<s>
In	O
statistics	O
and	O
machine	O
learning	O
,	O
lasso	B-Algorithm
(	O
least	B-Algorithm
absolute	I-Algorithm
shrinkage	I-Algorithm
and	I-Algorithm
selection	I-Algorithm
operator	I-Algorithm
;	O
also	O
Lasso	B-Algorithm
or	O
LASSO	B-Algorithm
)	O
is	O
a	O
regression	O
analysis	O
method	O
that	O
performs	O
both	O
variable	B-General_Concept
selection	I-General_Concept
and	O
regularization	O
in	O
order	O
to	O
enhance	O
the	O
prediction	O
accuracy	O
and	O
interpretability	O
of	O
the	O
resulting	O
statistical	O
model	O
.	O
</s>
<s>
Lasso	B-Algorithm
was	O
originally	O
formulated	O
for	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
.	O
</s>
<s>
These	O
include	O
its	O
relationship	O
to	O
ridge	O
regression	O
and	O
best	O
subset	O
selection	O
and	O
the	O
connections	O
between	O
lasso	B-Algorithm
coefficient	O
estimates	O
and	O
so-called	O
soft	O
thresholding	O
.	O
</s>
<s>
It	O
also	O
reveals	O
that	O
(	O
like	O
standard	O
linear	B-General_Concept
regression	I-General_Concept
)	O
the	O
coefficient	O
estimates	O
do	O
not	O
need	O
to	O
be	O
unique	O
if	O
covariates	O
are	O
collinear	O
.	O
</s>
<s>
Though	O
originally	O
defined	O
for	O
linear	B-General_Concept
regression	I-General_Concept
,	O
lasso	B-Algorithm
regularization	O
is	O
easily	O
extended	O
to	O
other	O
statistical	O
models	O
including	O
generalized	O
linear	O
models	O
,	O
generalized	O
estimating	O
equations	O
,	O
proportional	O
hazards	O
models	O
,	O
and	O
M-estimators	O
.	O
</s>
<s>
Lasso	B-Algorithm
's	O
ability	O
to	O
perform	O
subset	O
selection	O
relies	O
on	O
the	O
form	O
of	O
the	O
constraint	O
and	O
has	O
a	O
variety	O
of	O
interpretations	O
including	O
in	O
terms	O
of	O
geometry	O
,	O
Bayesian	O
statistics	O
and	O
convex	O
analysis	O
.	O
</s>
<s>
The	O
LASSO	B-Algorithm
is	O
closely	O
related	O
to	O
basis	O
pursuit	O
denoising	O
.	O
</s>
<s>
Lasso	B-Algorithm
was	O
introduced	O
in	O
order	O
to	O
improve	O
the	O
prediction	O
accuracy	O
and	O
interpretability	O
of	O
regression	O
models	O
.	O
</s>
<s>
Lasso	B-Algorithm
was	O
developed	O
independently	O
in	O
geophysics	O
literature	O
in	O
1986	O
,	O
based	O
on	O
prior	O
work	O
that	O
used	O
the	O
penalty	O
for	O
both	O
fitting	O
and	O
penalization	O
of	O
the	O
coefficients	O
.	O
</s>
<s>
Prior	O
to	O
lasso	B-Algorithm
,	O
the	O
most	O
widely	O
used	O
method	O
for	O
choosing	O
covariates	O
was	O
stepwise	O
selection	O
.	O
</s>
<s>
Ridge	O
regression	O
improves	O
prediction	O
error	O
by	O
shrinking	O
the	O
sum	O
of	O
the	O
squares	O
of	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
to	O
be	O
less	O
than	O
a	O
fixed	O
value	O
in	O
order	O
to	O
reduce	O
overfitting	B-Error_Name
,	O
but	O
it	O
does	O
not	O
perform	O
covariate	O
selection	O
and	O
therefore	O
does	O
not	O
help	O
to	O
make	O
the	O
model	O
more	O
interpretable	O
.	O
</s>
<s>
Lasso	B-Algorithm
achieves	O
both	O
of	O
these	O
goals	O
by	O
forcing	O
the	O
sum	O
of	O
the	O
absolute	O
value	O
of	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
to	O
be	O
less	O
than	O
a	O
fixed	O
value	O
,	O
which	O
forces	O
certain	O
coefficients	O
to	O
zero	O
,	O
excluding	O
them	O
from	O
impacting	O
prediction	O
.	O
</s>
<s>
This	O
idea	O
is	O
similar	O
to	O
ridge	O
regression	O
,	O
which	O
also	O
shrinks	O
the	O
size	O
of	O
the	O
coefficients	O
;	O
however	O
,	O
ridge	O
regression	O
does	O
not	O
set	O
coefficients	O
to	O
zero	O
(	O
and	O
,	O
thus	O
,	O
does	O
not	O
perform	O
variable	B-General_Concept
selection	I-General_Concept
)	O
.	O
</s>
<s>
Some	O
basic	O
properties	O
of	O
the	O
lasso	B-Algorithm
estimator	O
can	O
now	O
be	O
considered	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
lasso	B-Algorithm
estimates	O
share	O
features	O
of	O
both	O
ridge	O
and	O
best	O
subset	O
selection	O
regression	O
since	O
they	O
both	O
shrink	O
the	O
magnitude	O
of	O
all	O
the	O
coefficients	O
,	O
like	O
ridge	O
regression	O
and	O
set	O
some	O
of	O
them	O
to	O
zero	O
,	O
as	O
in	O
the	O
best	O
subset	O
selection	O
case	O
.	O
</s>
<s>
Additionally	O
,	O
while	O
ridge	O
regression	O
scales	O
all	O
of	O
the	O
coefficients	O
by	O
a	O
constant	O
factor	O
,	O
lasso	B-Algorithm
instead	O
translates	O
the	O
coefficients	O
towards	O
zero	O
by	O
a	O
constant	O
value	O
and	O
sets	O
them	O
to	O
zero	O
if	O
they	O
reach	O
it	O
.	O
</s>
<s>
Then	O
the	O
values	O
of	O
and	O
that	O
minimize	O
the	O
lasso	B-Algorithm
objective	O
function	O
are	O
not	O
uniquely	O
determined	O
.	O
</s>
<s>
In	O
fact	O
,	O
if	O
some	O
in	O
which	O
,	O
then	O
if	O
replacing	O
by	O
and	O
by	O
,	O
while	O
keeping	O
all	O
the	O
other	O
fixed	O
,	O
gives	O
a	O
new	O
solution	O
,	O
so	O
the	O
lasso	B-Algorithm
objective	O
function	O
then	O
has	O
a	O
continuum	O
of	O
valid	O
minimizers	O
.	O
</s>
<s>
Several	O
variants	O
of	O
the	O
lasso	B-Algorithm
,	O
including	O
the	O
Elastic	O
net	O
regularization	O
,	O
have	O
been	O
designed	O
to	O
address	O
this	O
shortcoming	O
.	O
</s>
<s>
Lasso	B-Algorithm
regularization	O
can	O
be	O
extended	O
to	O
other	O
objective	O
functions	O
such	O
as	O
those	O
for	O
generalized	O
linear	O
models	O
,	O
generalized	O
estimating	O
equations	O
,	O
proportional	O
hazards	O
models	O
,	O
and	O
M-estimators	O
.	O
</s>
<s>
Lasso	B-Algorithm
can	O
set	O
coefficients	O
to	O
zero	O
,	O
while	O
the	O
superficially	O
similar	O
ridge	O
regression	O
cannot	O
.	O
</s>
<s>
but	O
with	O
respect	O
to	O
different	O
constraints	O
:	O
for	O
lasso	B-Algorithm
and	O
for	O
ridge	O
.	O
</s>
<s>
The	O
figure	O
shows	O
that	O
the	O
constraint	O
region	O
defined	O
by	O
the	O
norm	O
is	O
a	O
square	O
rotated	O
so	O
that	O
its	O
corners	O
lie	O
on	O
the	O
axes	O
(	O
in	O
general	O
a	O
cross-polytope	O
)	O
,	O
while	O
the	O
region	O
defined	O
by	O
the	O
norm	O
is	O
a	O
circle	O
(	O
in	O
general	O
an	O
n-sphere	O
)	O
,	O
which	O
is	O
rotationally	B-General_Concept
invariant	O
and	O
,	O
therefore	O
,	O
has	O
no	O
corners	O
.	O
</s>
<s>
The	O
lasso	B-Algorithm
can	O
be	O
rescaled	O
so	O
that	O
it	O
becomes	O
easy	O
to	O
anticipate	O
and	O
influence	O
the	O
degree	O
of	O
shrinkage	O
associated	O
with	O
a	O
given	O
value	O
of	O
.	O
</s>
<s>
Let	O
represent	O
the	O
hypothesized	O
regression	B-General_Concept
coefficients	I-General_Concept
and	O
let	O
refer	O
to	O
the	O
data-optimized	O
ordinary	O
least	O
squares	O
solutions	O
.	O
</s>
<s>
A	O
rescaled	O
version	O
of	O
the	O
adaptive	O
lasso	B-Algorithm
of	O
can	O
be	O
obtained	O
by	O
setting	O
.	O
</s>
<s>
These	O
results	O
can	O
be	O
compared	O
to	O
a	O
rescaled	O
version	O
of	O
the	O
lasso	B-Algorithm
by	O
defining	O
,	O
which	O
is	O
the	O
average	O
absolute	O
deviation	O
of	O
from	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
inclusion	O
of	O
irrelevant	O
regressors	O
delays	O
the	O
moment	O
that	O
relevant	O
regressors	O
are	O
activated	O
by	O
this	O
rescaled	O
lasso	B-Algorithm
.	O
</s>
<s>
The	O
adaptive	O
lasso	B-Algorithm
and	O
the	O
lasso	B-Algorithm
are	O
special	O
cases	O
of	O
a	O
'	O
1ASTc	O
 '	O
estimator	O
.	O
</s>
<s>
Just	O
as	O
ridge	O
regression	O
can	O
be	O
interpreted	O
as	O
linear	B-General_Concept
regression	I-General_Concept
for	O
which	O
the	O
coefficients	O
have	O
been	O
assigned	O
normal	O
prior	O
distributions	O
,	O
lasso	B-Algorithm
can	O
be	O
interpreted	O
as	O
linear	B-General_Concept
regression	I-General_Concept
for	O
which	O
the	O
coefficients	O
have	O
Laplace	O
prior	O
distributions	O
.	O
</s>
<s>
This	O
provides	O
an	O
alternative	O
explanation	O
of	O
why	O
lasso	B-Algorithm
tends	O
to	O
set	O
some	O
coefficients	O
to	O
zero	O
,	O
while	O
ridge	O
regression	O
does	O
not	O
.	O
</s>
<s>
Lasso	B-Algorithm
can	O
also	O
be	O
viewed	O
as	O
a	O
convex	O
relaxation	O
of	O
the	O
best	O
subset	O
selection	O
regression	O
problem	O
,	O
which	O
is	O
to	O
find	O
the	O
subset	O
of	O
covariates	O
that	O
results	O
in	O
the	O
smallest	O
value	O
of	O
the	O
objective	O
function	O
for	O
some	O
fixed	O
,	O
where	O
n	O
is	O
the	O
total	O
number	O
of	O
covariates	O
.	O
</s>
<s>
Therefore	O
,	O
since	O
p	O
=	O
1	O
is	O
the	O
smallest	O
value	O
for	O
which	O
the	O
"	O
norm	O
"	O
is	O
convex	O
(	O
and	O
therefore	O
actually	O
a	O
norm	O
)	O
,	O
lasso	B-Algorithm
is	O
,	O
in	O
some	O
sense	O
,	O
the	O
best	O
convex	O
approximation	O
to	O
the	O
best	O
subset	O
selection	O
problem	O
,	O
since	O
the	O
region	O
defined	O
by	O
is	O
the	O
convex	O
hull	O
of	O
the	O
region	O
defined	O
by	O
for	O
.	O
</s>
<s>
Lasso	B-Algorithm
variants	O
have	O
been	O
created	O
in	O
order	O
to	O
remedy	O
limitations	O
of	O
the	O
original	O
technique	O
and	O
to	O
make	O
the	O
method	O
more	O
useful	O
for	O
particular	O
problems	O
.	O
</s>
<s>
Group	O
lasso	B-Algorithm
allows	O
groups	O
of	O
related	O
covariates	O
to	O
be	O
selected	O
as	O
a	O
single	O
unit	O
,	O
which	O
can	O
be	O
useful	O
in	O
settings	O
where	O
it	O
does	O
not	O
make	O
sense	O
to	O
include	O
some	O
covariates	O
without	O
others	O
.	O
</s>
<s>
Further	O
extensions	O
of	O
group	O
lasso	B-Algorithm
perform	O
variable	B-General_Concept
selection	I-General_Concept
within	O
individual	O
groups	O
(	O
sparse	O
group	O
lasso	B-Algorithm
)	O
and	O
allow	O
overlap	O
between	O
groups	O
(	O
overlap	O
group	O
lasso	B-Algorithm
)	O
.	O
</s>
<s>
Fused	O
lasso	B-Algorithm
can	O
account	O
for	O
the	O
spatial	O
or	O
temporal	O
characteristics	O
of	O
a	O
problem	O
,	O
resulting	O
in	O
estimates	O
that	O
better	O
match	O
system	O
structure	O
.	O
</s>
<s>
Lasso-regularized	O
models	O
can	O
be	O
fit	O
using	O
techniques	O
including	O
subgradient	B-Algorithm
methods	I-Algorithm
,	O
least-angle	O
regression	O
(	O
LARS	O
)	O
,	O
and	O
proximal	B-Algorithm
gradient	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Determining	O
the	O
optimal	O
value	O
for	O
the	O
regularization	O
parameter	O
is	O
an	O
important	O
part	O
of	O
ensuring	O
that	O
the	O
model	O
performs	O
well	O
;	O
it	O
is	O
typically	O
chosen	O
using	O
cross-validation	B-Application
.	O
</s>
<s>
When	O
p>n	O
(	O
the	O
number	O
of	O
covariates	O
is	O
greater	O
than	O
the	O
sample	O
size	O
)	O
lasso	B-Algorithm
can	O
select	O
only	O
n	O
covariates	O
(	O
even	O
when	O
more	O
are	O
associated	O
with	O
the	O
outcome	O
)	O
and	O
it	O
tends	O
to	O
select	O
one	O
covariate	O
from	O
any	O
set	O
of	O
highly	O
correlated	O
covariates	O
.	O
</s>
<s>
So	O
the	O
result	O
of	O
the	O
elastic	O
net	O
penalty	O
is	O
a	O
combination	O
of	O
the	O
effects	O
of	O
the	O
lasso	B-Algorithm
and	O
ridge	O
penalties	O
.	O
</s>
<s>
Returning	O
to	O
the	O
general	O
case	O
,	O
the	O
fact	O
that	O
the	O
penalty	O
function	O
is	O
now	O
strictly	O
convex	O
means	O
that	O
if	O
,	O
,	O
which	O
is	O
a	O
change	O
from	O
lasso	B-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
highly	O
correlated	O
covariates	O
tend	O
to	O
have	O
similar	O
regression	B-General_Concept
coefficients	I-General_Concept
,	O
with	O
the	O
degree	O
of	O
similarity	O
depending	O
on	O
both	O
and	O
,	O
which	O
is	O
different	O
from	O
lasso	B-Algorithm
.	O
</s>
<s>
This	O
phenomenon	O
,	O
in	O
which	O
strongly	O
correlated	O
covariates	O
have	O
similar	O
regression	B-General_Concept
coefficients	I-General_Concept
,	O
is	O
referred	O
to	O
as	O
the	O
grouping	O
effect	O
.	O
</s>
<s>
Grouping	O
is	O
desirable	O
since	O
,	O
in	O
applications	O
such	O
as	O
tying	O
genes	O
to	O
a	O
disease	O
,	O
finding	O
all	O
the	O
associated	O
covariates	O
is	O
preferable	O
,	O
rather	O
than	O
selecting	O
one	O
from	O
each	O
set	O
of	O
correlated	O
covariates	O
,	O
as	O
lasso	B-Algorithm
often	O
does	O
.	O
</s>
<s>
In	O
addition	O
,	O
selecting	O
only	O
one	O
from	O
each	O
group	O
typically	O
results	O
in	O
increased	O
prediction	O
error	O
,	O
since	O
the	O
model	O
is	O
less	O
robust	O
(	O
which	O
is	O
why	O
ridge	O
regression	O
often	O
outperforms	O
lasso	B-Algorithm
)	O
.	O
</s>
<s>
In	O
2006	O
,	O
Yuan	O
and	O
Lin	O
introduced	O
the	O
group	O
lasso	B-Algorithm
to	O
allow	O
predefined	O
groups	O
of	O
covariates	O
to	O
jointly	O
be	O
selected	O
into	O
or	O
out	O
of	O
a	O
model	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
group	O
lasso	B-Algorithm
can	O
ensure	O
that	O
all	O
the	O
variables	O
encoding	O
the	O
categorical	O
covariate	O
are	O
included	O
or	O
excluded	O
together	O
.	O
</s>
<s>
where	O
the	O
design	B-Algorithm
matrix	I-Algorithm
and	O
covariate	O
vector	O
have	O
been	O
replaced	O
by	O
a	O
collection	O
of	O
design	O
matrices	O
and	O
covariate	O
vectors	O
,	O
one	O
for	O
each	O
of	O
the	O
J	O
groups	O
.	O
</s>
<s>
If	O
each	O
covariate	O
is	O
in	O
its	O
own	O
group	O
and	O
,	O
then	O
this	O
reduces	O
to	O
the	O
standard	O
lasso	B-Algorithm
,	O
while	O
if	O
there	O
is	O
only	O
a	O
single	O
group	O
and	O
,	O
it	O
reduces	O
to	O
ridge	O
regression	O
.	O
</s>
<s>
However	O
,	O
because	O
the	O
penalty	O
is	O
the	O
sum	O
over	O
the	O
different	O
subspace	O
norms	O
,	O
as	O
in	O
the	O
standard	O
lasso	B-Algorithm
,	O
the	O
constraint	O
has	O
some	O
non-differential	O
points	O
,	O
which	O
correspond	O
to	O
some	O
subspaces	O
being	O
identically	O
zero	O
.	O
</s>
<s>
However	O
,	O
it	O
is	O
possible	O
to	O
extend	O
the	O
group	O
lasso	B-Algorithm
to	O
the	O
so-called	O
sparse	O
group	O
lasso	B-Algorithm
,	O
which	O
can	O
select	O
individual	O
covariates	O
within	O
a	O
group	O
,	O
by	O
adding	O
an	O
additional	O
penalty	O
to	O
each	O
group	O
subspace	O
.	O
</s>
<s>
Another	O
extension	O
,	O
group	O
lasso	B-Algorithm
with	O
overlap	O
allows	O
covariates	O
to	O
be	O
shared	O
across	O
groups	O
,	O
e.g.	O
,	O
if	O
a	O
gene	O
were	O
to	O
occur	O
in	O
two	O
pathways	O
.	O
</s>
<s>
In	O
2005	O
,	O
Tibshirani	O
and	O
colleagues	O
introduced	O
the	O
fused	O
lasso	B-Algorithm
to	O
extend	O
the	O
use	O
of	O
lasso	B-Algorithm
to	O
this	O
type	O
of	O
data	O
.	O
</s>
<s>
The	O
first	O
constraint	O
is	O
the	O
lasso	B-Algorithm
constraint	O
,	O
while	O
the	O
second	O
directly	O
penalizes	O
large	O
changes	O
with	O
respect	O
to	O
the	O
temporal	O
or	O
spatial	O
structure	O
,	O
which	O
forces	O
the	O
coefficients	O
to	O
vary	O
smoothly	O
to	O
reflect	O
the	O
system	O
's	O
underlying	O
logic	O
.	O
</s>
<s>
Clustered	O
lasso	B-Algorithm
is	O
a	O
generalization	O
of	O
fused	O
lasso	B-Algorithm
that	O
identifies	O
and	O
groups	O
relevant	O
covariates	O
based	O
on	O
their	O
effects	O
(	O
coefficients	O
)	O
.	O
</s>
<s>
Algorithms	O
exist	O
that	O
solve	O
the	O
fused	O
lasso	B-Algorithm
problem	O
,	O
and	O
some	O
generalizations	O
of	O
it	O
.	O
</s>
<s>
Lasso	B-Algorithm
,	O
elastic	O
net	O
,	O
group	O
and	O
fused	O
lasso	B-Algorithm
construct	O
the	O
penalty	O
functions	O
from	O
the	O
and	O
norms	O
(	O
with	O
weights	O
,	O
if	O
necessary	O
)	O
.	O
</s>
<s>
where	O
is	O
an	O
arbitrary	O
concave	O
monotonically	O
increasing	O
function	O
(	O
for	O
example	O
,	O
gives	O
the	O
lasso	B-Algorithm
penalty	O
and	O
gives	O
the	O
penalty	O
)	O
.	O
</s>
<s>
The	O
adaptive	O
lasso	B-Algorithm
was	O
introduced	O
by	O
Zou	O
in	O
2006	O
for	O
linear	B-General_Concept
regression	I-General_Concept
and	O
by	O
Zhang	O
and	O
Lu	O
in	O
2007	O
for	O
proportional	O
hazards	O
regression	O
.	O
</s>
<s>
The	O
prior	O
lasso	B-Algorithm
was	O
introduced	O
for	O
generalized	O
linear	O
models	O
by	O
Jiang	O
et	O
al	O
.	O
</s>
<s>
In	O
prior	O
lasso	B-Algorithm
,	O
such	O
information	O
is	O
summarized	O
into	O
pseudo	O
responses	O
(	O
called	O
prior	O
responses	O
)	O
and	O
then	O
an	O
additional	O
criterion	O
function	O
is	O
added	O
to	O
the	O
usual	O
objective	O
function	O
with	O
a	O
lasso	B-Algorithm
penalty	O
.	O
</s>
<s>
the	O
usual	O
lasso	B-Algorithm
objective	O
function	O
with	O
the	O
responses	O
being	O
replaced	O
by	O
a	O
weighted	O
average	O
of	O
the	O
observed	O
responses	O
and	O
the	O
prior	O
responses	O
(	O
called	O
the	O
adjusted	O
response	O
values	O
by	O
the	O
prior	O
information	O
)	O
.	O
</s>
<s>
In	O
prior	O
lasso	B-Algorithm
,	O
the	O
parameter	O
is	O
called	O
a	O
balancing	O
parameter	O
,	O
in	O
that	O
it	O
balances	O
the	O
relative	O
importance	O
of	O
the	O
data	O
and	O
the	O
prior	O
information	O
.	O
</s>
<s>
In	O
the	O
extreme	O
case	O
of	O
,	O
prior	O
lasso	B-Algorithm
is	O
reduced	O
to	O
lasso	B-Algorithm
.	O
</s>
<s>
If	O
,	O
prior	O
lasso	B-Algorithm
will	O
solely	O
rely	O
on	O
the	O
prior	O
information	O
to	O
fit	O
the	O
model	O
.	O
</s>
<s>
Prior	O
lasso	B-Algorithm
is	O
more	O
efficient	O
in	O
parameter	O
estimation	O
and	O
prediction	O
(	O
with	O
a	O
smaller	O
estimation	O
error	O
and	O
prediction	O
error	O
)	O
when	O
the	O
prior	O
information	O
is	O
of	O
high	O
quality	O
,	O
and	O
is	O
robust	O
to	O
the	O
low	O
quality	O
prior	O
information	O
with	O
a	O
good	O
choice	O
of	O
the	O
balancing	O
parameter	O
.	O
</s>
<s>
The	O
loss	O
function	O
of	O
the	O
lasso	B-Algorithm
is	O
not	O
differentiable	O
,	O
but	O
a	O
wide	O
variety	O
of	O
techniques	O
from	O
convex	O
analysis	O
and	O
optimization	O
theory	O
have	O
been	O
developed	O
to	O
compute	O
the	O
solutions	O
path	O
of	O
the	O
lasso	B-Algorithm
.	O
</s>
<s>
These	O
include	O
coordinate	O
descent	O
,	O
subgradient	B-Algorithm
methods	I-Algorithm
,	O
least-angle	O
regression	O
(	O
LARS	O
)	O
,	O
and	O
proximal	B-Algorithm
gradient	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Subgradient	B-Algorithm
methods	I-Algorithm
are	O
the	O
natural	O
generalization	O
of	O
traditional	O
methods	O
such	O
as	O
gradient	B-Algorithm
descent	I-Algorithm
and	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
to	O
the	O
case	O
in	O
which	O
the	O
objective	O
function	O
is	O
not	O
differentiable	O
at	O
all	O
points	O
.	O
</s>
<s>
LARS	O
is	O
a	O
method	O
that	O
is	O
closely	O
tied	O
to	O
lasso	B-Algorithm
models	O
,	O
and	O
in	O
many	O
cases	O
allows	O
them	O
to	O
be	O
fit	O
efficiently	O
,	O
though	O
it	O
may	O
not	O
perform	O
well	O
in	O
all	O
circumstances	O
.	O
</s>
<s>
The	O
choice	O
of	O
method	O
will	O
depend	O
on	O
the	O
particular	O
lasso	B-Algorithm
variant	O
,	O
the	O
data	O
and	O
the	O
available	O
resources	O
.	O
</s>
<s>
Choosing	O
the	O
regularization	O
parameter	O
(	O
)	O
is	O
a	O
fundamental	O
part	O
of	O
lasso	B-Algorithm
.	O
</s>
<s>
A	O
good	O
value	O
is	O
essential	O
to	O
the	O
performance	O
of	O
lasso	B-Algorithm
since	O
it	O
controls	O
the	O
strength	O
of	O
shrinkage	O
and	O
variable	B-General_Concept
selection	I-General_Concept
,	O
which	O
,	O
in	O
moderation	O
can	O
improve	O
both	O
prediction	O
accuracy	O
and	O
interpretability	O
.	O
</s>
<s>
Cross-validation	B-Application
is	O
often	O
used	O
to	O
find	O
the	O
regularization	O
parameter	O
.	O
</s>
<s>
Information	O
criteria	O
such	O
as	O
the	O
Bayesian	B-General_Concept
information	I-General_Concept
criterion	I-General_Concept
(	O
BIC	B-General_Concept
)	O
and	O
the	O
Akaike	O
information	O
criterion	O
(	O
AIC	O
)	O
might	O
be	O
preferable	O
to	O
cross-validation	B-Application
,	O
because	O
they	O
are	O
faster	O
to	O
compute	O
and	O
their	O
performance	O
is	O
less	O
volatile	O
in	O
small	O
samples	O
.	O
</s>
<s>
LASSO	B-Algorithm
has	O
been	O
applied	O
in	O
economics	O
and	O
finance	O
,	O
and	O
was	O
found	O
to	O
improve	O
prediction	O
and	O
to	O
select	O
sometimes	O
neglected	O
variables	O
,	O
for	O
example	O
in	O
corporate	O
bankruptcy	O
prediction	O
literature	O
,	O
or	O
high	O
growth	O
firms	O
prediction	O
.	O
</s>
