<s>
Coordinate	B-Algorithm
descent	I-Algorithm
is	O
an	O
optimization	O
algorithm	O
that	O
successively	O
minimizes	O
along	O
coordinate	O
directions	O
to	O
find	O
the	O
minimum	O
of	O
a	O
function	O
.	O
</s>
<s>
A	O
line	B-Algorithm
search	I-Algorithm
along	O
the	O
coordinate	O
direction	O
can	O
be	O
performed	O
at	O
the	O
current	O
iterate	O
to	O
determine	O
the	O
appropriate	O
step	O
size	O
.	O
</s>
<s>
Coordinate	B-Algorithm
descent	I-Algorithm
is	O
applicable	O
in	O
both	O
differentiable	O
and	O
derivative-free	O
contexts	O
.	O
</s>
<s>
Coordinate	B-Algorithm
descent	I-Algorithm
is	O
based	O
on	O
the	O
idea	O
that	O
the	O
minimization	O
of	O
a	O
multivariable	O
function	O
can	O
be	O
achieved	O
by	O
minimizing	O
it	O
along	O
one	O
direction	O
at	O
a	O
time	O
,	O
i.e.	O
,	O
solving	O
univariate	O
(	O
or	O
at	O
least	O
much	O
simpler	O
)	O
optimization	O
problems	O
in	O
a	O
loop	O
.	O
</s>
<s>
In	O
the	O
simplest	O
case	O
of	O
cyclic	O
coordinate	B-Algorithm
descent	I-Algorithm
,	O
one	O
cyclically	O
iterates	O
through	O
the	O
directions	O
,	O
one	O
at	O
a	O
time	O
,	O
minimizing	O
the	O
objective	O
function	O
with	O
respect	O
to	O
each	O
coordinate	O
direction	O
at	O
a	O
time	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
this	O
sequence	O
has	O
similar	O
convergence	O
properties	O
as	O
steepest	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
No	O
improvement	O
after	O
one	O
cycle	O
of	O
line	B-Algorithm
search	I-Algorithm
along	O
coordinate	O
directions	O
implies	O
a	O
stationary	O
point	O
is	O
reached	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
continuously	O
differentiable	O
function	O
,	O
a	O
coordinate	B-Algorithm
descent	I-Algorithm
algorithm	O
can	O
be	O
sketched	B-Language
as	O
:	O
</s>
<s>
The	O
step	O
size	O
can	O
be	O
chosen	O
in	O
various	O
ways	O
,	O
e.g.	O
,	O
by	O
solving	O
for	O
the	O
exact	O
minimizer	O
of	O
(	O
i.e.	O
,	O
with	O
all	O
variables	O
but	O
fixed	O
)	O
,	O
or	O
by	O
traditional	O
line	B-Algorithm
search	I-Algorithm
criteria	O
.	O
</s>
<s>
Coordinate	B-Algorithm
descent	I-Algorithm
has	O
two	O
problems	O
.	O
</s>
<s>
The	O
following	O
picture	O
shows	O
that	O
coordinate	B-Algorithm
descent	I-Algorithm
iteration	O
may	O
get	O
stuck	O
at	O
a	O
non-stationary	O
point	O
if	O
the	O
level	O
curves	O
of	O
a	O
function	O
are	O
not	O
smooth	O
.	O
</s>
<s>
While	O
this	O
example	O
shows	O
that	O
coordinate	B-Algorithm
descent	I-Algorithm
is	O
not	O
necessarily	O
convergent	O
to	O
the	O
optimum	O
,	O
it	O
is	O
possible	O
to	O
show	O
formal	O
convergence	O
under	O
reasonable	O
conditions	O
.	O
</s>
<s>
Since	O
the	O
nature	O
of	O
coordinate	B-Algorithm
descent	I-Algorithm
is	O
to	O
cycle	O
through	O
the	O
directions	O
and	O
minimize	O
the	O
objective	O
function	O
with	O
respect	O
to	O
each	O
coordinate	O
direction	O
,	O
coordinate	B-Algorithm
descent	I-Algorithm
is	O
not	O
an	O
obvious	O
candidate	O
for	O
massive	O
parallelism	O
.	O
</s>
<s>
Recent	O
research	O
works	O
have	O
shown	O
that	O
massive	O
parallelism	O
is	O
applicable	O
to	O
coordinate	B-Algorithm
descent	I-Algorithm
by	O
relaxing	O
the	O
change	O
of	O
the	O
objective	O
function	O
with	O
respect	O
to	O
each	O
coordinate	O
direction	O
.	O
</s>
<s>
Coordinate	B-Algorithm
descent	I-Algorithm
algorithms	O
are	O
popular	O
with	O
practitioners	O
owing	O
to	O
their	O
simplicity	O
,	O
but	O
the	O
same	O
property	O
has	O
led	O
optimization	O
researchers	O
to	O
largely	O
ignore	O
them	O
in	O
favor	O
of	O
more	O
interesting	O
(	O
complicated	O
)	O
methods	O
.	O
</s>
<s>
An	O
early	O
application	O
of	O
coordinate	B-Algorithm
descent	I-Algorithm
optimization	O
was	O
in	O
the	O
area	O
of	O
computed	O
tomography	O
where	O
it	O
has	O
been	O
found	O
to	O
have	O
rapid	O
convergence	O
and	O
was	O
subsequently	O
used	O
for	O
clinical	O
multi-slice	O
helical	O
scan	O
CT	O
reconstruction	O
.	O
</s>
<s>
A	O
cyclic	O
coordinate	B-Algorithm
descent	I-Algorithm
algorithm	O
(	O
CCD	O
)	O
has	O
been	O
applied	O
in	O
protein	O
structure	O
prediction	O
.	O
</s>
<s>
Moreover	O
,	O
there	O
has	O
been	O
increased	O
interest	O
in	O
the	O
use	O
of	O
coordinate	B-Algorithm
descent	I-Algorithm
with	O
the	O
advent	O
of	O
large-scale	O
problems	O
in	O
machine	O
learning	O
,	O
where	O
coordinate	B-Algorithm
descent	I-Algorithm
has	O
been	O
shown	O
competitive	O
to	O
other	O
methods	O
when	O
applied	O
to	O
such	O
problems	O
as	O
training	O
linear	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
(	O
see	O
LIBLINEAR	B-Language
)	O
and	O
non-negative	O
matrix	O
factorization	O
.	O
</s>
