible	O
units	O
are	O
those	O
that	O
receive	O
information	O
from	O
the	O
'	O
environment	O
 '	O
,	O
i.e.	O
</s>
<s>
The	O
distribution	O
over	O
global	O
states	O
converges	O
as	O
the	O
Boltzmann	B-Algorithm
machine	I-Algorithm
reaches	O
thermal	O
equilibrium	O
.	O
</s>
<s>
A	O
gradient	B-Algorithm
descent	I-Algorithm
algorithm	O
over	O
,	O
changes	O
a	O
given	O
weight	O
,	O
by	O
subtracting	O
the	O
partial	O
derivative	O
of	O
with	O
respect	O
to	O
the	O
weight	O
.	O
</s>
<s>
Boltzmann	B-Algorithm
machine	I-Algorithm
training	O
involves	O
two	O
alternating	O
phases	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
connection	O
(	O
synapse	B-Application
,	O
biologically	O
)	O
does	O
not	O
need	O
information	O
about	O
anything	O
other	O
than	O
the	O
two	O
neurons	O
it	O
connects	O
.	O
</s>
<s>
This	O
is	O
more	O
biologically	O
realistic	O
than	O
the	O
information	O
needed	O
by	O
a	O
connection	O
in	O
many	O
other	O
neural	O
network	O
training	O
algorithms	O
,	O
such	O
as	O
backpropagation	B-Algorithm
.	O
</s>
<s>
The	O
training	O
of	O
a	O
Boltzmann	B-Algorithm
machine	I-Algorithm
does	O
not	O
use	O
the	O
EM	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
is	O
heavily	O
used	O
in	O
machine	O
learning	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
training	O
procedure	O
performs	O
gradient	B-Algorithm
ascent	I-Algorithm
on	O
the	O
log-likelihood	O
of	O
the	O
observed	O
data	O
.	O
</s>
<s>
This	O
is	O
in	O
contrast	O
to	O
the	O
EM	B-Algorithm
algorithm	I-Algorithm
,	O
where	O
the	O
posterior	O
distribution	O
of	O
the	O
hidden	O
nodes	O
must	O
be	O
calculated	O
before	O
the	O
maximization	O
of	O
the	O
expected	O
value	O
of	O
the	O
complete	O
data	O
likelihood	O
during	O
the	O
M-step	O
.	O
</s>
<s>
Theoretically	O
the	O
Boltzmann	B-Algorithm
machine	I-Algorithm
is	O
a	O
rather	O
general	O
computational	O
medium	O
.	O
</s>
<s>
Unfortunately	O
,	O
Boltzmann	B-Algorithm
machines	I-Algorithm
experience	O
a	O
serious	O
practical	O
problem	O
,	O
namely	O
that	O
it	O
seems	O
to	O
stop	O
learning	O
correctly	O
when	O
the	O
machine	O
is	O
scaled	O
up	O
to	O
anything	O
larger	O
than	O
a	O
trivial	O
size	O
.	O
</s>
<s>
Although	O
learning	O
is	O
impractical	O
in	O
general	O
Boltzmann	B-Algorithm
machines	I-Algorithm
,	O
it	O
can	O
be	O
made	O
quite	O
efficient	O
in	O
a	O
restricted	B-Algorithm
Boltzmann	I-Algorithm
machine	I-Algorithm
(	O
RBM	B-Algorithm
)	O
which	O
does	O
not	O
allow	O
intralayer	O
connections	O
between	O
hidden	O
units	O
and	O
visible	O
units	O
,	O
i.e.	O
</s>
<s>
After	O
training	O
one	O
RBM	B-Algorithm
,	O
the	O
activities	O
of	O
its	O
hidden	O
units	O
can	O
be	O
treated	O
as	O
data	O
for	O
training	O
a	O
higher-level	O
RBM	B-Algorithm
.	O
</s>
<s>
This	O
method	O
of	O
stacking	O
RBMs	B-Algorithm
makes	O
it	O
possible	O
to	O
train	O
many	O
layers	O
of	O
hidden	O
units	O
efficiently	O
and	O
is	O
one	O
of	O
the	O
most	O
common	O
deep	B-Algorithm
learning	I-Algorithm
strategies	O
.	O
</s>
<s>
An	O
extension	O
to	O
the	O
restricted	B-Algorithm
Boltzmann	I-Algorithm
machine	I-Algorithm
allows	O
using	O
real	O
valued	O
data	O
rather	O
than	O
binary	O
data	O
.	O
</s>
<s>
One	O
example	O
of	O
a	O
practical	O
RBM	B-Algorithm
application	O
is	O
in	O
speech	B-Application
recognition	I-Application
.	O
</s>
<s>
A	O
deep	O
Boltzmann	B-Algorithm
machine	I-Algorithm
(	O
DBM	O
)	O
is	O
a	O
type	O
of	O
binary	O
pairwise	O
Markov	O
random	O
field	O
(	O
undirected	O
probabilistic	O
graphical	O
model	O
)	O
with	O
multiple	O
layers	O
of	O
hidden	O
random	O
variables	O
.	O
</s>
<s>
No	O
connection	O
links	O
units	O
of	O
the	O
same	O
layer	O
(	O
like	O
RBM	B-Algorithm
)	O
.	O
</s>
<s>
In	O
a	O
DBN	O
only	O
the	O
top	O
two	O
layers	O
form	O
a	O
restricted	B-Algorithm
Boltzmann	I-Algorithm
machine	I-Algorithm
(	O
which	O
is	O
an	O
undirected	O
graphical	O
model	O
)	O
,	O
while	O
lower	O
layers	O
form	O
a	O
directed	O
generative	O
model	O
.	O
</s>
<s>
Like	O
DBNs	B-Algorithm
,	O
DBMs	O
can	O
learn	O
complex	O
and	O
abstract	O
internal	O
representations	O
of	O
the	O
input	O
in	O
tasks	O
such	O
as	O
object	O
or	O
speech	B-Application
recognition	I-Application
,	O
using	O
limited	O
,	O
labeled	O
data	O
to	O
fine-tune	O
the	O
representations	O
built	O
using	O
a	O
large	O
set	O
of	O
unlabeled	O
sensory	O
input	O
data	O
.	O
</s>
<s>
However	O
,	O
unlike	O
DBNs	B-Algorithm
and	O
deep	B-Architecture
convolutional	I-Architecture
neural	I-Architecture
networks	I-Architecture
,	O
they	O
pursue	O
the	O
inference	O
and	O
training	O
procedure	O
in	O
both	O
directions	O
,	O
bottom-up	O
and	O
top-down	O
,	O
which	O
allow	O
the	O
DBM	O
to	O
better	O
unveil	O
the	O
representations	O
of	O
the	O
input	O
structures	O
.	O
</s>
<s>
Another	O
option	O
is	O
to	O
use	O
mean-field	O
inference	O
to	O
estimate	O
data-dependent	O
expectations	O
and	O
approximate	O
the	O
expected	O
sufficient	O
statistics	O
by	O
using	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
(	O
MCMC	O
)	O
.	O
</s>
<s>
The	O
need	O
for	O
deep	B-Algorithm
learning	I-Algorithm
with	O
real-valued	O
inputs	O
,	O
as	O
in	O
Gaussian	B-Application
RBMs	B-Algorithm
,	O
led	O
to	O
the	O
spike-and-slab	O
RBM	B-Algorithm
(	O
ssRBM	O
)	O
,	O
which	O
models	O
continuous-valued	O
inputs	O
with	O
binary	O
latent	O
variables	O
.	O
</s>
<s>
Similar	O
to	O
basic	O
RBMs	B-Algorithm
and	O
its	O
variants	O
,	O
a	O
spike-and-slab	O
RBM	B-Algorithm
is	O
a	O
bipartite	O
graph	O
,	O
while	O
like	O
GRBMs	B-Algorithm
,	O
the	O
visible	O
units	O
(	O
input	O
)	O
are	O
real-valued	O
.	O
</s>
<s>
In	O
deep	B-Algorithm
learning	I-Algorithm
the	O
Boltzmann	O
distribution	O
is	O
used	O
in	O
the	O
sampling	O
distribution	O
of	O
stochastic	O
neural	O
networks	O
such	O
as	O
the	O
Boltzmann	B-Algorithm
machine	I-Algorithm
.	O
</s>
<s>
The	O
Boltzmann	B-Algorithm
machine	I-Algorithm
is	O
based	O
on	O
a	O
spin-glass	O
model	O
of	O
Sherrington-Kirkpatrick	O
'	O
s	O
stochastic	O
Ising	O
Model	O
.	O
</s>
<s>
The	O
idea	O
of	O
applying	O
the	O
Ising	O
model	O
with	O
annealed	O
Gibbs	B-Algorithm
sampling	I-Algorithm
is	O
present	O
in	O
Douglas	O
Hofstadter	O
's	O
Copycat	B-Language
project	O
.	O
</s>
<s>
The	O
explicit	O
analogy	O
drawn	O
with	O
statistical	O
mechanics	O
in	O
the	O
Boltzmann	B-Algorithm
Machine	I-Algorithm
formulation	O
led	O
to	O
the	O
use	O
of	O
terminology	O
borrowed	O
from	O
physics	O
(	O
e.g.	O
,	O
"	O
energy	O
"	O
rather	O
than	O
"	O
harmony	O
"	O
)	O
,	O
which	O
became	O
standard	O
in	O
the	O
field	O
.	O
</s>
<s>
The	O
various	O
proposals	O
to	O
use	O
simulated	B-Algorithm
annealing	I-Algorithm
for	O
inference	O
were	O
apparently	O
independent	O
.	O
</s>
<s>
Ising	O
models	O
became	O
considered	O
to	O
be	O
a	O
special	O
case	O
of	O
Markov	O
random	O
fields	O
,	O
which	O
find	O
widespread	O
application	O
in	O
linguistics	O
,	O
robotics	O
,	O
computer	B-Application
vision	I-Application
and	O
artificial	B-Application
intelligence	I-Application
.	I-Application
</s>
