<s>
Covariance	B-Algorithm
matrix	I-Algorithm
adaptation	I-Algorithm
evolution	I-Algorithm
strategy	I-Algorithm
(	O
CMA-ES	B-Algorithm
)	O
is	O
a	O
particular	O
kind	O
of	O
strategy	O
for	O
numerical	O
optimization	O
.	O
</s>
<s>
Evolution	B-Algorithm
strategies	I-Algorithm
(	O
ES	O
)	O
are	O
stochastic	O
,	O
derivative-free	B-Algorithm
methods	I-Algorithm
for	O
numerical	O
optimization	O
of	O
non-linear	O
or	O
non-convex	O
continuous	O
optimization	O
problems	O
.	O
</s>
<s>
They	O
belong	O
to	O
the	O
class	O
of	O
evolutionary	B-Algorithm
algorithms	I-Algorithm
and	O
evolutionary	O
computation	O
.	O
</s>
<s>
An	O
evolutionary	B-Algorithm
algorithm	I-Algorithm
is	O
broadly	O
based	O
on	O
the	O
principle	O
of	O
biological	O
evolution	O
,	O
namely	O
the	O
repeated	O
interplay	O
of	O
variation	O
(	O
via	O
recombination	O
and	O
mutation	O
)	O
and	O
selection	O
:	O
in	O
each	O
generation	O
(	O
iteration	O
)	O
new	O
individuals	O
(	O
candidate	O
solutions	O
,	O
denoted	O
as	O
)	O
are	O
generated	O
by	O
variation	O
,	O
usually	O
in	O
a	O
stochastic	O
way	O
,	O
of	O
the	O
current	O
parental	O
individuals	O
.	O
</s>
<s>
In	O
an	O
evolution	B-Algorithm
strategy	I-Algorithm
,	O
new	O
candidate	O
solutions	O
are	O
sampled	O
according	O
to	O
a	O
multivariate	O
normal	O
distribution	O
in	O
.	O
</s>
<s>
The	O
covariance	B-Algorithm
matrix	I-Algorithm
adaptation	I-Algorithm
(	O
CMA	O
)	O
is	O
a	O
method	O
to	O
update	O
the	O
covariance	O
matrix	O
of	O
this	O
distribution	O
.	O
</s>
<s>
This	O
is	O
particularly	O
useful	O
if	O
the	O
function	O
is	O
ill-conditioned	B-Algorithm
.	O
</s>
<s>
Adaptation	O
of	O
the	O
covariance	O
matrix	O
amounts	O
to	O
learning	O
a	O
second	O
order	O
model	O
of	O
the	O
underlying	O
objective	O
function	O
similar	O
to	O
the	O
approximation	O
of	O
the	O
inverse	O
Hessian	O
matrix	O
in	O
the	O
quasi-Newton	B-Algorithm
method	I-Algorithm
in	O
classical	O
optimization	O
.	O
</s>
<s>
Two	O
main	O
principles	O
for	O
the	O
adaptation	O
of	O
parameters	O
of	O
the	O
search	O
distribution	O
are	O
exploited	O
in	O
the	O
CMA-ES	B-Algorithm
algorithm	O
.	O
</s>
<s>
Also	O
,	O
in	O
consequence	O
,	O
the	O
CMA	O
conducts	O
an	O
iterated	O
principal	B-Application
components	I-Application
analysis	I-Application
of	O
successful	O
search	O
steps	O
while	O
retaining	O
all	O
principal	O
axes	O
.	O
</s>
<s>
Estimation	O
of	O
distribution	O
algorithms	O
and	O
the	O
Cross-Entropy	B-Algorithm
Method	I-Algorithm
are	O
based	O
on	O
very	O
similar	O
ideas	O
,	O
but	O
estimate	O
(	O
non-incrementally	O
)	O
the	O
covariance	O
matrix	O
by	O
maximizing	O
the	O
likelihood	O
of	O
successful	O
solution	O
points	O
instead	O
of	O
successful	O
search	O
steps	O
.	O
</s>
<s>
One	O
path	O
is	O
used	O
for	O
the	O
covariance	B-Algorithm
matrix	I-Algorithm
adaptation	I-Algorithm
procedure	O
in	O
place	O
of	O
single	O
successful	O
search	O
steps	O
and	O
facilitates	O
a	O
possibly	O
much	O
faster	O
variance	O
increase	O
of	O
favorable	O
directions	O
.	O
</s>
<s>
In	O
the	O
following	O
the	O
most	O
commonly	O
used	O
(μ/	O
μw	O
,	O
λ	O
)	O
-CMA-ES	O
is	O
outlined	O
,	O
where	O
in	O
each	O
iteration	O
step	O
a	O
weighted	O
combination	O
of	O
the	O
μ	O
best	O
out	O
of	O
λ	O
new	O
candidate	O
solutions	O
is	O
used	O
to	O
update	O
the	O
distribution	O
parameters	O
.	O
</s>
<s>
A	O
pseudocode	B-Language
of	O
the	O
algorithm	O
looks	O
as	O
follows	O
.	O
</s>
<s>
More	O
considerations	O
on	O
the	O
update	O
equations	O
of	O
CMA-ES	B-Algorithm
are	O
made	O
in	O
the	O
following	O
.	O
</s>
<s>
The	O
CMA-ES	B-Algorithm
implements	O
a	O
stochastic	O
variable-metric	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
For	O
selection	O
ratio	O
(	O
and	O
hence	O
population	O
size	O
)	O
,	O
the	O
selected	O
solutions	O
yield	O
an	O
empirical	O
covariance	O
matrix	O
reflective	O
of	O
the	O
inverse-Hessian	O
even	O
in	O
evolution	B-Algorithm
strategies	I-Algorithm
without	O
adaptation	O
of	O
the	O
covariance	O
matrix	O
.	O
</s>
<s>
The	O
update	O
equations	O
for	O
mean	O
and	O
covariance	O
matrix	O
maximize	O
a	O
likelihood	O
while	O
resembling	O
an	O
expectation-maximization	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
See	O
estimation	B-General_Concept
of	I-General_Concept
covariance	I-General_Concept
matrices	I-General_Concept
for	O
details	O
on	O
the	O
derivation	O
.	O
</s>
<s>
without	O
step-size	O
control	O
and	O
rank-one	O
update	O
,	O
CMA-ES	B-Algorithm
can	O
thus	O
be	O
viewed	O
as	O
an	O
instantiation	O
of	O
Natural	B-Algorithm
Evolution	I-Algorithm
Strategies	I-Algorithm
(	O
NES	O
)	O
.	O
</s>
<s>
finally	O
found	O
a	O
rigorous	O
derivation	O
for	O
the	O
weights	O
,	O
,	O
as	O
they	O
are	O
defined	O
in	O
the	O
CMA-ES	B-Algorithm
.	O
</s>
<s>
The	O
weights	O
are	O
an	O
asymptotically	O
consistent	O
estimator	O
of	O
the	O
CDF	O
of	O
at	O
the	O
points	O
of	O
the	O
th	O
order	B-General_Concept
statistic	I-General_Concept
,	O
as	O
defined	O
above	O
,	O
where	O
,	O
composed	O
with	O
a	O
fixed	O
monotonically	O
decreasing	O
transformation	O
,	O
that	O
is	O
,	O
</s>
<s>
That	O
means	O
,	O
setting	O
,	O
the	O
CMA-ES	B-Algorithm
updates	O
descend	O
in	O
direction	O
of	O
the	O
approximation	O
of	O
the	O
natural	O
gradient	O
while	O
using	O
different	O
step-sizes	O
(	O
learning	O
rates	O
1	O
and	O
)	O
for	O
the	O
orthogonal	O
parameters	O
and	O
respectively	O
.	O
</s>
<s>
The	O
most	O
recent	O
version	O
of	O
CMA-ES	B-Algorithm
also	O
use	O
a	O
different	O
function	O
for	O
and	O
with	O
negative	O
values	O
only	O
for	O
the	O
latter	O
(	O
so-called	O
active	O
CMA	O
)	O
.	O
</s>
<s>
It	O
is	O
comparatively	O
easy	O
to	O
see	O
that	O
the	O
update	O
equations	O
of	O
CMA-ES	B-Algorithm
satisfy	O
some	O
stationarity	O
conditions	O
,	O
in	O
that	O
they	O
are	O
essentially	O
unbiased	O
.	O
</s>
<s>
The	O
following	O
invariance	O
properties	O
have	O
been	O
established	O
for	O
CMA-ES	B-Algorithm
.	O
</s>
<s>
Invariance	O
under	O
rotation	O
of	O
the	O
search	O
space	O
in	O
that	O
for	O
any	O
and	O
any	O
the	O
behavior	O
on	O
is	O
independent	O
of	O
the	O
orthogonal	B-Algorithm
matrix	I-Algorithm
,	O
given	O
.	O
</s>
<s>
More	O
general	O
,	O
the	O
algorithm	O
is	O
also	O
invariant	O
under	O
general	O
linear	B-Architecture
transformations	I-Architecture
when	O
additionally	O
the	O
initial	O
covariance	O
matrix	O
is	O
chosen	O
as	O
.	O
</s>
<s>
A	O
prominent	O
example	O
with	O
the	O
same	O
invariance	O
properties	O
is	O
the	O
Nelder	B-Algorithm
–	I-Algorithm
Mead	I-Algorithm
method	I-Algorithm
,	O
where	O
the	O
initial	O
simplex	O
must	O
be	O
chosen	O
respectively	O
.	O
</s>
<s>
Conceptual	O
considerations	O
like	O
the	O
scale-invariance	O
property	O
of	O
the	O
algorithm	O
,	O
the	O
analysis	O
of	O
simpler	O
evolution	B-Algorithm
strategies	I-Algorithm
,	O
and	O
overwhelming	O
empirical	O
evidence	O
suggest	O
that	O
the	O
algorithm	O
converges	O
on	O
a	O
large	O
class	O
of	O
functions	O
fast	O
to	O
the	O
global	O
optimum	O
,	O
denoted	O
as	O
.	O
</s>
<s>
Empirically	O
,	O
the	O
fastest	O
possible	O
convergence	O
rate	O
in	O
for	O
rank-based	O
direct	O
search	O
methods	O
can	O
often	O
be	O
observed	O
(	O
depending	O
on	O
the	O
context	O
denoted	O
as	O
linear	B-Architecture
or	O
log-linear	O
or	O
exponential	O
convergence	O
)	O
.	O
</s>
<s>
The	O
actual	O
linear	B-Architecture
dependencies	O
in	O
and	O
are	O
remarkable	O
and	O
they	O
are	O
in	O
both	O
cases	O
the	O
best	O
one	O
can	O
hope	O
for	O
in	O
this	O
kind	O
of	O
algorithm	O
.	O
</s>
<s>
The	O
covariance	O
matrix	O
defines	O
a	O
bijective	B-Algorithm
transformation	O
(	O
encoding	O
)	O
for	O
all	O
solution	O
vectors	O
into	O
a	O
space	O
,	O
where	O
the	O
sampling	O
takes	O
place	O
with	O
identity	O
covariance	O
matrix	O
.	O
</s>
<s>
Because	O
the	O
update	O
equations	O
in	O
the	O
CMA-ES	B-Algorithm
are	O
invariant	O
under	O
linear	B-Architecture
coordinate	O
system	O
transformations	O
,	O
the	O
CMA-ES	B-Algorithm
can	O
be	O
re-written	O
as	O
an	O
adaptive	O
encoding	O
procedure	O
applied	O
to	O
a	O
simple	O
evolution	B-Algorithm
strategy	I-Algorithm
with	O
identity	O
covariance	O
matrix	O
.	O
</s>
<s>
This	O
adaptive	O
encoding	O
procedure	O
is	O
not	O
confined	O
to	O
algorithms	O
that	O
sample	O
from	O
a	O
multivariate	O
normal	O
distribution	O
(	O
like	O
evolution	B-Algorithm
strategies	I-Algorithm
)	O
,	O
but	O
can	O
in	O
principle	O
be	O
applied	O
to	O
any	O
iterative	O
search	O
method	O
.	O
</s>
<s>
In	O
contrast	O
to	O
most	O
other	O
evolutionary	B-Algorithm
algorithms	I-Algorithm
,	O
the	O
CMA-ES	B-Algorithm
is	O
,	O
from	O
the	O
user	O
's	O
perspective	O
,	O
quasi-parameter-free	O
.	O
</s>
<s>
The	O
CMA-ES	B-Algorithm
has	O
been	O
empirically	O
successful	O
in	O
hundreds	O
of	O
applications	O
and	O
is	O
considered	O
to	O
be	O
useful	O
in	O
particular	O
on	O
non-convex	O
,	O
non-separable	O
,	O
ill-conditioned	B-Algorithm
,	O
multi-modal	O
or	O
noisy	O
objective	O
functions	O
.	O
</s>
<s>
Assuming	O
a	O
black-box	O
optimization	O
scenario	O
,	O
where	O
gradients	O
are	O
not	O
available	O
(	O
or	O
not	O
useful	O
)	O
and	O
function	O
evaluations	O
are	O
the	O
only	O
considered	O
cost	O
of	O
search	O
,	O
the	O
CMA-ES	B-Algorithm
method	O
is	O
likely	O
to	O
be	O
outperformed	O
by	O
other	O
methods	O
in	O
the	O
following	O
conditions	O
:	O
</s>
<s>
on	O
low-dimensional	O
functions	O
,	O
say	O
,	O
for	O
example	O
by	O
the	O
downhill	B-Algorithm
simplex	I-Algorithm
method	I-Algorithm
or	O
surrogate-based	O
methods	O
(	O
like	O
kriging	O
with	O
expected	O
improvement	O
)	O
;	O
</s>
<s>
on	O
separable	O
functions	O
without	O
or	O
with	O
only	O
negligible	O
dependencies	O
between	O
the	O
design	O
variables	O
in	O
particular	O
in	O
the	O
case	O
of	O
multi-modality	O
or	O
large	O
dimension	O
,	O
for	O
example	O
by	O
differential	B-Algorithm
evolution	I-Algorithm
;	O
</s>
<s>
on	O
(	O
nearly	O
)	O
convex-quadratic	O
functions	O
with	O
low	O
or	O
moderate	O
condition	B-Algorithm
number	I-Algorithm
of	O
the	O
Hessian	O
matrix	O
,	O
where	O
BFGS	B-Algorithm
or	O
NEWUOA	O
or	O
SLSQP	O
are	O
typically	O
at	O
least	O
ten	O
times	O
faster	O
;	O
</s>
<s>
on	O
functions	O
that	O
can	O
already	O
be	O
solved	O
with	O
a	O
comparatively	O
small	O
number	O
of	O
function	O
evaluations	O
,	O
say	O
no	O
more	O
than	O
,	O
where	O
CMA-ES	B-Algorithm
is	O
often	O
slower	O
than	O
,	O
for	O
example	O
,	O
NEWUOA	O
or	O
Multilevel	B-Algorithm
Coordinate	I-Algorithm
Search	I-Algorithm
(	O
MCS	O
)	O
.	O
</s>
<s>
On	O
separable	O
functions	O
,	O
the	O
performance	O
disadvantage	O
is	O
likely	O
to	O
be	O
most	O
significant	O
in	O
that	O
CMA-ES	B-Algorithm
might	O
not	O
be	O
able	O
to	O
find	O
at	O
all	O
comparable	O
solutions	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
on	O
non-separable	O
functions	O
that	O
are	O
ill-conditioned	B-Algorithm
or	O
rugged	O
or	O
can	O
only	O
be	O
solved	O
with	O
more	O
than	O
function	O
evaluations	O
,	O
the	O
CMA-ES	B-Algorithm
shows	O
most	O
often	O
superior	O
performance	O
.	O
</s>
<s>
The	O
(	O
1+1	O
)	O
-CMA-ES	O
generates	O
only	O
one	O
candidate	O
solution	O
per	O
iteration	O
step	O
which	O
becomes	O
the	O
new	O
distribution	O
mean	O
if	O
it	O
is	O
better	O
than	O
the	O
current	O
mean	O
.	O
</s>
<s>
For	O
the	O
(	O
1+1	O
)	O
-CMA-ES	O
is	O
a	O
close	O
variant	O
of	O
Gaussian	O
adaptation	O
.	O
</s>
<s>
Some	O
Natural	B-Algorithm
Evolution	I-Algorithm
Strategies	I-Algorithm
are	O
close	O
variants	O
of	O
the	O
CMA-ES	B-Algorithm
with	O
specific	O
parameter	O
settings	O
.	O
</s>
<s>
Natural	B-Algorithm
Evolution	I-Algorithm
Strategies	I-Algorithm
do	O
not	O
utilize	O
evolution	O
paths	O
(	O
that	O
means	O
in	O
CMA-ES	B-Algorithm
setting	O
)	O
and	O
they	O
formalize	O
the	O
update	O
of	O
variances	O
and	O
covariances	O
on	O
a	O
Cholesky	O
factor	O
instead	O
of	O
a	O
covariance	O
matrix	O
.	O
</s>
<s>
The	O
CMA-ES	B-Algorithm
has	O
also	O
been	O
extended	O
to	O
multiobjective	O
optimization	O
as	O
MO-CMA-ES	O
.	O
</s>
